Miller, Reiner, Sturmfels - Geometric Combinatorics (Ias Park City Mathematics Series, AMS) (696p) PDF
Miller, Reiner, Sturmfels - Geometric Combinatorics (Ias Park City Mathematics Series, AMS) (696p) PDF
Miller, Reiner, Sturmfels - Geometric Combinatorics (Ias Park City Mathematics Series, AMS) (696p) PDF
Preface xv
Bibliography 17
Alexander Barvinok
Lattice Points, Polyhedra, and Complexity 19
Introduction 21
Bibliography 61
vii
viii CONTENTS
Bibliography 129
Robin Forman
Topics in Combinatorial Differential Topology and Geometry 133
Bibliography 201
Lecture 2. Catalan Numbers, Trees, Lagrange Inversion, and their q-Analogs 217
2.1. Catalan Numbers 217
2.2. Rooted Trees 218
2.3. The Lagrange Inversion Formula 219
2.4. q-Analogs 220
2.5. q-Lagrange Inversion 222
2.6. Exercises 226
Bibliography 247
Dmitry N. Kozlov
Chromatic Numbers, Morphism Complexes, and Stiefel-Whitney
Characteristic Classes 249
Preamble 251
Bibliography 311
Robert MacPherson
Equivariant Invariants and Linear Geometry 317
Introduction 319
0.1. Spaces with a Torus Action 320
0.2. Linear Graphs 322
0.3. Rings and Modules 323
Bibliography 387
Richard P. Stanley
An Introduction to Hyperplane Arrangements 389
Bibliography 495
Introduction 499
Bibliography 605
xiv CONTENTS
Günter M. Ziegler
Convex Polytopes: Extremal Constructions and f -Vector Shapes 617
Introduction 619
Lecture 1. Constructing 3-Dimensional Polytopes 621
1.1. The Cone of f -vectors 623
1.2. The Steinitz Theorem 625
1.3. Steinitz’ Theorem via Circle Packings 628
Lecture 2. Shapes of f -Vectors 643
2.1. Unimodality Conjectures 644
2.2. Basic Examples 644
2.3. Global Constructions 647
2.4. Local Constructions 649
Lecture 3. 2-Simple 2-Simplicial 4-Polytopes 653
3.1. Examples 654
3.2. 2-Simple 2-Simplicial 4-Polytopes 657
3.3. Deep Vertex Truncation 659
3.4. Constructing DVT(Stack(n, 4)) 661
Lecture 4. f -Vectors of 4-Polytopes 665
4.1. The f -Vector Cone 666
4.2. Fatness and the Upper Bound Problem 669
4.3. The Lower Bound Problem 671
Lecture 5. Projected Products of Polygons 673
5.1. Products and Deformed Products 673
5.2. Computing the f -Vector 674
5.3. Deformed Products 674
5.4. Surviving a Generic Projection 678
5.5. Construction 678
Appendix: A Short Introduction to polymake
(by Thilo Schröder and Nikolaus Witte) 681
A.1. Getting Started 681
A.2. The polymake System 684
Bibliography 687
Preface
The IAS/Park City Mathematics Institute (PCMI) was founded in 1991 as part of
the “Regional Geometry Institute” initiative of the National Science Foundation.
In mid 1993 the program found an institutional home at the Institute for Advanced
Study (IAS) in Princeton, New Jersey.
The IAS/Park City Mathematics Institute encourages both research and ed-
ucation in mathematics and fosters interaction between the two. The three-week
summer institute offers programs for researchers and postdoctoral scholars, gradu-
ate students, undergraduate students, high school teachers, undergraduate faculty,
and researchers in mathematics education. One of PCMI’s main goals is to make
all of the participants aware of the total spectrum of activities that occur in math-
ematics education and research: we wish to involve professional mathematicians
in education and to bring modern concepts in mathematics to the attention of
educators. To that end the summer institute features general sessions designed
to encourage interaction among the various groups. In-year activities at the sites
around the country form an integral part of the High School Teachers Program.
Each summer a different topic is chosen as the focus of the Research Program
and Graduate Summer School. Activities in the Undergraduate Summer School
deal with this topic as well. Lecture notes from the Graduate Summer School are
being published each year in this series. The first fourteen volumes are:
John C. Polking
Series Editor
April 2007
IAS/Park City Mathematics Series
Volume 14, 2004
1. Polytopes
A popular class of examples are the convex polytopes, that is, convex hulls of finite
point sets in Rd . These form the main topic of the graduate course by Ziegler, but
also play prominent roles in the undergraduate courses by Swartz and Thomas, and
in the undergraduate faculty course by Su (as well as making cameo appearances
in the graduate courses by Barvinok, Fomin, Forman, MacPherson, and Wachs!).
In R2 , convex polytopes are polygons such as triangles, quadrilaterals, pen-
tagons, hexagons, etc. In R3 they can be more interesting, such as the triangular
prism depicted in Figure 1(a).
d f
e
e
b
d b
f
a c a c
(a) (b)
1 School
of Mathematics, University of Minnesota, Minneapolis MN, 55455.
E-mail address: ezra@math.umn.edu, reiner@math.umn.edu.
c
2007 American Mathematical Society
1
2 EZRA MILLER AND VICTOR REINER, OVERVIEW
ab ac bc ad be cf de ef df
a b c d e f
Figure 2. The Hasse diagram for the poset of faces of the prism in Figure 1.
2. Characterizing f -vectors
What kinds of combinatorial questions about convex polytopes might we ask? One
that has been considered often is the following.
Question 1. Which (non-negative) vectors (f0 , f1 , . . . , fd−1 ) in Zd can actually
arise as the f -vector of a d-dimensional convex polytope?
EZRA MILLER AND VICTOR REINER, OVERVIEW 3
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Figure 3. The digon and the monogon: two valid CW -balls, obeying the
topological constraint f0 = f1 . The digon has 2 vertices and 2 edges, while
the monogon has 1 vertex and 1 edge.
From now on, when we speak of a “d-dimensional” polytope, we will assume that
it is fully d-dimensional in the sense that its points affinely span a d-dimensional
space. For d = 2, Question 1 has an obvious answer.
Proposition 2. A vector (f0 , f1 ) ∈ Z2 is the f -vector of a 2-dimensional convex
polytope (polygon) if and only if
(i) f0 = f1 , and
(ii) f0 , f1 ≥ 3.
In spite of its simplicity, this answer foreshadows some important issues arising in
higher dimensions. Note that the equation constraint (i) is really a consequence of
topology: the boundary of a convex polygon is homeomorphic to a one-dimensional
sphere. The same equation (i) would hold—without any polytopality assumption—
for any CW -complex homeomorphic to a 2-dimensional ball, e.g. the digon or mono-
gon depicted in Figure 3.
On the other hand, the inequality (ii) is really a consequence of polytopality. It
highlights the importance of clarifying in which category we work when studying f -
vectors (such as CW -spheres, regular CW -spheres, P L-spheres, polytopal spheres,
etc.) as this can have a dramatic effect on the answers and the difficulty level for
questions about f -vectors.
Question 1 for d = 3 is also not hard, and was answered by Steinitz roughly a
century ago.
Theorem 3. A vector (f0 , f1 , f2 ) ∈ Z3 is the f -vector of a 3-dimensional convex
polytope if and only if
(i) f0 − f1 + f2 = 2 (Euler’s relation),
(ii) f0 , f2 ≥ 4, and
(iii) 2f1 ≥ 3f2 , 2f1 ≥ 3f0 .
Again, the equational constraint (i) is a familiar consequence of topology. Poly-
topality provides us with the first inequality f0 ≥ 4 in (ii), since we have assumed
that our polytope affinely spans R3 and hence must have at least 4 affinely indepen-
dent vertices.2 The condition f2 ≥ 4 then follows from the important tool of polar
duality: every convex polytope P in Rd has a (polar) dual polytope P ♦ , whose
faces correspond bijectively with those of P , but in an inclusion-reversing and
dimension-reversing fashion. Thus for a 3-dimensional polytope P with f -vector
2Actually, both inequalities in (ii) already follow from (i) and (iii), and hence are redundant, but
we have included them anyway.
4 EZRA MILLER AND VICTOR REINER, OVERVIEW
Figure 4. A pair of Platonic solids, which are polar dual to each other: the
icosahedron and the dodecahedron. Their f -vectors (f0 , f1 , f2 ) are related by
reversal, namely (12, 30, 20) and (20, 30, 12), respectively.
(f0 , f1 , f2 ), its polar dual P ♦ will have f -vector (f2 , f1 , f0 ). Two classic examples
of dual Platonic solids, the icosahedron and dodecahedron are shown in Figure 4.
The remaining inequalities (iii) in the above theorem are another consequence
of convexity that follows from counting the edges in the polytope after “blowing
apart” the facets, as depicted in Figure 5. Combining the fact that every edge
lies in exactly two facets with the fact that each facet has at least three boundary
edges, one is led to the inequality 2f1 ≥ 3f2 . The second inequality in (iii) then
follows from polar duality. This shows the necessity of Steinitz’s conditions; the
sufficiency can be shown by constructing 3-dimensional polytopes with specified
f -vectors via some relatively simple constructions (start with a pyramid having an
arbitrary polygonal base, and iterate the operation of shaving off a vertex, or its
polar dual operation of stellarly subdividing a facet).
What about Question 1 for d ≥ 4? In dimension 4 there are only partial answers
(see Ziegler’s course), and in higher dimensions, the question is wide open.
EZRA MILLER AND VICTOR REINER, OVERVIEW 5
Figure 6. The area of a lattice triangle having i = 1 interior lattice point and
b = 4 boundary lattice points is i + 12 b − 1 = 2.
3. Lattice points
There is even more combinatorial structure attached to lattice polytopes, the topic
of the graduate course by Barvinok, appearing also in the undergraduate course by
Thomas as well as the undergraduate faculty course by Su. A lattice polytope is
a convex polytope whose vertices lie in Zd . Here there are non-trivial results even
for d = 2, that is for lattice polygons! The most famous is probably Pick’s formula
for the area of a lattice polygon.
Theorem 4. (Pick [6]) Let P be a lattice polygon with i lattice points in its interior
and b lattice points on its boundary. Then the area of P is i + 21 b − 1.
Figure 6 illustrates this result for a certain lattice triangle. In fact, Pick’s Theorem
holds even for lattice polygons which are not convex.
The theory of lattice polytopes becomes more interesting in higher dimensions,
including the theory of Ehrhart polynomials. It is a subject that has seen many
advances within the last decade that have greatly increased our ability for explicit
computations. One such advance is Brion’s formula, which says how to list the
lattice points in a lattice polytope. More precisely, let P be a polytope in Rd with
integer vertices. If a = (a1 , . . . , ad ) ∈ Zd is a lattice point, then write ta = ta1 1 · · · tadd
for the corresponding Laurent monomial. The generating function for the lattice
points in P is the sum of all Laurent monomials ta for a ∈ Zd ∩ P . It is a rational
function because it has only finitely many terms. In contrast, consider the tangent
cone Tv to the polytope at the vertex v, which is the translate by v of cone generated
over the positive real numbers by P − v. The generating function for the lattice
points Zd ∩ Tv in a tangent cone is not a finite sum, but it is still expressible as a
rational function Cv (t). Brion’s formula breaks the lattice point enumerator of P
into a sum over the vertices of P :
X X
ta = Cv (t).
a∈Zd ∩P vertices v of P
This counter-intuitive result looks like it counts each lattice point in P once for
each vertex of P , and furthermore counts all of the lattice points outside of P some
number of times, as well. But when that wild-looking generating function (sup-
ported on all of Zd ) is expressed as a single rational function, the over-counting
inside of P and parts outside of P vanish. Brion’s formula is important for compu-
tation because it provides a “short” way to represent the set of lattice points in P .
6 EZRA MILLER AND VICTOR REINER, OVERVIEW
= + + +
st2 t2 1 s
= + + +
(1 − s−1 )(1− t−1 ) (1 − s)(1 − t−1 ) (1 − s)(1 − t) (1 − s−1 )(1 − t)
s t − t + 1 − s2
2 3 3
=
(1 − s)(1 − t)
(1 − t3 )(1 − s2 )
=
(1 − s)(1 − t)
= (1 + t + t2 )(1 + s)
= 1 + t + t2 + s + st + st2
with vertex set {(0, 0), (1, 0), (2, 0), (1, 2)}. The lattice point enumerator of P , writ-
ten in variables (s, t) = (t1 , t2 ), is 1 + t + t2 + s + st + st2 . The lattice points in
the tangent cone at (say) the vertex (1, 2) of P consist of all integer vectors (a, b)
such that a ≤ 1 and b ≤ 2. The generating function for these lattice points is
st2 /(1 − s−1 )(1 − t−1 ). The statement of Brion’s formula in this case is verified in
the calculation appearing in Figure 7.
4. Hyperplane arrangements
Another interesting example of geometric objects with combinatorial structure are
arrangements of hyperplanes, the subject of Stanley’s graduate course, and other
(affine or) linear subspaces of a vector space, which form part of the subject of
Wachs’s graduate course. Figure 8 illustrates an affine arrangement of hyperplanes
(lines) in R2 , along with a central arrangement of hyperplanes in R3 depicted via
their intersections with the unit sphere.
Hyperplanes dissect Rd into open regions (or chambers), which can be bounded
or unbounded, and which one can attempt to count. When one complexifies real
hyperplanes or subspaces by considering them inside Cd , they “poke holes” in the
space, creating non-trivial topology one can try to measure, e.g. by computing
homotopy invariants such as homology or homotopy groups, or cohomology rings.
When the hyperplanes or subspaces are defined over Z, one can consider their
EZRA MILLER AND VICTOR REINER, OVERVIEW 7
reductions mod p as arrangements in vector spaces Fdp over finite fields, and then
count points lying on or off the arrangement. It turns out that almost all of this
enumerative or topological analysis comes down to understanding the topology of
another poset: the lattice of intersections of the subspaces, ordered by inclusion.
In particular, one learns that it is important to associate a simplicial complex
(and hence a topological space) to this poset, via the ubiquitous order complex
or nerve construction. We also find ourselves in need of a wide array of tools,
provided in the graduate course on poset topology by Wachs, for understanding
the homotopy or homeomorphism type of the various kinds of simplicial complexes
that arise in this way.
5. Symmetry
Many of the examples of combinatorial geometric objects cropping up all over
mathematics, such as in the geometry and representation theory of Lie groups
and algebras, are those possessing a high degree of symmetry. Such objects are the
subject of the graduate course by Fomin, and also play a prominent role in the part
of Wachs’s course that deals with the equivariant theory of poset topology.
To give some flavor of Fomin’s course, let’s look briefly at the classical topic of
regular polytopes. A regular polytope is one in which every maximal flag of faces
vertex ⊂ edge ⊂ · · · ⊂ facet
“looks” the same, meaning that the group of linear symmetries preserving the poly-
tope acts transitively on all such flags. The 3-dimensional regular polytopes are
exactly the Platonic solids, depicted in Figure 4. Classical results in the theory as-
sociate to every regular polytope P a certain well-studied and well-behaved hyper-
plane arrangement: the symmetry group of a regular polytope is always generated
by reflection symmetries, and one simply takes the associated reflecting hyperplanes
for all such symmetries. For the regular tetrahedron, the associated dissection by
reflecting hyperplanes and the hyperplane arrangement are shown in Figure 9. Not
only do these reflection arrangements play a central role in Fomin’s course, but
they show up as key motivating examples, along with some of their well-behaved
deformations, in Stanley’s course as well.
8 EZRA MILLER AND VICTOR REINER, OVERVIEW
6. Moment graphs
Geometric combinatorics does not only concern structures arising from spaces that
feel discrete. Smooth spaces often have underlying combinatorics, as well. Many
smooth spaces can be considered from the point of view in MacPherson’s course,
EZRA MILLER AND VICTOR REINER, OVERVIEW 9
S 1 × S 1 action:
CP1 × CP1 = ×
where the combinatorics takes the form of a graph drawn with straight edges in Rn .
The setup is as follows.
An algebraic torus is a group of the form T = (C∗ )n , where C∗ = C \ {0} is
the set of nonzero complex numbers, considered as a group under multiplication.
Inside of the algebraic torus T is an honest compact torus TR = (S 1 )n , the product
of n copies of the unit circle group. MacPherson’s course concerns spaces X with
an action of T . More precisely, let X be a smooth compact complex algebraic
variety of dimension d; thus X is a real manifold of dimension 2d with some extra
structure to make it a manifold over C. We require that the action T : X → X has
finitely many
• T -fixed points and
• complex 1-dimensional orbits.
An orbit of complex dimension 1 has real dimension 2, and is necessarily isomorphic
to a copy of C∗ . Since X is compact, the closure of such an orbit is an isomor-
phic copy of the Riemann sphere (projective complex line) P1 : add an origin 0 and
a point ∞ at infinity (both of which will be T -fixed points) to the copy of C∗ .
The union of the T -fixed points and the 1-dimensional orbits is a configuration,
called a balloon sculpture, of finitely many Riemann spheres in X joined at some
of their poles. The moment graph is a real 1-dimensional shadow of the complex
1-dimensional balloon sculpture. It is obtained from the balloon sculpture by iden-
tifying together all points in each orbit of the compact torus TR .
Example 6. Let X = CP1 ×CP1 be a product of two Riemann spheres. This space
comes with an action of T = C∗ × C∗ , so n = 2 in the preceding notation. The
compact torus TR = S 1 × S 1 is the familiar real 2-dimensional doughnut. The two
copies of S 1 spin the corresponding spheres CP1 around their axes, each leaving the
other sphere fixed pointwise, as depicted in Figure 10. The balloon sculpture in X
consists of four spheres joined pole-to-pole in a cycle, as in Figure 11. The circles
of latitude in the four balloons are TR orbits, as are each of the poles. Collapsing
each of these orbits to a point yields the moment graph of CP1 × CP1 : a square.
In the above example, the quotient of all of X by TR is the entire square—
including the interior, over which the TR orbits are 2-dimensional tori. More gener-
ally, for every lattice polytope P there is a toric variety XP whose moment graph is
the edge graph of P , and whose quotient by TR is all of P . Although toric varieties
constitute a very important class of examples—they are the simplest spaces with
moment graphs—they aren’t the only spaces with moment graphs.
10 EZRA MILLER AND VICTOR REINER, OVERVIEW
balloon sculpture X
Figure 11. The balloon sculpture of X = CP1 × CP1 and its map to the moment graph
Figure 13. The convex hull of the moment graph of the flag manifold F ℓ3 is a permutohedron
λ = (7, 4, 2, 2, 1) ←→
y5
xy 4
x2 y 2
x4 y
x7
Figure 14. Monomials in the nooks immediately outside of the partition λ = (7, 4, 2, 2, 1)
8. Morse theory
Localization theorems are powerful ways to reconstruct topological invariants from
knowledge of local data near fixed points. However, even to speak of fixed points we
must have a group action. In the preceding situations, such actions were natural,
in that they were fundamental to the smooth spaces under consideration. The flag
manifold, for instance, is the quotient of a Lie group by a closed subgroup, and
hence obviously has lots of Lie group actions on it; and a toric variety is (by some
definitions) the closure of a dense torus orbit. But what if our smooth space doesn’t
come with a natural Lie group action? Make a group action from scratch!
Suppose that X is a real manifold with a Riemannian metric. Any real-valued
function f : X → R yields a gradient flow on X: each point goes in the direction
of steepest descent. Gradient flow can be viewed as an action of the Lie group R
14 EZRA MILLER AND VICTOR REINER, OVERVIEW
Figure 15. Four critical points on a torus, with the negative flow directions
(thought of as parametrizing time) on X. The fixed points of the flow are the
critical points of f , where the derivative of f vanishes; these points are ambivalent
about which direction to go, so they stay put. See Figure 15 for an example with
four critical points on a torus. When f is generic, we can define the index of a
critical point x to be the number of independent directions at x in which the flow
points away from x—that is, the limit is x as time approaches −∞.
Topological invariants are extracted from this (more or less) combinatorial data
of critical points, indexes, and downward flow submanifolds by constructing a cell
decomposition of X. There is one cell for each critical point, and the dimension of
the cell is the index of the critical point. From Figure 15, we see that a torus can
be constructed from a vertex (the bottom critical point), two edges (the middle two
critical points), and one 2-cell (the top critical point). The manner in which the
downward flow submanifold from one critical point approaches the other critical
points determines how to glue the cells.
Gradient flow is all well and good if we’re given a smooth manifold. But what if,
in the spirit of how this Overview started, we’re given a discrete geometric object,
such as a collection of polytopes or a simplicial complex ∆? The answer lies in
Forman’s lectures: use discrete Morse theory. The idea is strikingly simple. Let P
be the Hasse diagram of the face poset of ∆. Orient all of the edges of P downward.
A Morse flow in this context is a (partial) matching on P such that reversing the
edges in the matching does not ruin the directed acylicity property of the directed
graph P . This mirrors the stipulation that our Morse function f mapped X to
the real numbers, and not (for example) to the circle. The critical simplices of the
Morse flow are the unmatched elements of P . In analogy with the smooth case,
the critical simplices correspond to cells in a complex that is homotopy-equivalent
to ∆, so the topological invariants have not changed. Discrete Morse theory is an
extremely useful tool in making explicit calculations. It is also a key theoretical
tool for poset homology, which leads to Wachs’s course.
Going beyond Morse theory, it is possible to combinatorialize a number of
other notions from differential geometry. Forman’s fourth and fifth lectures, for
example, discuss combinatorializations of curvature, and the purely combinatorial
EZRA MILLER AND VICTOR REINER, OVERVIEW 15
9. Further topics
We have tried in this Overview to give an idea of what “geometric combinatorics”
might mean, although (for obvious reasons) we have done so mostly in the con-
text of the courses at PCMI 2004. But this summer’s offerings are by no means
comprehensive! There are vast numbers of ways combinatorial structure arises in
geometry. Here, for example, is a small list of keywords.
• Tropicalization: polyhedral structures reflect the geometry of complex
algebraic varieties.
• Degeneration: replace a manifold or variety, such as a Schubert variety, by
a degenerate version that has several components, each of which is simpler.
• Stratification: different strata, as in moduli spaces of curves, can represent
collections of geometric objects with identical combinatorial properties.
• Branch point data (Hurwitz schemes and ramified covers): counting meth-
ods rely on combinatorics of the symmetric group.
• Generating functions: for example, Gromov–Witten theory leads to mul-
tivariate hypergeometric series.
• Characteristic classes: for example, functorial approaches to graph color-
ing and Tverberg-type theorems.
Some of the above items were hot topics at the 2004 PCMI Research Program: the
Clay lecture by Sturmfels was one of many talks about tropical geometry and its
applications, and the research talk by (for example) Vakil concerned recent advances
using degeneration. The last item on the list was expanded by Kozlov to a survey
paper that is included in this volume. The survey concerns graph complexes and
functorial approaches to graph coloring. More precisely, in 1978 Lovász proved a
subtle conjecture of Kneser in graph theory using functoriality: a proper vertex-
coloring of a graph is intepreted as a morphism in a certain category of graphs.
This leads to a morphism between two topological spaces with free Z2 -actions,
to which the Borsuk–Ulam theorem can be applied. Recently these techniques of
graph complexes and characteristic classes have been greatly extended, culminating
in Babson and Kozlov’s proof of a conjecture of Lovász.
Keeping in mind that the above list is incomplete, it should be clear that there
would never be enough time to cover all of the relevant topics. The only remedy
would be another summer school on Geometric Combinatorics.
16 EZRA MILLER AND VICTOR REINER, OVERVIEW
BIBLIOGRAPHY
17
Lattice Points, Polyhedra, and
Complexity
Alexander Barvinok
IAS/Park City Mathematics Series
Volume 14, 2004
Alexander Barvinok
Introduction
The central topic of these lectures is efficient counting of integer points in polyhe-
dra. Consequently, various structural results about polyhedra and integer points
are ultimately discussed with an eye on computational complexity and algorithms.
This approach is one of many possible and it suggests some new analogies and
connections. For example, we consider unimodular decompositions of cones as a
higher-dimensional generalization of the classical construction of continued frac-
tions. There is a well recognized difference between the theoretical computational
complexity of an algorithm and the performance of a computational procedure in
practice. Recent computational advances [L+04], [V+04] demonstrate that many
of the theoretical ideas described in these notes indeed work fine in practice. On the
other hand, some other theoretically efficient algorithms look completely “unimple-
mentable”, a good example is given by some algorithms of [BW03]. Moreover,
there are problems for which theoretically efficient algorithms are not available at
the time. In our view, this indicates the current lack of understanding of some
important structural issues in the theory of lattice points and polyhedra. It shows
that the theory is very much alive and open for explorations.
Exercises constitute an important part of these notes. They are assembled at
the end of each lecture and classified as review problems, supplementary problems,
and preview problems.
Review problems ask the reader to complete a proof, to fill some gaps in a
proof, or to establish some necessary technical prerequisites. Problems of this kind
tend to be relatively straightforward. To be able to complete them is essential for
understanding.
Supplementary problems explore various topics in more depth and breadth.
Problems of this kind can be harder. They may use some general concepts which
are not formally introduced in the text, but which, nevertheless, are likely to be
familiar to the reader.
1 Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043.
E-mail address: barvinok@umich.edu.
This work is partially supported by the NSF grant DMS 0400617.
c
2007 American Mathematical Society
21
22 A. BARVINOK, LATTICE POINTS, POLYHEDRA, AND COMPLEXITY
Example 1.
n
1 − xn+1
xm = .
m=0
1−x
We observe that the long polynomial on the left-hand-side of the equation sums up
to a short rational function on the right-hand-side.
Geometrically, we do the following: we take the interval [0, n], for every integer
point m in the interval we write the monomial xm , and then take the sum over the
integer points in the interval, see Figure 1.
xm
0 m n
We observe that the thus obtained “long” polynomial (it contains n + 1 mono-
mials) can be written as a “short” rational function (it is expressed in terms of only
4 monomials).
Naturally, we ask what happens if we replace the interval by something higher-
dimensional. Let us, for example, draw a big triangle in the plane, for each integer
point m = (m1 , m2 ) in the triangle let us write the bivariate monomial xm =
xm 1 m2
1 x2 , and then let us try to write the sum over all integer points in the triangle
as some simple rational function in x1 and x2 , see Figure 2. If the triangle is really
large, we get a really long polynomial this way. Later, we will see how to write it
as a short rational function.
Our second inspiration comes from the formula for the sum of the infinite
geometric series.
23
24 A. BARVINOK, LATTICE POINTS, POLYHEDRA, AND COMPLEXITY
x1m1x 2m 2
(m1,m 2 )
Example 2.
+∞
1
xm = .
m=0
1−x
This formula makes sense because the series on the left-hand-side converges for all
|x| < 1 to the function on the right-hand-side. Similarly,
0
1 −x
xm = −1
=
m=−∞
1−x 1−x
makes sense because the series converges for all |x| > 1.
How do we make sense of
+∞
xm ?
m=−∞
This sum does not converge for any x, so we take the easiest route and say that
the sum is 0. This may look bizarre but there is some consistence in the way we
define the sums: the inclusion-exclusion principle seems to be respected. Indeed,
we get the set of all integers if we take all non-negative integers, add all non-positive
integers, and subtract 0, as it was double-counted:
+∞
+∞
0
xm = xm + xm − x0 .
m=−∞ m=0 m=−∞
Naturally, we ask what happens in higher dimensions. Let us draw three lines
in general position in the plane: each line splits the plane into two halfplanes, every
two lines form four angles, and there are various other regions (one triangle, the
whole plane, and some nameless unbounded polygonal regions), see Figure 4.
xm
and with the integer point lattice Zd ⊂ Rd , consisting of the points x with integer
coordinates. A polyhedron P ⊂ Rd is the set of solutions to finitely many linear
inequalities,
d
P = x ∈ Rd : aij xj ≤ bi , i = 1, . . . , m .
i=1
If all aij , bi are integers, the polyhedron is rational. The main object in these notes
is the set P ∩ Zd of integer points in a rational polyhedron P .
What can we do with polyhedra? The intersection of finitely many (rational)
polyhedra is a (rational) polyhedron. The union doesn’t have to be but may happen
to be a polyhedron. To account for all possible relations among polyhedra, we
introduce the algebra of polyhedra.
Defintion 2. For a set A ⊂ Rd , let [A] : Rd −→ R be the indicator of A. Thus [A]
is the function on Rd defined by
1 if x ∈ A
[A](x) =
0 if x ∈
/ A.
The algebra of polyhedra P(Rd ) is the vector space spanned by the indicators [P ]
for all polyhedra P ⊂ Rd . The coefficient field does not matter much: it can be
Q, R, or C.
26 A. BARVINOK, LATTICE POINTS, POLYHEDRA, AND COMPLEXITY
Valuations
Let V be a vector space. A linear transformation P(Rd ), P(Qd ) −→ V is called
a valuation. Basically, this course is about the existence and properties of one
particular valuation P(Qd ) −→ C(x1 , . . . , xd ), where C(x1 , . . . , xd ) is the space of
d-variate rational functions. We saw a glimpse of this valuation in Examples 1 and
2.
To warm up, we introduce one of the simplest and most useful valuations.
Theorem 1. There exists a unique valuation χ : P(Rd ) −→ R, called the Euler
characteristic, such that χ([P ]) = 1 for any non-empty polyhedron P ⊂ Rd .
Sketch of proof. Uniqueness of χ, if it exists, is clear: there is at most one way to
extend the definition χ([P ]) = 1 linearly on the whole algebra P(Rd ). Because the
indicators [P ] are linearly dependent, it is not at all obvious that such an extension
exists. To establish existence, we use induction on the dimension d. If d = 0, we
define χ(f ) = f (0) and it works.
Suppose that d > 0. First, we prove the existence of χ on the subspace of
P(Rd ) spanned by the indicators of bounded polyhedra, also known as polytopes.
Let us slice Rd into copies of Rd−1 by the value of the last coordinate of a point.
That is, we define Ht to be the hyperplane xd = t. Then Ht looks like Rd−1 and by
the induction hypothesis there is the Euler characteristic χt there. Given a function
f ∈ P(Rd ), we define its restriction ft onto Ht . One can easily check that if f is
a linear combination of indicators of bounded polyhedra in Rd then ft is a linear
combination of indicators of bounded polyhedra in Ht . Hence, we can define χt (ft ).
Now, the key observation is that the one-sided limit
lim χt− (ft− )
−→+0
always exists and that for all but finitely many t’s it is equal to χt (ft ). In fact, if
f= αi [Pi ],
i
then
lim χt− (ft− ) = χt (ft )
−→+0
unless t is the minimum value of the last coordinate on one of the polyhedra Pi in
the support of f , see Figure 5.
LECTURE 1. INSPIRATIONAL EXAMPLES. VALUATIONS 27
Although the sum is infinite, only finitely many terms are non-zero.
One can check that χ satisfies the required properties.
Now, we extend χ to the whole algebra P(Rd ). Let us take Pt to be the cube
|xi | ≤ t for i = 1, . . . , d and let us define
Problems
Review problems.
1. Let A1 , . . . , An ⊂ Rd be sets. Prove the inclusion-exclusion formula
n
|I|−1
Ai = (−1) Ai ,
i=1 I i∈I
where the sum is taken over all non-empty subsets I ⊂ {1, . . . , n} and |I| is the
cardinality of I.
2. Fill in the gaps in the proof of Theorem 1.
3. Show that the Euler characteristic can be extended to the space spanned by
the indicators [A] of closed convex sets A ⊂ Rd so that χ([A]) = 1 if A is a
non-empty closed convex set (a set A is called convex if, for every pair of points
x, y ∈ A it contains the interval [x, y] = {αx + (1 − α)y : 0 ≤ α ≤ 1}).
A supplementary problem.
1. Let P ⊂ Rd be a bounded polyhedron with a non-empty interior int P . Show
that [int P ] ∈ P(Rd ) and that χ([int P ]) = (−1)d . Deduce the Euler-Poincaré
d
formula: if P is a d-dimensional polytope (bounded polyhedron), then k=0 (−1)k fk =
1, where fk is the number of k-dimensional faces of P (including the polytope
itself).
28 A. BARVINOK, LATTICE POINTS, POLYHEDRA, AND COMPLEXITY
Preview problems.
1. Let P ⊂ Rd be a polyhedron and let T : Rd −→ Rk be a linear transformation.
Prove that T (P ) is a polyhedron.
2. We know that whenever there is an Euler characteristic, there must be an
underlying cohomology theory. What is the underlying cohomology theory for
the Euler characteristic in Theorem 1?
One problem is that the Euler characteristic of Theorem 1 is not a topo-
logical invariant: we have χ([A]) = 1 = −1 = χ([B]), where A is a line and B
is an open interval. Hence the underlying cohomology theory must somehow
distinguish between bounded and unbounded sets.
Remarks: Theorem 1 and its proof is due to H. Hadwiger, see also Section I.7 of
[Ba02] for more detail.
LECTURE 2
Identities in the Algebra of Polyhedra
What can we do with polyhedra? One important observation is that the image of
a polyhedron under a linear transformation is a polyhedron.
Theorem 1. Let P ⊂ Rd be a polyhedron and let T : Rd −→ Rk be a linear
transformation. Then T (P ) ⊂ Rk is a polyhedron. Furthermore, if P is a rational
polyhedron and T is a rational linear transformation (that is, the matrix of T is
rational), then T (P ) is a rational polyhedron.
The crucial step in the proof. Let us consider the following particular case:
k = d − 1 and T is the projection onto the first (d − 1) coordinates: (x1 , . . . , xd )
−→
(x1 , . . . , xd−1 ). Suppose that the polyhedron P is defined by a system of linear
inequalities:
d
aij xj ≤ bi for i = 1, . . . , m.
j=1
bi aij
d−1
xd ≤ − xj for i ∈ I+
aid j=1 aid
(2)
bi aij
d−1
xd ≥ − xj for i ∈ I−
aid j=1 aid
Conditions (1) are some linear inequalities needed to describe T (P ), but not all
of them. We get the complete set of linear inequalities by majorizing every lower
bound by every upper bound in (2), see Figure 6:
bi1 ai j
d−1
bi ai j
d−1
− 1
xj ≥ 2 − 2
xj for every pair i1 ∈ I+ , i2 ∈ I− .
ai1 d j=1 ai1 d ai2 d j=1 ai2 d
29
30 A. BARVINOK, LATTICE POINTS, POLYHEDRA, AND COMPLEXITY
Upper bounds
xd
Lower bounds
transformations T but starting to look less obvious for projections, see Figure 7 for
a simple example.
D B
A
T
C
Now we need to take a closer look at polyhedra. Some polyhedra have vertices,
some don’t.
Defintion 1. Let P ⊂ Rd be a polyhedron. A point v ∈ P is called a vertex of P
if whenever v = (x + y)/2 for some x, y ∈ P , we must have x = y = v. If v is a
point in P , we define the tangent cone of P at v as follows:
co(P, v) = x ∈ Rd : x + (1 − )v ∈ P for all sufficiently small > 0 .
A
C
B
C
B
Not all polyhedra have vertices. In fact, a non-empty polyhedron has a vertex
if and only if it does not contain a line.
Defintion 2. We say that a polyhedron P contains a line if there are points x
and y such that y = 0 and x + ty ∈ P for all t ∈ R. Finally, let P0 (Rd ) ⊂ P(Rd ),
P0 (Qd ) ⊂ P(Qd ) be the subspace spanned by the indicators of (rational) polyhedra
that contain lines.
It turns out that modulo polyhedra with lines, every polyhedron is just the
sum of its tangent cones.
32 A. BARVINOK, LATTICE POINTS, POLYHEDRA, AND COMPLEXITY
holds for the image T (P ). Indeed, by Theorem 2 the transformation T gives rise
to the transformation T on the algebra of polyhedra. Let us apply T to both
sides of the identity. We have T [P ] = [T (P )] and T [co(P, v)] = [T (co(P, v))] =
[co(T (P ), T (v))], cf. Review Problem 10.
We have to be somewhat careful with g: we know that g is a linear combination
of indicators of polyhedra with lines. If we are unlucky, the kernel of T may “eat
up” some of those lines and T (g) will not lie in P0 (Rk ). This is the reason why we
chose T to be “generic”. Thus if we prove the theorem for some “model” polyhedra
P , we can extend it (with some care) to polyhedra obtained from P by linear
transformations.
Now, we show that the result holds for a simplex, which we define as a compact
polyhedron Δ ⊂ Rd that is the non-empty intersection of d + 1 sufficiently generic
halfspaces H1 , . . . , Hd+1 . We notice that
[H1 ∪ . . . ∪ Hd+1 ] = [Rd ]
and expanding [H1 ∪ . . . ∪ Hd+1 ] by the inclusion-exclusion formula we represent
[Rd ] as the alternating sum of the indicators [Hi1 ∩ . . . ∩ Hik ] of intersections of
halfspaces. All such intersections contain lines except for the simplex Δ = [H1 ∩
. . . ∩ Hd+1 ] itself (the intersection of all d + 1 halfspaces) and the tangent cones
[H1 ∩ . . . ∩ Hi−1 ∩ Hi+1 ∩ . . . ∩ Hd+1 ] (the intersections of all but one halfspace) at
the vertices of Δ, see Figure 9.
It follows now that the result holds for all projections of simplices, that is for
polytopes (bounded polyhedra). To obtain the formula for a general polyhedron,
one needs some structural results about unbounded polyhedra, namely that every
unbounded polyhedron is the Minkowski sum of its recession cone and a polytope,
see Review Problem 11 and Supplementary Problem 3.
Defintion 3. Let A ⊂ Rd be a non-empty set. The set
A◦ = y ∈ Rd : x, y ≤ 1 for all x ∈ A
is called the polar of A.
It is easy to see that A◦ is a non-empty closed convex set containing the origin.
◦
The Bipolar Theorem asserts that (A◦ ) = A provided A is a closed convex set
LECTURE 2. IDENTITIES IN THE ALGEBRA OF POLYHEDRA 33
= + +
− − −
+
Figure 9. A triangle is the sum of the angles at its vertices minus the half-
planes based on its sides plus the whole plane.
containing the origin. One can show that if P is a (rational) polyhedron then P ◦
is a (rational) polyhedron, see Figure 10.
0
0
0 0
Figure 10. Some (bounded and unbounded) polyhedra and their polars.
For f ∈ P(Rd ), P(Qd ) and y ∈ Rd , let gy, (x) = f (x)G (x, y). One can check
that gy, ∈ P(Rd ), P(Qd ), so we can apply the Euler characteristic χ to gy, . Let
us define h = D (f ) by h (y) = χ(gy, ). Finally, we define h = D(f ) by h(y) =
lim−→0+ h (y). One can check then that D satisfies the desired properties.
34 A. BARVINOK, LATTICE POINTS, POLYHEDRA, AND COMPLEXITY
C
0 = + _ 0
A 0B 0 D
o
C
0 = + _
0 0 0
o o
A B
o
D
Problems
Review problems.
1. Complete the proof of Theorem 1.
2. In Theorem 1, suppose that P ⊂ Rd is defined by m linear inequalities. Esti-
mate the number of inequalities needed to define T (P ).
3. Check the proof of Theorem 2.
4. Let P ⊂ Rd be a polyhedron defined by m linear inequalities
d
aij xj ≤ bi for i = 1, . . . , m.
j=1
Supplementary problems.
1. For sets A,
B ⊂ R d
, we define their Minkowski sum A + B = x+y : x ∈
A, y ∈ B . Prove that the Minkowski sum of polyhedra is a polyhedron and
that the Minkowski sum of rational polyhedra is a rational polyhedron.
2. Prove that there exists a bilinear operation, called convolution, : P(Rd ) ×
P(Rd ) −→ P(Rd ) such that [P ][Q] = [P +Q] for any two polyhedra P, Q ⊂ Rd .
This gives P(Rd ) another (more interesting) commutative algebra structure.
Note that [0] plays the role of the identity, so f [0] = [0] f = f for all
f ∈ P(Rd ).
36 A. BARVINOK, LATTICE POINTS, POLYHEDRA, AND COMPLEXITY
where the sum is taken over all faces F = P of P , including the empty face (cf.
Supplementary Problem 1 to Lecture 1).
8. Prove that the polar of a (rational) polyhedron is a (rational polyhedron) and
◦
that (A◦ ) = A if A is closed, convex, and contains 0.
9. Complete the proof of Theorem 4.
10. Show that if we apply the polarity transform D to both sides of the identity in
Problem 7 above, we get the Gram-Brianchon identity of Problem 6.
Preview problems.
1. A polyhedron K ⊂ Rd is called a (polyhedral) cone if 0 ∈ K and λx ∈ K for
all x ∈ K and all λ ≥ 0 (note that the tangent cone of Definition 1 is not
necessarily a cone in the sense of this definition, since the vertex of the tangent
cone is not necessarily the origin). Prove that if K is a cone then K ◦ is a cone
◦
and that (K ◦ ) = K.
2. Let K1 , K2 ⊂ Rd be polyhedral cones. Prove that [K1 ∩ K2 ]◦ = [K1 + K2 ],
where “+” is the Minkowski sum, see Supplementary Problem 1.
3. Let D be the transform of Theorem 4 and let f1 , f2 ∈ P(Rd ) be linear com-
binations of indicator functions of polyhedral cones. Prove that D(f1 f2 ) =
LECTURE 2. IDENTITIES IN THE ALGEBRA OF POLYHEDRA 37
+ ∩Z
m∈Rd d m1 =0 md =0
d
1
= provided |xi | < 1 for i = 1, . . . , d.
i=1
1 − xi
39
40 A. BARVINOK, LATTICE POINTS, POLYHEDRA, AND COMPLEXITY
u1
0 u2
u1
0 u2
Proof. The proof consists of the observation that every point m ∈ K ∩ Zd can be
uniquely written as m = m1 + m2 , where m1 ∈ Π ∩ Zd and m2 is a non-negative
integer combination of u1 , . . . , ud . Indeed, since u lies in the cone K, it can be
written in the form
d
m= αi ui for some real numbers αi ≥ 0.
i=1
Let
α denote the largest integer not exceeding α (a.k.a the integer part of α) and
let {α} = α −
α (the fractional part of α). Then
d
d
m1 = {αi }ui and m2 =
αi ui .
i=1 i=1
To prove uniqueness, suppose that we have two decompositions m = m1 + m2 and
m = m1 + m2 , where m1 and m2 are integer points from the parallelepiped Π and
m2 and m2 are non-negative integer combinations of u1 , . . . , ud . Then we can write
LECTURE 3. GENERATING FUNCTIONS AND CONES. CONTINUED FRACTIONS 41
Theorem 1 provides us with a finite formula for an infinite series, but there
is still something unsatisfactory about it. Namely, the sum over integer points in
the fundamental parallelepiped is not very explicit and, although finite, can be
quite large. Although the set of integer points lying in the parallelepiped can be
complicated, we can tell the number of such points exactly.
Theorem 2. The number of integer points in the fundamental parallelepiped is
equal to the volume of the parallelepiped.
Sketch of proof. Let Λ be the set of all integer combinations of u1 , . . . , ud :
d
Λ= αi ui : αi ∈ Z for i = 1, . . . , d .
i=1
(2 ,3)
( 3,1)
0
Figure 13. The number of integer points in a fundamental parallelogram is
equal to the area of the parallelogram.
d
1
f (K, x) = .
i=1
1 − xui
Continued Fractions
Let us choose a number a ∈ R. The following procedure produces what is called
the continued fraction expansion [a0 ; a1 , . . . , an , . . .] of a. First, we write
Now, if {a} = 0, we stop. Otherwise, 0 < {a} < 1, we let b = 1/{a}, so b > 1. We
write
b =
b + {b} and let a1 =
b.
If {b} = 0 we stop. Otherwise, we let new b := 1/{old b}, and continue. In the
end, we get the expansion
1
a = a0 + .
1
a1 +
1
a2 +
...
The expansion can be finite (if a is rational) or infinite (if a is irrational). We define
the k-th convergent [a0 ; a1 , . . . , ak ] by cutting the expansion at ak . For example,
the 4-th convergent [a0 ; a2 , a3 , a4 ] is
1
a = a0 + .
1
a1 +
1
a2 +
1
a3 +
a4
LECTURE 3. GENERATING FUNCTIONS AND CONES. CONTINUED FRACTIONS 43
cut
paste
cut
paste
we get:
1 1 1
f (K, x) = − +
(1 − x1 )(1 − x2 ) (1 − x2 )(1 − x1 x2 ) 1 − x1 x52
5
1 1
+ −
(1 − x1 x52 )(1 − x31 x16
2 ) 1 − x1 x52
1 1
− 7 x37 ) + 1 − x7 x37
(1 − x31 x16
2 )(1 − x 1 2 1 2
1 1
+ − ,
(1 − x1 x2 )(1 − x1 x2 ) 1 − x71 x37
7 37 31 164
2
so finally,
1 1 1
f (K, x) = − +
(1 − x1 )(1 − x2 ) (1 − x2 )(1 − x1 x52 ) (1 − x1 x52 )(1 − x31 x16
2 )
1 1
− 7 x37 ) + (1 − x7 x37 )(1 − x31 x164 ) .
(1 − x31 x16
2 )(1 − x1 2 1 2 1 2
The formula is reasonably short.
Given an arbitrary 2-dimensional rational cone generated by u1 , u2 ∈ Z2 , we can
always change the coordinates by applying a linear transformation which preserves
Z2 so that u2 becomes equal to (1, 0). Suppose that u1 = (q, p) for integers p and
q > 0. To compute the generating function f (K, x), we compute the continued
fraction expansion of p/q and obtain K by cutting and pasting the unimodular
cones computed from the convergents of p/q. If the k-th convergent is pk /qk , we
cut or paste, depending on the parity of k > 1, the cone generated by (qk , pk ) and
(qk−1 , pk−1 ), which is always unimodular, see Review Problems 4 and 5.
to write the coordinates of its generators, we need about log |p| + log |q| + O(1)
bits (or digits) since to write an integer a we need about log |a| + O(1) bits (or
digits). Thus we say that the input size of the problem of computing f (K, x)
is about log |p| + log |q| + O(1). The number of operations required to compute
f (K, x) via continued fractions is about O(log2 |p| + log2 |q| + 1), that is, bounded
by a polynomial in the input size. In contrast, the number operations required to
compute f (K, x) via Theorem 1 (and even to write down the answer) is exponential
in the input size of K. In Lecture 5, for any dimension d (fixed in advance), we
present a polynomial time algorithm, which, given a rational cone K ⊂ Rd as an
input, computes f (K, x) as a rational function.
Problems
Review problems.
1. Check the proof of Theorem 1.
2. Make the proof of Theorem 2 rigorous.
3. Let K be the 2-dimensional simple cone generated by u1 = (1, 0) and u2 = (1, n)
for some positive integer n. Compute f (K, x).
4. Let [a0 ; a1 , . . . , an . . .] be the continued fraction expansion of a real number a
and let pk /qk = [a0 ; a1 , . . . , ak ] be the k-th convergent (we assume that pk and
qk are coprime). Prove that for k ≥ 2
pk = ak pk−1 + pk−2 and qk = ak qk−1 + qk−2 .
Deduce that
pk−1 qk − pk qk−1 = (−1)k−1 for k ≥ 0.
5. Justify the procedure of computing f (K, x) for the cone K generated by (1, 0)
and (q, p) via continued fractions.
6. Let K ⊂ Rd be the set defined by
K = x ∈ Rd : ui , x ≤ 0 for i = 1, . . . , d
Supplementary problems.
1. Let u1 , . . . , ud be linearly independent vectors in Zd . Let K be the cone gener-
ated by u1 , . . . , ud and let
d
int K = αi ui : αi > 0 for i = 1, . . . , d
i=1
46 A. BARVINOK, LATTICE POINTS, POLYHEDRA, AND COMPLEXITY
Figure 15. The number of integer points in the triangle (8) is equal to the area
of the triangle (5) plus half of the number of integer points on the boundary
(2) plus 1.
Preview problems.
1. Let K ⊂ Rd be a unimodular cone generated by integer vectors u1 , . . . , ud and
let K + v be the translation of K by a rational vector v ∈ Qd . Prove that
d
1
d
f (K + v, x) = xw with w = v, u∗i ui ,
i=1
1 − xui i=1
where u∗1 , . . . , u∗d are defined by u∗i , uj = δij .
2. Construct an efficient (polynomial time) algorithm to sample a random integer
point in a given fundamental parallelepiped Π from the uniform distribution
on Π ∩ Zd (the dimension d needs not to be fixed in advance).
Remarks: For generating functions and rational cones, see Section 4.6 of [St97]
and Section VIII.1 of [Ba02]. A classical reference for continued fractions is [Kh97].
For the theory of computational complexity, see [Pa94].
LECTURE 4
Rational Polyhedra and Rational Functions
where pi (x) are Laurent polynomials in x and uij ∈ Zd are non-zero vectors.
A plausible argument. Since 0 is a vertex of K, there is a vector c ∈ Rd , c =
(c1 , . . . , cd ) such that c, x < 0 for all x ∈ K \ {0}. Now, for any
x from a
sufficiently small neighborhood U of x0 = (ec1 , . . . , ecd ) the series m∈K∩Zd xm
converges absolutely and uniformly on compact subsets of U . It seems intuitively
obvious and indeed correct that K can be cut into simple rational cones, so we
can deduce the formula for f (K, x) from Theorem 1, Lecture 3, and the inclusion-
exclusion formula. It takes some time though to make the proof rigorous, cf. Figure
16.
47
48 A. BARVINOK, LATTICE POINTS, POLYHEDRA, AND COMPLEXITY
0 0 0 0
= + −
= + −
Figure 16. The indicator of a cone with a square base can be written as the
sum of the indicators of cones with triangular bases minus the indicator of a
flat cone based on the interval.
where pi (x) are Laurent polynomials in x and uij ∈ Zd are non-zero vectors.
Sketch of proof. The idea is to consider P as a section of a pointed rational cone
K ⊂ Rd+1 . We think of Rd as the affine hyperplane xd+1 = 1 in Rd+1 . Given the
inequalities defining P ,
d
aij xj ≤ bi for i = 1, . . . , m,
j=1
P
K
P
0
Figure 17. Representing a d-dimensional polyhedron P as a hyperplane sec-
tion of a (d + 1)-dimensional cone K.
of indicators of rational polyhedra Pi without lines, we must have the same linear
dependence
(2) αi f (Pi , x) = 0
i∈I
of their generating functions. Suppose for a moment that in (1) there exists a
non-empty open set U ⊂ Cd such that for x ∈ U , each of the series m∈Pi ∩Zd xm
converges absolutely to f (Pi , x). Then (2) follows by a standard argument from
analysis. The problem is that there may not be a single set U which works for
50 A. BARVINOK, LATTICE POINTS, POLYHEDRA, AND COMPLEXITY
all polyhedra Pi in (1). To handle this difficulty, we break the global identity (1)
into small “local” pieces, prove (2) for every such piece and then “glue” the global
identity (2) from the local pieces.
Let us choose a representation
[Rd ] = βj [Qj ],
j∈J
where {Qj } is a finite family of rational polyhedra without lines and βj are numbers.
One way to obtain the representation is to cut Rd by the coordinate hyperplanes
and express [Rd ] as a linear combination of indicators of coordinate orthants and
their intersections using the inclusion-exclusion formula. Multiplying the above
formula by [Pi ], we get
[Pi ] = βj [Pi ∩ Qj ] for all i ∈ I.
j∈J
Let us fix some i ∈ I. Then Pi is a rational polyhedron without lines and Pi ∩Qj are
rational polyhedral pieces of Pi . Therefore, there is a non-empty open set Ui ⊂ Cd
such that for all x ∈ Ui all the series defining f (Pi , x) and f (Pi ∩ Qj ) converge and
so we have the identity
(3) f (Pi , x) = βj f (Pi ∩ Qj , x) for all i ∈ I.
j∈J
From (3) and (4) we get (2). This completes the first step of the proof.
Thus we are able to extend F to a valuation on P(Qd ). It remains to prove
Part (3) of the Theorem. One can show that if P is a rational polyhedron with
lines, then there exists a non-zero m ∈ Zd such that P + m = P (there is a non-
zero integer translation of P which maps P onto itself). On the other hand, from
elementary analysis we deduce that we must have f (P + m, x) = xm f (P, x) for any
rational polyhedron P without lines. By linearity, F [P + m] = xm F [P ] for any
rational polyhedron P . Hence, if P + m = P , we must have F [P ] = xm F [P ], from
which F [P ] = 0.
Suppose that P ⊂ Rd is a rational polyhedron without lines (maybe even
bounded) and that we want to compute a short formula for the rational generating
function f (P, x). Theorem 3 allows us to employ various identities in the algebra
P(Qd ) of rational polyhedra, including those that involve polyhedra with lines. In
particular, we get the following result, first obtained by M. Brion in 1988.
Theorem 4. Let P ⊂ Rd be a rational polyhedron with vertices. Then
f (P, x) = f co(P, v), x ,
v
LECTURE 4. RATIONAL POLYHEDRA AND RATIONAL FUNCTIONS 51
where the sum is taken over all vertices v of P and co(P, v) is the tangent cone of
P at v.
Proof. The proof follows by Theorem 3 of this lecture and Theorem 3 of Lecture
2.
Note that the tangent cone co(P, v) is not a rational cone per se, but a rational
translation of a rational cone.
Example 1. Let d = 1 and let P be the interval [0, n] ⊂ R1 for some positive integer
n. Then P is a rational polyhedron with the vertices at 0 and n, see Figure 18.
The tangent cone co(P, 0) at 0 is the ray [0, +∞) and the corresponding generating
function is
+∞
1
xm = .
m=0
1−x
The tangent cone co(P, n) at n is the ray (−∞, n] and the corresponding generating
function is
n
xn −xn+1
xm = −1
= .
m=−∞
1−x 1−x
Note that there is not a single value of x for which both series converge. Never-
theless, Theorem 4 predicts that the sum of the two functions gives the generating
function for P :
n
1 xn+1
xm = − ,
m=0
1−x 1−x
which is indeed the case.
0 n
Problems
Review problems
1. Complete the proof of Theorem 2.
2. Let P ⊂ Rd be a rational polyhedron with a line. Prove that there exists a
non-zero vector m ∈ Zd such that P + m = P .
3. Complete the proof of Theorem 3.
4. Check Theorem 4 for the triangle in the plane with the vertices (0, 0), (0, 1),
and (1, 0).
52 A. BARVINOK, LATTICE POINTS, POLYHEDRA, AND COMPLEXITY
A supplementary problem
1. Let K ⊂ Rd be a pointed rational cone with non-empty interior int K. Prove
the reciprocity relation f (int K, x−1 ) = (−1)d f (K, x).
non-empty interior int K.
Preview problems
1. Prove that the polar of a unimodular cone is a unimodular cone.
2. Let K ⊂ R2 be the cone generated by u1 = (1, 0) and u2 = (q, p) for some pos-
itive integers p and q. Compare the following two ways of computing f (K, x).
The first way is the continued fractions method of Lecture 3. The second way
is as follows: consider the polar K ◦ (check that K ◦ is the cone generated by
(−p, q) and (0, −1)). Represent [K ◦ ] as a linear combination of the indicators
of unimodular cones using the continued fractions method. Apply Theorem
◦
4 of Lecture 1 to obtain a unimodular decomposition of K = (K ◦ ) . Com-
pute f (K, x) from that decomposition. What kind of identities do we get for
f (K, x)?
This question was asked by one of the attendees.
which is just a Laurent polynomial in x, we can get the number of integer points
|P ∩ Zd | by substituting x = (1, . . . , 1). Our technique allows us to compute f (P, x)
as a reasonably short rational function of the type
pi (x)
f (P, x) = ,
i
(1 − x ) · · · (1 − xuid )
ui1
where pi (x) are Laurent polynomials in x. This seems to pose a little problem since
x = (1, . . . , 1) is a pole of every fraction. Nevertheless, the poles cancel each other,
as in the model example
n
1 xn+1
xm = − .
m=0
1−x 1−x
We deal with singularities by approaching the point (1, . . . , 1) via some curve
and computing the appropriate limit. One of the standard choices is the curve
x(t) = (etc1 , . . . , etcd ), where c = (c1 , . . . , cd ) is a sufficiently generic vector: we
need c, uij = 0 for all i, j. Then x(0) = (1, . . . , 1) and the limit f (P, x(t)) as
t −→ 0 can be computed by using standard analysis techniques.
Generating functions help to solve integer programming problems, that is the
problems of optimizing a given linear function on the set P ∩ Zd of integer points
in a given rational polytope. In short, generating functions f (P, x) encode all the
information about the set of integer points in P in a compact form. One remarkable
fact is that to find a short formula for f (P, x) for a bounded polyhedron, we employ
the full power of the algebra P(Qd ) and identities in the algebra involving unbounded
polyhedra and even polyhedra with lines (Theorems 3 and 4 of Lecture 4).
53
54 A. BARVINOK, LATTICE POINTS, POLYHEDRA, AND COMPLEXITY
The input size of P is the number of bits needed to write down the inequalities,
assuming that aij and bi are integers written in the binary system. For example, to
write an integer a, we need about log |a|+O(1) bits. Thus we say that the algorithm
for computing f (P, x) is reasonably fast and the resulting formula is reasonably
short if the time we need to compute f (P, x) and the space we need to write it
down grows only modestly when the input size of P grows. More precisely, we say
that we have a polynomial time algorithm for a particular class of rational polyhedra
if there is a polynomial poly such that the running time of the algorithm on every
polyhedron P from the class does not exceed poly(input size of P ). One example
of a polynomial time algorithm is provided by the continued fraction method for
computing f (K, x) where K is a 2-dimensional rational cone, see Lecture 3.
It is probably hopeless to search for a polynomial time algorithm in the class
of all rational polyhedra. However, once the dimension d is fixed such algorithms
exist.
Theorem 1. Let us fix d. Then there exists a polynomial time algorithm, which,
given a rational polyhedron P ⊂ Rd , computes the generating function f (P, x) in
the form
xvi
f (P, x) = αi ,
i
(1 − x i1 ) · · · (1 − xuid )
u
Preliminaries
The main result we need is Minkowski’s Convex Body Theorem. Let A ⊂ Rd be a
set, such that
(1) A two points x, y ∈ A, the interval [x, y] =
is convex, that is, for every
αx + (1 − α)y : 0 ≤ α ≤ 1 also lies in A;
(2) A is symmetric about the origin, that is, for every x ∈ A, the point −x
also lies in A;
(3) A has a sufficiently large volume: vol A > 2d .
LECTURE 5. COMPUTING GENERATING FUNCTIONS FAST 55
K
v
0 u1
A
u2 u2 u 2
v v v
= − +
0 u1 0 u 0 0
1
Figure 20. Writing the cone as a linear combination of cones with smaller
indices for d = 2.
0 0 0 0 0 0
= + − − +
v v v v
u1 u u1 u1 u1
2
u2 u2 u2
u3 u3 u3 u3
= + − − +
Figure 21. Writing the cone as a linear combination of cones with smaller
indices for d = 3 (the sections of the cones by a plane are shown below).
There are certain similarities between the described procedure and the unimod-
ular decomposition obtained from the continued fractions method in dimension 2.
There are differences, too. In the method just described, there is a certain flexibility
in choosing vector v, while the continued fractions method is quite rigid. This is, of
course, due to the fact that we know much more about integer points in dimension
2 than in higher dimensions. On the other hand, there is a version of our algorithm
that reduces to the continued fractions method in dimension 2.
be the polar of K, see Lecture 2. It is not hard to prove that K ◦ is a simple rational
cone, that K ◦ is unimodular if and only if K is unimodular, and that (K ◦ )◦ = K.
Thus we modify the above procedure as follows.
Given a simple rational cone K, we compute the polar K ◦ . Then we apply the
unimodular decomposition and get
[K ◦ ] = αi [Ki ] + indicators of lower-dimensional cones,
i
where Ki are unimodular cones. Next, we compute Ki◦ and observe that
[K] = αi [Ki◦ ] + indicators of cones with lines,
i
see Theorem 4 and Review Problem 14 of Lecture 1.
By Theorem 3 of Lecture 4,
f (K, x) = αi f (Ki◦ , x),
i
since we can ignore polyhedra with lines.
Problems
Review problems
1. Check that the procedure of computing f (K, x) for a simple rational cone
K ⊂ R2 via continued fractions (see Lecture 3) indeed runs in polynomial
time.
2. Prove Minkowski’s Convex Body Theorem.
3. Check that the algorithm for the unimodular decomposition of a cone indeed
works. 4. Let K ⊂ Rd be a unimodular cone. Prove that K ◦ is a unimodular
cone.
Supplementary problems
1. Let a1 and a2 be positive coprime integers and let S ⊂ Z be the set of all
non-negative integer combinations of a1 and a2 . Prove that
1 − xa1 a2
xm = .
(1 − xa1 )(1 − xa2 )
m∈S
2. Let a1 , a2 and a3 be positive coprime integers and let S ⊂ Z be the set of all
non-negative integer combinations of a1 , a2 and a3 . Prove that
1 − xb1 − xb2 − xb3 + xb4 + xb5
xm = ,
(1 − xa1 )(1 − xa2 )(1 − xa3 )
m∈S
Concluding Remarks
The algorithmic theory of counting lattice points in polyhedra is discussed in
[BP99]; some of the algorithms suggested there are implemented, see [L+04] and
[V+04]. For other algorithmic questions concerning lattice points, see [G+93]. For
Minkowski’s Theorems and other topics in the geometry of numbers, see [GL87].
We conclude these lectures by discussing various related topics and open questions.
Something curvilinear?
Is it possible to extend the developed theory onto something non-polyhedral, such
as Euclidean balls? Probably not, as it appears to be in the realm of totally different
forces, more akin to theta functions than to rational functions. For example, let
4
B = (x1 , . . . , x4 ) : x2i ≤ n
i=1
√
be the standard Euclidean ball of radius n in dimension 4. Suppose for a moment
that we can efficiently enumerate integer points in B. Then we can count integer
points on the sphere x21 + x22 + x23 + x24 = n. However, the number of such points,
that is, the number of ways to represent n as a sum of four squares of integers, by
Jacobi’s formula is equal to
8 p
4 | p | n
(in words: eight times the sum of the divisors of n that are not divisible by 4). Thus
we gain some insight into divisors of n, and, pushing it a bit further, we can come
up with an efficient algorithm for factoring integers, see [B+86] and [Dy91]. The
existence of such an algorithm is not entirely impossible, but somewhat doubtful.
Irrational polyhedra?
How can we enumerate integer points in irrational polyhedra? There are some
obvious difficulties with generating functions.√Consider, for example, a cone K ⊂ R2
defined by the inequalities x1 ≥ 0 and x2 ≤ 2x1 . Just as before, we can write the
generating function
f (K, x) = xm .
m∈K∩Z2
transformation T . In [BW03], it is proved that for such sets S (obtained from the
set of integer points P ∩ Zd in a rational polyhedron P ⊂ Rd by a projection) the
generating function
f (S; x) = xm
m∈S
admits a short representation as a rational function in x, which can be computed
in polynomial time when the dimension d is fixed.
There have been some advances towards the general theory of sets of integer
points encoded by short rational generating functions. For example, in [BW03] it
is proved that if two sets S1 , S2 ⊂ Zd are defined by their short rational generating
functions f (S1 , x) and f (S2 , x), then the generating function f (S, x) of S = S1 ∩ S2
can be computed in polynomial time as a short rational function.
However, we are still quite far from having a full-fledged theory for sets S with
short rational generating functions. Suppose, for example, that S is the projection
of the difference X \ Y , where X and Y are the projections of the sets of integer
points P ∩ Zk1 and Q ∩ Zk2 in some rational polyhedra P and Q. We don’t know
how to handle such a set S (our lack of understanding is mitigated by the lack
of interesting examples of such complicated constructions). Also, algorithms of
[BW03] seem to be outrageously impractical.
60 A. BARVINOK, LATTICE POINTS, POLYHEDRA, AND COMPLEXITY
[Ba93] A. Barvinok, Computing the volume, counting integral points, and expo-
nential sums, Discrete Comput. Geom., 10, (1993), pp. 123-141.
[Ba02] A. Barvinok, A Course in Convexity, Graduate Studies in Mathematics,
vol 54, Amer. Math. Soc., Providence, RI, 2002.
[Br88] M. Brion, Points entiers dans les polyédres convexes (French) , Ann. Sci.
École Norm. Sup. (4) 21 (1988), pp. 653–663.
[BP99] A. Barvinok and J. Pommersheim, An algorithmic theory of lattice points
in polyhedra, New Perspectives in Algebraic Combinatorics (Berkeley,
CA, 1996–97), Math. Sci. Res. Inst. Publ., vol 38, Cambridge Univ.
Press, Cambridge, 1999, pp. 91–147.
[BW03] A. Barvinok and K. Woods, Short rational generating functions for lat-
tice point problems, J. Amer. Math. Soc. 16 (2003), pp. 957–979
[B+86] E. Bach, G. Miller, and J. Shallit, Sums of divisors, perfect numbers
and factoring, SIAM J. Comput. 15 (1986), pp. 1143–1154.
[B+05] M. Beck and F. Sottile Irrational proofs of three theorems of Stan-
ley (preprint) arXiv math.CO/0501359, European Journal of Combi-
natorics, to appear.
[Dy91] M. Dyer, On counting lattice points in polyhedra, SIAM J. Comput. 20
(1991), pp. 695–707.
[Dy03] M. Dyer, Approximate counting by dynamic programming, Proceedings
of the 35th Annual ACM Symposium on the Theory of Computing
(STOC 2003), 2003, pp. 693–699.
[GL87] P.M. Gruber and C.G. Lekkerkerker, Geometry of Numbers. Second
edition, North-Holland Mathematical Library, vol. 37, North-Holland,
Amsterdam, 1987.
[G+93] M. Grötschel, L. Lovász, and A. Schrijver, Geometric Algorithms and
Combinatorial Optimization. Second edition, Algorithms and Combina-
torics, vol. 2, Springer-Verlag, Berlin, 1993.
[Kh97] A. Ya. Khinchin, Continued Fractions, Translated from the third (1961)
Russian edition. Reprint of the 1964 translation, Dover Publications,
Inc., Mineola, NY, 1997.
[KR97] D. Klain and G.-C. Rota, Introduction to Geometric Probability, Lezioni
Lincee, Cambridge Univ. Press, Cambridge, 1997.
61
62 A. BARVINOK, LATTICE POINTS, POLYHEDRA, AND COMPLEXITY
[KP92] A.G. Khovanskii and A.V. Pukhlikov, The Riemann-Roch theorem for
integrals and sums of quasipolynomials on virtual polytopes. (Russian),
translation in St. Petersburg Math. J. 4 (1993), no. 4, pp. 789–812,
Algebra i Analiz 4, no. 4 (1992), pp. 188–216.
[KP93] A.G. Khovanskii and A.V. Pukhlikov, Integral transforms based on Euler
characteristic and their applications, Integral Transform. Spec. Funct. 1
(1993), pp. 19–26.
[La91] J. Lawrence, Rational-function-valued valuations on polyhedra, Discrete
and Comput. Geometry (New Brunswick, NJ, 1989/1990), DIMACS Ser.
Discrete Math. Theoret. Comput. Sci., vol. 6, Amer. Math. Soc., Provi-
dence, RI, 1991, pp. 199–208.
[L+04] J.A. De Loera, R. Hemmecke, J. Tauzer, and R. Yoshida, Effective
lattice point counting in rational convex polytopes, Journal of Symbolic
Computation 38 (2004), pp. 1273–1302.
see also http://www.math.ucdavis.edu/∼latte/.
[Pa94] C.H. Papadimitriou, Computational Complexity, Addison-Wesley, addr
Reading, MA, 1994.
[St97] R.P. Stanley, Enumerative Combinatorics. Vol. 1, Corrected reprint of
the 1986 original. Cambridge Studies in Advanced Mathematics, vol. 49,
Cambridge Univ. Press, Cambridge, 1997.
[V+04] S. Verdoolaege, R. Seghir, K. Beyls, V. Loechner, and M. Bruynooghe,
Analytical computation of Ehrhart polynomials: enabling more compiler
analyses and optimizations, Proceedings of the 2004 International Con-
ference on Compilers, Architecture, and Synthesis for Embedded Sys-
tems (CASES 2004), 2004, pp. 248–258.
see also http://www.kotnet.org/∼skimo/barvinok/.
[Zi95] G. Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics, vol.
152, Springer-Verlag, New York, 1995.
Root Systems and
Generalized Associahedra
2007
c Sergey Fomin and Nathan Reading
65
66 FOMIN AND READING, ROOTS AND ASSOCIAHEDRA
Acknowledgments
We thank Christos Athanasiadis, Jim Stasheff and Andrei Zelevinsky for careful
readings of earlier versions of these notes and for a number of editorial suggestions,
which led to the improvement of the paper.
S.F.: I am grateful to the organizers of the 2004 Graduate Summer School at Park
City (Ezra Miller, Vic Reiner, and Bernd Sturmfels) for the invitation to deliver
these lectures, and for their support, understanding, and technical help.
Sections 3.3-3.4 and Lecture 4 present results of an ongoing joint project with
Andrei Zelevinsky centered around cluster algebras.
N.R.: I would like to thank Vic Reiner for teaching the course which sparked my
interest in Coxeter groups; Anders Björner and Francesco Brenti for making a
preliminary version of their forthcoming book available to the students in Reiner’s
course; and John Stembridge, whose course and lecture notes have deepened my
knowledge of Coxeter groups and root systems.
Some of the figures in these notes are inspired by figures produced by Satyan
Devadoss, Vic Reiner and Rodica Simion. Several figures were borrowed from
[13, 19, 20, 21, 23].
LECTURE 1. REFLECTIONS AND ROOTS 67
LECTURE 1
Reflections and Roots
Both τ1 and τ2 are involutions: τ12 = τ22 = 1, where 1 denotes the identity map. The
5-periodicity of the recurrence (1) translates into the identity (τ2 τ1 )5 = 1. That is,
the group generated by τ1 and τ2 is a dihedral group with 10 elements.
Let us now consider a similar but simpler pair of maps. Throw away the +1’s
that occur in the definitions of τ1 and τ2 , and take logarithms. We then obtain a
pair of linear maps
x y−x x x
s1 : −→ and s2 : −→ .
y y y x−y
A (linear) hyperplane in a vector space V is a linear subspace of codimension 1.
A (linear) reflection is a map that fixes all the points in some linear hyperplane,
and has an eigenvalue of −1. The maps s1 and s2 are linear reflections satisfying
(s2 s1 )3 = 1. Thus, the group s1 , s2 is a dihedral group with 6 elements.
We are led to wonder if the dihedral behavior of τ1 , τ2 is related to, or even
explained by the dihedral behavior of s1 , s2 . To test this unlikely-sounding hy-
pothesis, let us try to find similar examples. What other pairs (s, s ) of linear
1The discovery of this recurrence and its 5-periodicity are sometimes attributed to R. C. Lyness
(1942); see, e.g., [15]. It was probably already known to N. H. Abel. This recurrence is closely
related to (and easily deduced from) the famous “pentagonal identity” for the dilogarithm function,
first obtained by W. Spence (1809) and rediscovered by Abel (1830) and C. H. Hill (1830). See,
e.g., [37].
68 FOMIN AND READING, ROOTS AND ASSOCIAHEDRA
Calculations show that (τ3 τ1 )6 = 1, and the group τ1 , τ3 is dihedral with 12
elements. We can think of τ1 and τ3 as defining a “moving window” for the sequence
y + 1 x2 + (y + 1)2 x2 + y + 1 x2 + 1
(4) x, y, , , , , x, y, . . .
x x2 y xy y
Notice that the Laurent phenomenon holds: these rational functions are Laurent
polynomials—again, with nonnegative integer coefficients.
Likewise, (τ4 τ1 )8 = 1, the group τ1 , τ4 is dihedral with 16 elements, and τ1
and τ4 define an 8-periodic sequence of Laurent polynomials.
In the first two lectures, we will develop the basic theory of finite reflection
groups that will include their complete classification. This theory will later help
explain the periodicity and Laurentness of the sequences discussed above, and pro-
vide appropriate algebraic and combinatorial tools for the study of other similar
recurrences.
two reflections s and t whose reflecting lines make an angle of π/5. It consists of 5
reflections, 4 rotations, and the identity element. In Figure 1.1, each of the 5 lines
is labeled by the corresponding reflection.
s t
sts 1
s t tst
st ts
ststs = tstst
sts tst
stst tsts
ststs
tstst
x=y
(0, 1, 0) (1, 0, 0)
x=z
y=z
(0, 0, 1)
Example 1.8. The n-crosspolytope is the convex hull of (the endpoints of) the
vectors ±e1 , ±e2 , . . . , ±en in Rn . For example, the 3-crosspolytope is the regular
octahedron. The symmetry group of this polytope is the hyperoctahedral group Bn .
As in the previous examples, it is generated by the reflections it contains.
The special case n = 3 (the symmetry group B3 of a regular octahedron) is
shown in Figure 1.3. The dotted lines show the intersections of reflecting hyper-
planes with the front surface of the octahedron. Each edge of the octahedron is
also contained in a reflecting plane.
There are two types of reflections in the symmetry group of the crosspolytope.
One type of reflection transposes a vertex with its negative and fixes all other
vertices. Also, for each pair i = j, there is a reflection which transposes ei and ej ,
transposes −ei and −ej , and fixes all other vertices.
To construct a minimal set of reflections generating Bn , take the minimal gener-
ating set for An−1 given in Example 1.7 and adjoin the reflection that interchanges
e1 and −e1 .
The group Bn is also the symmetry group of the n-dimensional cube.
Example 1.9. The symmetry group of a regular dodecahedron (or a regular icosa-
hedron) is the reflection group H3 . Figure 1.4 shows the dodecahedron and a
minimal set of three reflections generating its symmetry group. The dotted lines
show the intersections of the corresponding three hyperplanes with the front surface
of the dodecahedron.
Example 1.10. In 4-space, there are six types of regular polytopes. The obvious
three are the 4-simplex, the 4-cube, and the 4-crosspolytope. There are two regular
polytopes whose symmetry group is the reflection group called H4 . One of these,
the 120-cell, has 600 vertices and 120 dodecahedral faces; the other, the 600-cell,
has 120 vertices and 600 tetrahedral faces. The remaining regular 4-dimensional
polytope is the 24-cell, with 24 vertices and 24 octahedral faces. Its symmetry
group is a reflection group denoted by F4 .
Not every reflection group is the symmetry group of a regular polytope. A
counterexample is constructed as follows.
Example 1.11. Let n ≥ 3. Returning to the crosspolytope, ignore the reflections
which transpose an opposite pair of vertices. The remaining reflections generate a
reflection group called Dn , which is a proper subgroup of Bn . The reflections of
D3 are represented by the dotted lines in Figure 1.3. We note that the Coxeter
arrangements of types A3 and D3 are related by an orthogonal transformation, so
the reflection groups A3 and D3 are isomorphic to each other.
Remark 1.12. It can be shown that, for n ≥ 4, the group Dn is not a symmetry
group of a regular polytope. See Section 2.3 for further details.
LECTURE 1. REFLECTIONS AND ROOTS 73
For any α ∈ Φ, the coefficients ci in the expansion α = i∈I ci αi are called the
simple root coordinates of α. The set Φ+ of positive roots consists of all roots whose
simple root coordinates are all non-negative. The negative roots Φ− are those with
non-positive simple root coordinates.
Lemma 1.14. Φ is the disjoint union of Φ+ and Φ− .
In these lectures, we focus on the study of the important class of finite crystal-
lographic root systems. These are the finite non-empty collections of vectors that,
in addition to the axioms (i)–(ii) above, satisfy the “crystallographic condition”
(iii) For any α, β ∈ Φ, we have σα (β) = β−aαβ α with aαβ ∈ Z. (See Figure 1.5.)
Equivalently, the simple root coordinates of any root are integers.
α
σα (β)
aαβ α
For the rest of these lectures, a “root system” will always be presumed finite
and crystallographic.
74 FOMIN AND READING, ROOTS AND ASSOCIAHEDRA
α2
A1 × A1
−1 0
σα1 =
0 1 α1
1 0
σα2 =
0 −1
α2 α1 + α2
A2
−1 1
σα1 =
0 1 α1
1 0
σα2 =
1 −1
α2 α1 + α2 2α1 + α2
B2
−1 2
σα1 =
0 1
α1
1 0
σα2 =
1 −1
3α1 + 2α2
α2 α1 + α2 2α1 + α2 3α1 + α2
G2
−1 3
σα1 =
0 1 α1
1 0
σα2 =
1 −1
For the root systems A2 , B2 and G2 , the reflections σα1 and σα2 have appeared
earlier in Section 1.1. (The matrices of these reflections in the basis (α1 , α2 ) of sim-
ple roots are shown in Figure 1.6.) In these three cases, the pair (σα1 , σα2 ) coincides
with (s2 , s1 ), (s3 , s1 ), and (s4 , s1 ), respectively, in the notation of Section 1.1.
α2 + α3 α1 + α2 + α3
α1
α3
α2
α1
α3
Figure 1.8. The root system B3 . The endpoints of the 9 positive roots are
shown as black circles on the cube’s front. The negative roots are not shown.
The root system Cn can be realized as the set of vectors in Rn of the form ±2ei
or ±ei ± ej . The vectors α0 = 2e1 and αi = ei+1 − ei form a set of simple roots.
The positive roots are 2ei and ei ± ej . See Figure 1.9.
α2
α1
α3
Figure 1.9. The root system C3 . The endpoints of the 9 positive roots are
shown on the front of the octahedron. The negative roots are not shown.
Example 2.6. For the root systems A1 × A1 and A2 , the 2 × 2 identity matrix
serves as D. For B2 and G2 , take D = 10 √02 and D = 10 √03 , respectively.
Lemma 2.7. A Cartan matrix of finite type (resp., a root system) is indecomposable
(resp., irreducible) if and only if its Dynkin diagram is connected.
An (n ≥ 1) t t t t t t t t
Bn (n ≥ 2) t t t t t t t t
Cn (n ≥ 3) t t t t t t t t
t
HH
Dn (n ≥ 4) Ht
t t t t t t
t
E6 t t t t t
E7 t t t t t t
E8 t t t t t t t
F4 t t t t
G2 t t
Root systems are just one example among a large number of mathematical
objects of “finite type” which are classified by (some class of) Dynkin diagrams. The
appearance of the ubiquitous Dynkin diagrams in a variety of seemingly unrelated
classification problems has fascinated several generations of mathematicians, and
helped establish nontrivial connections between different areas of mathematics. See
Section 2.3 and references therein.
The following theorem demonstrates that the notion of a Coxeter group indeed
captures the geometric essence of reflection groups.
Theorem 2.11. Let W be the group generated by the reflections {σβ : β ∈ Φ}. Let
An (n ≥ 1) t t t t t t t t
Bn (n ≥ 2) t 4 t t t t t t t
tH
HHt t t t t t t
Dn (n ≥ 4)
t
E6 t t t t t
E7 t t t t t t
E8 t t t t t t t
F4 t t 4 t t
H3 t 5 t t
H4 t 5 t t t
I2 (m) (m ≥ 5) t m t
Regular polytopes
By Theorem 1.5, the symmetry group of a regular polytope is a reflection group. In
fact, it is a Coxeter group whose Coxeter diagram is linear : the underlying graph
is a path with no branching points. This narrows down the possibilities, leading to
the conclusion that there are no other regular polytopes besides the ones described
in Section 1.2. In particular, there are no “exceptional” regular polytopes beyond
dimension 4: only simplices, cubes, and crosspolytopes. See [14].
Lie algebras
The original motivation for the Cartan-Killing classification of root systems came
from Lie theory. Complex finite-dimensional simple Lie algebras correspond nat-
urally, and one-to-one, to finite irreducible crystallographic root systems. There
exist innumerable expositions of this classical subject; see, e.g., [25].
82 FOMIN AND READING, ROOTS AND ASSOCIAHEDRA
Et cetera
And the list goes on: simple singularities, finite subgroups of SU (2), symmetric
matrices with nonnegative integer entries and eigenvalues between −2 and 2, etc.
For more, see [28, 33, 59]. In Section 4.2, we will present yet another classification
that is parallel to Cartan-Killing: the classification of the cluster algebras of finite
type.
one at a time, and cross each hyperplane at most once; two paths are equivalent if
they cross the same hyperplanes in the same order.
In order to make the correspondence between paths and reduced words more
explicit, one can restrict the paths to the edges of the W -permutohedron, a convex
polytope that we will now define. Fix a point x in the interior of R1 . The W -
permutohedron is the convex hull of the orbit of x under the action of W . The name
“permutohedron” comes from the fact that the vertices of an An -permutohedron
are obtained by permuting the coordinates of a generic point in Rn+1 .
In both pictures, the bottom vertex can be associated with the identity ele-
ment 1 ∈ W , so that the top vertex is w◦ . A reduced word for w corresponds to a
path along edges from 1 to w which moves up in a monotone fashion. There are 16
such paths from 1 to w◦ in the A3 -permutohedron; cf. Example 2.12.
Theorem 2.15. The number of reduced words for w◦ in the reflection group An is
n+1
2 !
.
1n 3n−1 5n−2 · · · (2n − 1)1
1An Archimedean solid is a non-regular polytope whose all facets are regular polygons, and whose
symmetry group acts transitively on vertices. In dimension 3, there are 13 Archimedean solids.
The permutohedra of types A3 , B3 , and H3 are also known as the truncated octahedron, great
rhombicuboctahedron, and great rhombicosidodecahedron, respectively. See, e.g., [55].
84 FOMIN AND READING, ROOTS AND ASSOCIAHEDRA
+r −r +r −r +r −r +r
r
−
Figure 2.5. Bi-partition of the nodes of the Coxeter diagram of type E8
2 More broadly, one often calls the product of the elements in S (in any order) a Coxeter element,
but for our present purposes the definition above will do.
LECTURE 2. DYNKIN DIAGRAMS AND COXETER GROUPS 85
Example 2.17. Figure 2.6 shows the Coxeter arrangement of type A3 and the
plane L fixed by the Coxeter element c = s2 s1 s3 (dotted). The great circles rep-
resent the intersections of the six reflecting hyperplanes with a unit hemisphere.
The sphere is opaque, so only half of each circle is visible, and appears either as a
half of an ellipse or as a straight line segment. (The “equator” does not represent
a hyperplane in the arrangement.) The restriction of c onto L has order 4, so the
Coxeter number for A3 is h = 4.
Example 2.18. Figure 2.7 is a similar picture for B3 , illustrating that the Coxeter
number for B3 is h = 6. In this picture, the equator does represent a hyperplane
in the arrangement.
s1 s3
s2
Figure 2.6. The Coxeter arrangement A3 and the plane fixed by the Coxeter element
s2 s3
s1
Figure 2.7. The Coxeter arrangement B3 and the plane fixed by the Coxeter element
86 FOMIN AND READING, ROOTS AND ASSOCIAHEDRA
type of Φ |Φ+ | h e 1 , . . . , en |W |
An n(n + 1)/2 n+1 1, 2, . . . , n (n + 1)!
Bn , Cn n2 2n 1, 3, 5, . . . , 2n − 1 2n n!
Dn n(n − 1) 2(n − 1) 1, 3, 5, . . . , 2n − 3, n − 1 2n−1 n!
E6 36 12 1, 4, 5, 7, 8, 11 27 34 5
E7 63 18 1, 5, 7, 9, 11, 13, 17 2 3 5·7
10 4
H4 60 30 1, 11, 19, 29 26 32 52
I2 (m) m m 1, m−1 2m
Figure 2.8. Number of positive roots, Coxeter number, exponents, and the order of W .
LECTURE 3
Associahedra and Mutations
3.1. Associahedron
We start by discussing two classical problems of combinatorial enumeration.
(i) Count the number of bracketings (parenthesizations) of a non-associative
product of n + 2 factors. Note that we need n pairs of brackets in order
to make the product unambiguous.
(ii) Count the number of triangulations of a convex (n+3)-gon by diagonals.
Note that each triangulation involves exactly n diagonals.
Example 3.1. In the special cases n = 1, 2, 3, there are, respectively:
• 2 bracketings (ab)c and a(bc) of a product of 3 factors;
• 5 bracketings ((ab)c)d, (a(bc))d, a((bc)d), (ab)(cd), and a(b(cd)) of a prod-
uct of 4 factors;
• 14 bracketings of a product of 5 factors (check!).
As to triangulations, there are:
• 2 triangulations of a convex quadrilateral (n = 1);
• 5 triangulations of a pentagon (n = 2, Figure 3.3);
• 14 triangulations of a hexagon (n = 3, Figure 3.4).
Theorem 3.2. Both bracketings
2n+2and
triangulations described above are enumerated
1
by the Catalan numbers n+2 n+1 .
There are a great many families of combinatorial objects enumerated by the
Catalan numbers; more than a hundred of those are listed in [50]. This list includes:
ballot sequences; Young diagrams and tableaux satisfying certain restrictions; non-
crossing partitions; trees of various kinds; Dyck paths; permutations avoiding pat-
terns of length 3; and much more. In Lecture 5, we will discuss several additional
members of the “Catalan family,” together with their analogues for arbitrary root
systems. (We will see that the ordinary Catalan numerology should be considered
as “type A.”)
A bijection between bracketings and triangulations is described in Figure 3.1.
For a fixed n, the bracketings naturally form the set of vertices of a graph whose
edges correspond to applications of the associativity axiom. Figure 3.2 shows this
graph for n = 2.
87
88 FOMIN AND READING, ROOTS AND ASSOCIAHEDRA
a f
b e
c d
a b c d e f
(( a ( b c ))( d e )) f
(ab)(cd)
((ab)c)d a(b(cd))
(a(bc))d a((bc)d)
The drawing of the exchange graph in Figure 3.4 fails to convey its crucial prop-
erty: this exchange graph is the 1-skeleton of a convex polytope, the 3-dimensional
associahedron. (Sometimes it is also called the Stasheff polytope, after J. Stash-
eff, who first defined it in [52].) Figure 3.5 shows a polytopal realization of this
associahedron.
LECTURE 3. ASSOCIAHEDRA AND MUTATIONS 89
It is not clear a priori that these complexes are topological spheres. But, as
already mentioned, more is true.
Theorem 3.4. The simplicial complex described in Definition 3.3 can be realized
as the boundary of an n-dimensional convex polytope.
vertices: triangulations
faces: partial triangulations
(6)
facets: diagonals
edges: diagonal flips
We note that we could have defined the associahedron directly, as a cell com-
plex whose cell structure is described by (6). (This would require resolving some
technical issues that we would rather avoid here.) The fact that these cell complexes
are polytopal—i.e., the fact that a combinatorially defined associahedron can be
realized as a convex polytope—is essentially equivalent to Theorem 3.4.
LECTURE 3. ASSOCIAHEDRA AND MUTATIONS 93
3.2. Cyclohedron
The n-dimensional cyclohedron (also known as the Bott-Taubes polytope [8]) is con-
structed similarly to the associahedron using centrally-symmetric triangulations of
a regular (2n + 2)-gon. Each edge of the cyclohedron represents either a diagonal
flip involving two diameters of the polygon, or a pair of two centrally-symmetric
diagonal flips. Figures 3.8 and 3.9 show the 2- and 3-dimensional cyclohedra re-
spectively. As these figures suggest, the cyclohedron is a convex polytope for any n.
Explicit polytopal realizations of cyclohedra were constructed by M. Markl [38] and
R. Simion [47]. Each face of a cyclohedron is a product of smaller cyclohedra and
associahedra.
Further details about the combinatorics of cyclohedra, and about their appear-
ance in the study of configuration spaces can be found in [17].
The geometry of associahedra and cyclohedra is related to the geometry of
permutohedra, as the following theorem (due to Tonks [54]) shows.
Theorem 3.6. The 1-skeleton of the n-dimensional associahedron (resp., cyclohe-
dron) can be obtained from the 1-skeleton of the permutohedron of type An (resp.,
type Bn ) by contraction of edges.
Theorem 3.6 is further discussed in Section 5.4 in connection with Theorem 5.11.
For n = 3, the theorem is illustrated in Figure 3.10. (Cf. Figures 2.3 and 2.4.)
In light of Theorem 3.6, the cyclohedron can be viewed as a “type B counter-
part” of the associahedron (which is a “type A” object).
94 FOMIN AND READING, ROOTS AND ASSOCIAHEDRA
r ⎡ ⎤
0 1
7 5 ⎢ −1 0 ⎥
⎢ ⎥
r r ⎢ 0 1 ⎥
⎢ ⎥ 0 1
1 2 B̃ = ⎢
⎢ −1 0 ⎥
⎥ B=
⎢ ⎥ −1 0
⎢ 0 −1 ⎥
4 3 ⎣ 1 −1 ⎦
1 0
r r
6
Lemma 3.8. Assume that B̃ and B̃ (resp., B and B ) are the edge-adjacency
matrices (resp., their principal parts) for two triangulations T and T obtained
from each other by flipping the diagonal numbered k; the rest of the labels are the
same in T and T . Then B̃ = μk (B̃) (resp., B = μk (B)).
Lemma 3.8 is illustrated in Figures 3.12 and 3.13. Note that the numbering of
diagonals used in defining the matrices B̃ and B can change as we move along the
exchange graph. For instance, the sequence of 5 flips shown in Figure 3.13 results
in switching the labels of the two diagonals.
⎡ ⎤
0 0 1 −1
1 4 ⎢ 0
⎢ 0 1 0 ⎥
⎥
3 ⎣ −1 −1 0 1 ⎦
1 0 −1 0
2
μ3
⎡ ⎤
0 0 −1 0
4 ⎢ 0
1
⎢ 0 −1 1 ⎥
⎥
3 ⎣ 1 1 0 −1 ⎦
2 0 −1 1 0
One can similarly define edge-adjacency matrices for centrally symmetric tri-
angulations (those matrices will have entries 0, ±1, and ±2), and verify that cyclo-
hedral flips translate precisely into matrix mutations.
0 1
1 2
−1 0
0 −1
1 2
1 0
2
0 1
1
−1 0
2
0 −1
1 1 0
0 1
2
1 −1 0
0 −1
2 1
1 0
Figure 3.13. Diagonal flips in a pentagon, and the corresponding matrix mutations
98 FOMIN AND READING, ROOTS AND ASSOCIAHEDRA
q q
HH c HH c
HH HH
HHq HHq
d d
x
x
−→
q
q
@ b @ b
a@ a@
@q @q
Lemma 3.9. The rational function xγ associated to each diagonal γ does not de-
pend on the particular choice of a sequence of flips that connects the initial trian-
gulation with another one containing γ.
Lemma 3.9 can be rephrased as saying that there are no “monodromies” asso-
ciated with sequences of flips that begin and end at the same triangulation.
To illustrate Lemma 3.9, consider the triangulations of a pentagon (i.e., n = 2).
We label the sides of the pentagon by the variables q1 , q2 , q3 , q4 , q5 , as shown in
Figure 3.15.
q
q5 q3
q q
q2 q1
q q
q4
We then label the diagonals incident to the top vertex by the variables y1 and y2 .
Thus, our initial triangulation T◦ appears at the top of Figure 3.16. The rational
functions y3 , y4 , y5 associated with the remaining three diagonals are then computed
from the exchange relations associated with the flips shown in Figure 3.16.
Starting at the top of Figure 3.16 and moving clockwise, we recursively express
y3 , y4 , . . . in terms of y1 , y2 and q1 , . . . , q5 :
q2 y2 + q4 q5
y3 = ,
y1
q3 y3 + q5 q1 q3 q2 y2 + q3 q4 q5 + q5 q1 y1
y4 = = ,
y2 y1 y2
q4 y4 + q1 q2 q3 q4 + q1 y1
y5 = = ··· = (check!),
y3 y2
and, finally,
q5 y5 + q2 q3
y1 = = · · · = y1 ,
y4
q1 y1 + q3 q4
y2 = = · · · = y2 ,
y5
LECTURE 3. ASSOCIAHEDRA AND MUTATIONS 99
y1 y2
y5 y2 = q1 y1 + q3 q4 y1 y3 = q2 y2 + q4 q5
y1 y2
y5 y3
y4 y1 = q5 y5 + q2 q3 y2 y4 = q3 y3 + q5 q1
y4 y4
y5 y3 y5 = q4 y4 + q1 q2 y3
In other words, the right-hand side of (9) is the sum of two monomials whose
exponents are the absolute values of the entries in the kth column of B̃, while the
sign of an entry determines which monomial the corresponding term contributes to.
Example 4.1. Let T be the triangulation of a pentagon in Figure 3.11, with its
edges labeled 1, . . . , 7 as shown. The exchange relations corresponding to flipping
101
102 FOMIN AND READING, ROOTS AND ASSOCIAHEDRA
Seed mutations generate the mutation equivalence relation on seeds: (x̃, B̃) ∼
(x̃ , B̃ ). Let S be an equivalence class for this relation. Thus, S is obtained by
repeated mutations of an arbitrary initial seed in all possible directions. This creates
an exchange graph. See Figure 4.1.
seed
seed
@
@ @
@
@
initial
seed
seed
@
@
Figure 4.1. Seed mutations and the exchange graph
Let X = X (S) be the union of all clusters for all the seeds in S. The elements
of X are called cluster variables. See Figure 4.2.
x1 , x2 , x3 x1 , x2 , x3
@
@ @
@
@
@
x1 , x2 , x3
x1 , x2 , x3
@@
We note that in deciding whether a cluster algebra is of finite type, the bottom
part of the matrix B̃ plays no role whatsoever: everything is determined by its
principal part B.
In the special cases where a cluster algebra has rank n = 2, is of finite type
(that is, one of the types A2 , B2 , and G2 ), and has no frozen variables (that is,
m = 2), Theorem 4.7 brings us back to the recurrences of Section 1.1. Indeed, these
recurrences are precisely given by the exchange relations in those cluster algebras.
The periodicity of the corresponding sequences is simply a reformulation of the
“finite type” property for cluster algebras.
Theorem 4.8 (Combinatorial criterion for finite type). A cluster algebra A is of
finite type if and only if the exchange matrix B = (bij ) for any seed of A satisfies
the inequalities |bij bji | ≤ 3 for all i, j ∈ {1, . . . , n}.
To rephrase, a mutation equivalence class of skew-symmetrizable n×n matrices
defines a class of cluster algebras of finite type if and only if, for each matrix B = (bij )
in this equivalence class, the inequality |bij bji | ≤ 3 holds for all i and j.
Combining Theorems 4.8 and 2.4 yields the following completely elementary
statement about integer matrices, no direct proof of which is known3.
Corollary 4.9. Let B be a mutation equivalence class of skew-symmetrizable in-
teger matrices, with the skew-symmetrizing matrix D. (Cf. Lemma 4.2.) The fol-
lowing are equivalent:
• any matrix B = (bij ) ∈ B satisfies the inequalities |bij bji | ≤ 3, for all i and j;
• there exists a matrix B = (bij ) ∈ B with the following property. Define
A = (aij ) by
−|bij | if i = j;
aij =
2 if i = j.
Then DAD−1 is positive definite.
Let Φ be an irreducible finite root system with Cartan matrix A, and let A
be a cluster algebra of the corresponding cluster type. Theorem 4.7 tells us that
the set X of cluster variables is finite. A more detailed description of this set is
provided by Theorem 4.10 below.
Let α1 , . . . , αn be the simple roots of Φ, and let {x1 , . . . , xn } be the cluster at
a seed in A with the exchange matrix B(A).
Let Φ≥−1 denote the set of roots in Φ which are either negative simple or
positive. Theorem 4.10 shows that the cluster variables in A are naturally labeled
by the roots in Φ≥−1 .
Theorem 4.10. For any root α = c1 α1 + · · · + cn αn ∈ Φ≥−1 , there is a unique
cluster variable x[α] given by
Pα (x1 , . . . , xm )
(10) x[α] = ,
xc11 · · · xcnn
where Pα is a polynomial in x1 , . . . , xm with nonzero constant term. The map
α → x[α] is a bijection between Φ≥−1 and X .
3Note added in revision. According to A. Zelevinsky, such a proof has been recently found in his
joint work with M. Barot and C. Geiss.
106 FOMIN AND READING, ROOTS AND ASSOCIAHEDRA
x3 x2
x1
This dual graph of the cluster complex is precisely the exchange graph of the
cluster algebra.
Theorem 4.11 below shows that the cluster complex is always spherical, and
moreover polytopal.
Recall that QR denotes the R-span of Φ. The Z-span of Φ is the root lattice,
denoted by Q.
Theorem 4.11. The n roots that label the cluster variables in a given cluster form
a Z-basis of the root lattice Q. The cones spanned by such n-tuples of roots form
a complete simplicial fan in the ambient real vector space QR (the “cluster fan”).
This fan is the normal fan4 of a simple n-dimensional convex polytope in the dual
space Q∗R .
This polytope is called the generalized associahedron of the corresponding type.
Thus, the cluster complex of a cluster algebra of finite type is canonically iso-
morphic to the dual simplicial complex of a generalized associahedron of the corre-
sponding type. Conversely, the dual graph of the cluster complex is the 1-skeleton
of the generalized associahedron.
4Let P ⊂ V ∼ Rn be an n-dimensional simple convex polytope. The support function F : V ∗ → R
=
of P is given by
F (γ) = maxz, γ.
z∈P
The normal fan N (P ) is a complete simplicial fan in the dual space V ∗ whose full-dimensional
cones are the domains of linearity for F . More precisely, each vertex z of P gives rise to the cone
{γ ∈ V ∗ : F (γ) = z, γ} in N (P ).
LECTURE 4. CLUSTER ALGEBRAS 107
α2 α1 +α2
−α1 α1
−α2
Theorem 4.12. The cluster complex is a clique complex for its 1-skeleton. In other
words, a subset S ⊂ Φ≥−1 is a simplex in the cluster complex if and only if every
2-element subset of S is a 1-simplex in this complex.
In type An , Theorem 4.12 reflects the basic property of the dual complex of an
associahedron: a collection of diagonals forms a simplex if and only if any two of
them do not cross.
In order to describe the cluster complex, we therefore need only to clarify which
pairs of roots label the edges of the cluster complex. Thus, we need to define the
root-theoretic analogue of the notion of “non-intersecting diagonals” that lies at
the heart of the combinatorial construction of an associahedron.
We will assume from now on that the root system Φ underlying a cluster alge-
bra A is irreducible. (The general case can be obtained by taking direct products.)
We retain the notation of Lecture 2. Thus, n is the rank of Φ (and A); I is the
n-element indexing set, which is partitioned into disconnected pieces I+ and I− ;
108 FOMIN AND READING, ROOTS AND ASSOCIAHEDRA
⎧
⎪
⎨ α if α = −αi , for i ∈ I−ε ;
τε (α) =
⎪
⎩ si (α) otherwise.
i∈Iε
For example, in type A2 , we get:
τ+ τ− τ+ τ−
−α1 ←→ α1 ←→ α1 + α2 ←→ α2 ←→ −α2
τ− τ+
The product τ− τ+ can be viewed as a deformation of the Coxeter element. Hence,
what is the counterpart of the Coxeter number?
Theorem 4.14. The order of τ− τ+ is (h + 2)/2 if w◦ = −1, and is h + 2 other-
wise. Every τ− , τ+ -orbit in Φ≥−1 has a nonempty intersection with −Π. These
intersections are precisely the −w◦ -orbits in (−Π).
Theorem 4.15. There is a unique binary relation (called “compatibility”) on Φ≥−1
that has the following two properties:
• τ− , τ+ -invariance: α and β are compatible if and only if τε α and τε β
are, for ε ∈ {+, −};
• a negative simple root −αi is compatible with a root β if and only if the
simple root expansion of β does not involve αi .
This compatibility relation is symmetric. The clique complex for the compatibility
relation is canonically isomorphic to the cluster complex.
In other words (cf. Theorem 4.12), a subset of roots in Φ≥−1 forms a simplex
in the cluster complex if and only if every pair of roots in this subset is compatible.
Example 4.16. In type An , the compatibility relation can be described in concrete
combinatorial terms using a particular identification of the roots in Φ≥−1 with the
diagonals of a regular (n + 3)-gon. Under this identification, the roots in −Π
correspond to the diagonals on the “snake” shown in Figure 4.5. Each positive root
αi + αi+1 + · · · + αj corresponds to the unique diagonal that crosses precisely the
diagonals −αi , −αi+1 , . . . , −αj from the snake (see Figure 4.6). It is easy to check
that the transformations τ+ and τ− act on the set of diagonals as if they were the
reflections generating the dihedral group of symmetries of the (n + 3)-gon. It then
follows that two roots are compatible if and only if the corresponding diagonals do
not cross each other (at an interior point).
r
H r
HH @
@
−αH H
5
HH @
@
HH
H@
r −α4 H
H
@r
P PP
PP
PP −α
PP3
PP
PP
PP
r −α2 PPr
H
@H
@ HH
@ HH
@ HH
−α1
@ HH
@r H
Hr
α1 + α2
r r
−α1 −α2
α1 α2
r r
half-sum of all positive coroots. (Coroots are the roots of the “dual” root system;
see [9, 34].)
Example 4.18. In type A3 , Theorem 4.17 is illustrated in Figure 4.7, which shows
a 3-dimensional associahedron given by the inequalities
Example 4.19. In type C3 , Theorem 4.17 is illustrated in Figure 4.8 that shows
a 3-dimensional cyclohedron given by the inequalities
max(−z1 , z1 , z1 + z2 , z2 + z3 ) ≤ 5/2 ,
max(−z2 , z2 , z1 + z2 + z3 , z1 + 2z2 + z3 ) ≤ 4 ,
max(−z3 , z3 , 2z2 + z3 , 2z1 + 2z2 + z3 ) ≤ 9/2 .
s s
α2 @
s s @
@
B
B @
B @
B @
B α2 + α3 @
B @
Bs @s
α1 + α2
@ @
@
@
@s
α1 +α2 +α3
s s
@ @
α3
s @ s
@
@s
α1
s s
r rH
r r HH
A HH
A Hr
@
Ar @
HH @
Hr
@
@
@
@
r @
H
HHr
@r
B
B
r B r
r r Br
HH @
H
HH @
Hr @r
@
@
@r
r r
3 1
3 @ @ @ 1
@ @ @
2 @@ @ @@ 2
@
2 @ @ @ 2
@ @ @
1 @ @@ @ 3
@ @
1 3
Figure 4.9. Double wiring diagram
Figure 4.9 by sliding the two leftmost crossings past each other, and also doing the
same for the two rightmost crossings.
1 3
3 @ @ @ 1
@ @ @
2 @@ @ @@ 2
@
2 @ @ @ 2
@ @ @
3 @ @@ @ 1
@ @
1 3
Figure 4.10. An isotopic double wiring diagram
B@ C B@ Z @ C
@ - @ @
A@ Y @ D A @ D
@ @ @
B@ C B@ Z @ C
@ - @ @
A@ Y @ D A @ D
@ @ @
B B
-
A@ Y @ C A@ Z @ C
@ @ @ @
D D
To illustrate Lemma 4.20, the double wiring diagram in Figure 4.9 allows 4 dif-
ferent local moves, all of which are of the kind shown at the bottom of Figure 4.11.
Two of these moves can be performed by first passing to the isotopic Figure 4.10.
To make each of the other two moves, slide the two innermost crossings in Figure 4.9
past each other; this will create two patterns of the form shown at the bottom of
Figure 4.11.
A chamber of a double wiring diagram is a connected component of the com-
plement to the union of the lines, with the exception of the “crumbs” made of
narrow horizontal isthmuses and small triangular regions; the large component at
LECTURE 4. CLUSTER ALGEBRAS 113
the very bottom is not included either. With these conventions, there are exactly
n2 chambers altogether (e.g., 9 chambers in Figure 4.9). We then assign to every
chamber a pair of subsets of the set [1, n] = {1, . . . , n}: each subset indicates which
lines of the corresponding color pass below that chamber; see Figure 4.12.
123,123
1 3
3 @ @ @ 1
23,12 @ 13,12 @ 13,23 @ 12,23
2 @@ @ @@ 2
@
2 @ @ @ 2
3,1 @ 3,2 @ 1,2 @ 1,3
3 @ @@ @ 1
@ @
1 3
Figure 4.12. Chamber minors
Lemma 4.23. Whenever two double wiring diagrams differ by a single local move
of one of the three types shown in Figure 4.11, the chamber minors appearing there
satisfy the identity AC + BD = Y Z.
Lemmas 4.21 and 4.23 suggest the existence of a cluster algebra structure asso-
ciated with n×n matrices. We next present one of several versions of this structure,
leaving out most of the technical details. The ambient field for our cluster alge-
bra is the field F of rational functions on GLn (C) introduced above. Each double
114 FOMIN AND READING, ROOTS AND ASSOCIAHEDRA
123,123
1 3
3 @ @ 1
23,12 @ 12,12 @ 12,23
2 @@ @ 2
@
2 @ @ @ @ 2
3,1 @ 2,1 @ 1,1 @ 1,2 @ 1,3
3 @@ @
@ @ @ 1
@ @
1 3
Figure 4.13. A double wiring diagram allowing 3 local moves
wiring diagram provides us with a seed whose cluster variables are the (n − 1)2
chamber minors associated with the bounded chambers; the frozen variables are
the 2n − 1 chamber minors associated with the unbounded chambers at the edges
of the diagram. It remains to define the matrices B̃.
Take any double wiring diagram in which every bounded chamber can be
“flipped” (such a diagram can be constructed for any n). Comparing the cor-
responding exchange relations AC + BD = Y Z with (9), determine the matrix
entries of B̃. It can be shown that exchanges associated with the local moves
on double wiring diagrams are compatible with the cluster algebra axioms. Fur-
thermore, applying these axioms uncovers hitherto hidden clusters which do not
correspond to any wiring diagrams. Each variable in these clusters is a regular
function on GLn (C) (a polynomial in the matrix entries). The resulting cluster
algebra coincides with the coordinate ring of the open double Bruhat cell Gw◦ ,w◦
in GLn (C). We refer to [4] for further details.
Example 4.24. The open double Bruhat cell Gw◦ ,w◦ ⊂ GL3 (C) consists of all
complex 3 × 3 matrices x = (xij ) whose minors
x x13 x21 x22
(11) x13 , 12 , x31 , , det(x)
x22 x23 x31 x32
are nonzero. (These 5 minors correspond to the unbounded chambers of any double
wiring diagram for GL3 (C).) The coordinate ring C[Gw◦ ,w◦ ] turns out to be a
cluster algebra of type D4 over the ground ring generated by the minors in (11)
and their inverses. Thus, the ring of rational functions on GL3 exhibits some quite
unexpected symmetries of type D4 .
This cluster algebra has 16 cluster variables, corresponding to the 16 roots
in Φ≥−1 . These variables are:
• 14 (among the 19 total) minors of x, namely, all except those listed in (11);
• two “hidden” variables: x12 x21 x33 − x12 x23 x31 − x13 x21 x32 + x13 x22 x31
and x11 x23 x32 − x12 x23 x31 − x13 x21 x32 + x13 x22 x31 .
These 16 variables form 50 clusters of size 4, one for each of the 50 vertices of the
type D4 associahedron.
For any n ≥ 4, the construction described above produces a cluster algebra of
infinite type.
LECTURE 5
Enumerative Problems
An Bn , Cn Dn E6 E7 E8 F4 G2
1
2n+2
2n
3n−2 2n−2
n+2 n+1 n n n−1 833 4160 25080 105 8
3α1 + 2α2
3α1 + α2
2α1 + α2 2α1 + α2
α1 + α2 α1 + α2 α1 + α2
α1 α2 α1 α2 α1 α2
Non-crossing partitions
The classical non-crossing partitions introduced by Kreweras are (unordered) par-
titions of the set [n + 1] = {1, . . . , n + 1} into non-empty subsets called blocks which
satisfy the following “non-crossing” condition:
• there does not exist an ordered quadruple (a < b < c < d) such that the
two-element sets {a, c} and {b, d} are contained in different blocks.
(1234)
(1)(2)(3)(4)
4 2
1
-3 2
-2 3
-1
are obtained by looking at the corresponding row of Pascal’s triangle on the left,
computing products of consecutive pairs of entries, and dividing them by n + 1.
1
1 1 1
1 2 1 1 1
1 3 3 1 1 3 1
1 4 6 4 1 1 6 6 1
1 5 10 10 5 1 1 10 20 10 1
Remarkably, the row sums in the triangle of Narayana numbers are the Catalan
numbers:
n
1 n+1 n+1 1 2n + 2
= .
n+1 k k+1 n+2 n+1
k=0
This suggests introducing a q-analogue of the Catalan numbers given by
n
1 n+1 n+1 k
(13) q .
n+1 k k+1
k=0
We will now explain the connection between this q-analogue and the classical
(type A) associahedron. This connection will lead us to an extension of the de-
finition to other root systems.
122 FOMIN AND READING, ROOTS AND ASSOCIAHEDRA
14 ? 14 1
21 ? ? 21 13 6
9 ? ? ? 9 8 7 6
1 1 1 1 1 1 1 1 1 1
to fn−1 , the num-
It is easy to see that the entries of an h-vector always add up
ber of top-dimensional faces in the simplicial complex. Thus, k Nk (Φ) = N (Φ).
Consequently, the generating function for the Narayana numbers of type Φ
n
N (Φ, q) = k=0 Nk (Φ)q k
provides a q-analogue of N (Φ) which generalizes (13). These generating functions
for the finite crystallographic root systems are tabulated in Figure 5.12.
The Narayana numbers provide refined counts for the various interpretations
of N (Φ) given in Section 5.1. These enumerative results are listed in Theorem 5.9
below; we elaborate on the items in the theorem in subsequent comments.
Theorem 5.9 is a combination of results in [2, 19, 39, 44, 48]; see [2] for a
historical overview, and for further generalizations.
LECTURE 5. ENUMERATIVE PROBLEMS 123
n
1 n+1 n+1 k
N (An , q) = q
n+1 k k+1
k=0
n 2
n
N (Bn , q) = qk
k
k=0
n2
n−1
n
n−1 n−1
N (Dn , q) = 1+q +n
− qk
k n−1 k−1 k
k=1
N (E6 , q) = 1 + 36q + 204q 2 + 351q 3 + 204q 4 + 36q 5 + q 6
N (E7 , q) = 1 + 63q + 546q 2 + 1470q 3 + 1470q 4 + 546q 5 + 63q 6 + q 7
N (E8 , q) = 1 + 120q + 1540q 2 + 6120q 3 + 9518q 4
+6120q 5 + 1540q 6 + 120q 7 + q 8
N (F4 , q) = 1 + 24q + 55q + 24q + q 4
2 3
N (G2 , q) = 1 + 6q + q 2
Theorem 5.9. The following numbers are equal to each other, and to Nk (Φ):
(i) the kth component of the h-vector for the dual complex of a generalized
associahedron of type Φ;
(ii) the number of elements of rank k in the non-crossing partition lattice
for W ;
(iii) the number of antichains of size k in the root poset for Φ;
(iv) the number of W -orbits in Q/(h + 1)Q consisting of elements whose sta-
bilizer has rank k;
(v) the components of the h-vector for the dual cell complex of the positive
part of the Shi arrangement.
Remark 5.10 (Comments on Theorem 5.9).
(i) This was our definition of Nk (Φ).
(ii) The lattice of non-crossing partitions of type Φ is graded, and Nk (Φ) is the
number of elements of rank k.
(iii) The h-vector of any simplicial polytope satisfies the Dehn-Sommerville
equations hi = hd−i . Thus interpretation (i) implies that Nk (Φ) = Nn−k (Φ). This
symmetry of the Narayana numbers is also apparent in the interpretation (ii) be-
cause the non-crossing partition lattices are self-dual. However, this symmetry is
not at all obvious in the interpretations (iii)–(v). In particular, no direct combina-
torial explanation is known for why the number of antichains of size k in the root
poset is the same as the number of antichains of size n − k.
(iv) The stabilizer of an element in Q/(h + 1)Q is a reflection subgroup of W .
The stabilizers of elements in the same W -orbit are conjugate, and therefore have
the same rank. Nk (Φ) is the number of orbits in which the stabilizers have rank k.
For example, in type A2 there is 1 orbit whose stabilizer has rank 2 (the unfilled
circle in Figure 5.5), 3 orbits whose stabilizers have rank 1 (each symbolized by a
triangle) and 1 orbit whose stabilizers have rank 0 (the filled circles), in agreement
with N (A2 , q) = 1 + 3q + q 2 .
124 FOMIN AND READING, ROOTS AND ASSOCIAHEDRA
(v) The positive regions of the Shi arrangement can be used to define a “dual”
cell complex. The vertices of this complex correspond to the positive regions of the
Shi arrangement. The faces of the complex correspond to those faces of the closures
of these regions that are not contained in the boundary of the positive cone. Accord-
ingly, the maximal faces correspond to the vertices of the arrangement which lie in
the interior of the positive cone. See Figure 5.13. Amazingly, this cell complex has
the same f -vector (hence the same h-vector) as the corresponding associahedron.
In the example of Figure 5.13, we get 5 vertices, 5 faces, and 1 two-dimensional
face, matching the numbers for the pentagon (the type A2 associahedron).
Figure 5.13. The dual complex for the positive part of the Shi arrangement of type A2 .
H3 H4 I2 (m)
32 280 m + 2
The statement of Theorem 5.11 does not specify which regions of the Coxeter
arrangement should be combined together to produce the maximal cones of the
transformed cluster fan. We next present a lattice-theoretic construction that,
conjecturally, answers this question.
The weak order on W is the partial order in which u ≤ v if and only if some
reduced word for u occurs as an initial segment of a reduced word for v. In partic-
ular, v covers u in the weak order if and only if u−1 v is a simple reflection, and the
length of v is greater than the length of u (necessarily by 1). Lemma 2.13 (see also
the paragraph that follows it) implies that the Hasse diagram of the weak order
can be identified with the 1-skeleton of a W -permutohedron.
Theorem 5.12 ([6]). The weak order on a finite Coxeter group is a lattice.
Example 5.13. The weak order of type An can be described in the language of
permutations of [n+1], written in one-line notation. Permutation v = (v1 , . . . , vn+1 )
covers u = (u1, . . . , un+1 ) if v is obtained from u by exchanging two entries ui and
ui+1 with ui < ui+1 . Figure 5.16 shows the weak order on A3 . (Cf. Figure 2.3.)
4321
1234
4321
1234
Conjecture 5.15. Two regions Ru and Rv of the Coxeter arrangement are con-
tained in the same maximal cone of the transformed cluster fan (see Theorem 5.11)
if and only if u ≡ v under the bipartite Cambrian congruence.
Conjecture 5.15 has been proved in types An and Bn . The proof makes explicit
the combinatorics of the Cambrian congruence and connects it to constructions
given by Billera and Sturmfels [5] (type A) and Reiner [42] (type B). The conjecture
implies in particular that the Hasse diagram of the quotient of the weak order by the
Cambrian congruence (called the Cambrian lattice) is isomorphic to the 1-skeleton
of the generalized associahedron.
More concretely, the Cambrian lattice is obtained as the induced subposet of
the weak order formed by taking the (unique) smallest element in each (Cambrian)
congruence class; see Figure 5.18. We omit the description of the bijection used to
128 FOMIN AND READING, ROOTS AND ASSOCIAHEDRA
translate the top picture in Figure 5.18 (the Cambrian lattice labeled by permuta-
tions) into the bottom one (the associahedron labeled by triangulations).
4321
1234
40. M. Picantin, Explicit presentations for the dual braid monoids, C. R. Math.
Acad. Sci. Paris 334 (2002), 843–848.
41. N. Reading, Cambrian Lattices, Adv. Math., 205 (2006), no. 2, 313-353.
42. V. Reiner, Equivariant fiber polytopes. Doc. Math. 7 (2002), 113–132.
43. V. Reiner, Non-crossing partitions for classical reflection groups, Discrete
Math. 177 (1997), 195–222.
44. V. Reiner and V. Welker, On the Charney-Davis and Neggers-Stanley Con-
jectures, J. Combin. Theory Ser. A 109 (2005), 247–280.
45. I. Reiten, Dynkin diagrams and the representation theory of algebras, Notices
Amer. Math. Soc. 44 (1997), no. 5, 546–556.
46. J.-Y. Shi, The number of ⊕-sign types, Quart. J. Math. Oxford 48 (1997),
93–105.
47. R. Simion, A type-B associahedron, Adv. in Appl. Math. 30 (2003), 2–25.
48. E. Sommers, B-stable ideals in the nilradical of a Borel subalgebra, Canad.
Math. Bull., to appear.
49. R. P. Stanley, On the number of reduced decompositions of elements of Cox-
eter groups, European J. Combin. 5 (1984), 359–372.
50. R. P. Stanley, Enumerative Combinatorics, vol.2, Cambridge University
Press, 1999, Exercise 6.19. See also the “Catalan addendum” posted at
http://www-math.mit.edu/~rstan/ec/.
51. R. P. Stanley, ibid., Exercise 6.34.
52. J. D. Stasheff, Homotopy associativity of H-spaces. I, II, Trans. Amer. Math.
Soc. 108 (1963), 275–292, 293–312.
53. J. Stasheff, What is . . . an operad? Notices Amer. Math. Soc. 51 (2004), no.
6, 630–631.
54. A. Tonks, Relating the associahedron and the permutohedron, in: Operads:
Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), 33–
36, Contemp. Math. 202, Amer. Math. Soc., Providence, RI, 1997.
55. E. W. Weisstein, Archimedean Solid, in: MathWorld–A Wolfram Web Re-
source, http://mathworld.wolfram.com/ArchimedeanSolid.html.
56. A. Zelevinsky, From Littlewood-Richardson coefficients to cluster algebras
in three lectures, Symmetric Functions 2001: Surveys of Developments and
Perspectives, S. Fomin, Ed., NATO Science Series II: Mathematics, Physics
and Chemistry, 74. Kluwer Academic Publishers, Dordrecht, 2002.
57. A. Zelevinsky, Cluster algebras: notes for 2004 IMCC (Chonju, Korea, August
2004), math.RT/0407414.
58. G. Ziegler, Lectures on Polytopes, Springer-Verlag, 1995.
59. J.-B. Zuber, CFT, BCFT, ADE and all that, in: Quantum symmetries in
theoretical physics and mathematics (Bariloche, 2000), 233–266, Contemp.
Math. 294, Amer. Math. Soc., Providence, RI, 2002.
Topics in Combinatorial Differential
Topology and Geometry
Robin Forman
IAS/Park City Mathematics Series
Volume 14, 2004
Robin Forman
Many questions from a variety of areas of mathematics lead one to the problem
of analyzing the topology or the combinatorial geometry of a simplicial complex.
We will see a number of examples in these notes. Some very general theories have
been developed for the investigation of similar questions for smooth manifolds. Our
goal in these lectures is to show that there is much to be gained in the world of
combinatorics from borrowing questions, tools, motivation, and even inspiration
from the theory of smooth manifolds.
These lectures center on two main topics which illustrate the dramatic impact
that ideas from the study of smooth manifolds have had on the study of combina-
torial spaces. The first topic has its origins in differential topology, and the second
in differential geometry.
One of the most powerful and useful tools in the study of the topology of smooth
manifolds is Morse theory. In the first three lectures we present a combinatorial
Morse theory that posesses many of the desirable properties of the smooth theory,
and which can be usefully applied to the study of very general combinatorial spaces.
In the first two lectures we present the basic theory along with numerous examples.
In the third lecture, we show that discrete Morse theory is a very natural tool for
the study of some questions in complexity theory.
Much of the study of global differential geometry is concerned with the rela-
tionship between the geometry of a Riemannian manifold and its topology. One
long conjectured, still unproved, relationship is the Hopf conjecture, which states
that if a manifold has nonpositive sectional curvature, then the sign of its Euler
characteristic depends only on its dimension. (See Lecture 4 for a more precise
statement.) In [15] Charney and Davis formulated a combinatorial analogue of
1 The Department of Mathematics, Rice University, Houston, TX, USA 77251.
E-mail address: forman@rice.edu.
The author was supported in part by the National Science Foundation. The author would also
like to thank Carsten Lange, who served as the TA for these lectures, created most of the figures
in these notes, and whose enthusiasm and attention to detail dramatically increased the com-
prehensibility of the text. The author expresses his gratitude to the organizers of the IAS/PC
Summer Institute for their tireless dedication and enthusiasm for all things organizational and
mathematical. Their support greatly improved the lectures and these notes.
c
2007 American Mathematical Society
135
136 R. FORMAN, COMB. DIFFERENTIAL TOPOLOGY AND GEOMETRY
this conjecture, and then observed that their conjecture is related to some of the
central questions in geometric combinatorics. There has been some fascinating re-
cent work on this subject, which has resulted in some very tantalizing, more general
conjectures. In Lectures 4 and 5 we present an introduction to the conjectures of
Charney and Davis, discuss some of the known partial results, and survey the most
recent developments.
LECTURE 1
Discrete Morse Theory
1. Introduction
There is a very close relationship between the topology of a smooth manifold M
and the critical points of a smooth function f on M . For example, if M is compact,
then f must achieve a maximum and a minimum. Morse theory is a far-reaching
extension of this fact. Milnor’s beautiful book [71] is the standard reference on
this subject. In these notes we present an adaptation of Morse theory that may be
applied to any simplicial complex (or more general cell complex). There have been
other adaptations of Morse Theory that can be applied to combinatorial spaces. For
example, a Morse Theory of piecewise linear functions appears in [59] and the very
powerful “Stratified Morse Theory” was developed by Goresky and MacPherson
[46],[47]. These theories, especially the latter, have each been successfully applied
to prove some very striking results.
We take a slightly different approach than that taken in these references. Rather
than choosing a suitable class of continuous functions on our spaces to play the role
of Morse functions, we will be working entirely with discrete structures. Hence, we
have chosen the name discrete Morse theory for the ideas we will describe. More-
over, in these notes, we will describe the theory entirely in terms of the (discrete)
gradient vector field, rather than an underlying function. We show that even with-
out introducing any continuity, one can recreate, in the category of combinatorial
spaces, a complete theory that captures many of the intricacies of the smooth the-
ory, and can be used as an effective tool for a wide variety of combinatorial and
topological problems.
The goal of these lectures is to present an overview of the subject of discrete
Morse theory that is sufficient both to understand the major applications of the
theory to combinatorics, and to apply the the theory to new problems. We will
not be presenting theorems in their most recent or most general form, and simple
examples will often take the place of proofs. Those interested in a more complete
presentation of the theory can consult the reference [32]. Earlier surveys of this
work have appeared in [31] and [35], and earlier, and similar, versions of some of
the sections in these notes appeared in [39] and [40].
137
138 R. FORMAN, COMB. DIFFERENTIAL TOPOLOGY AND GEOMETRY
(1) ∅ ⊂ X0 ⊂ X1 ⊂ · · · ⊂ Xn = X
such that for each i = 0, 1, 2, . . . , n, Xi is the result of attaching a cell to X(i−1) .
Note that this definition requires that X0 be a 0-cell. If X is a cell complex, we
refer to any sequence of spaces as in (1) as a cell decomposition of X. Suppose that
(i) (ii)
Figure 1. On the left a 1-cell is attached to a circle. This is not true for the
picture on the right.
LECTURE 1. DISCRETE MORSE THEORY 139
X0 X1 X2 X3
in the cell decomposition (1), of the n + 1 cells that are attached, exactly cd are
d-cells. Then we say that the cell complex X has a cell decomposition consisting
of cd d-cells for every d.
We note that a (closed) d-simplex is a d-cell. Thus a finite simplicial complex
is a cell complex, and has a cell decomposition in which the cells are precisely the
closed simplices.
In Figure 2 we demonstrate a cell decomposition of a 2-dimensional torus which,
beginning with the 0-cell, requires attaching two 1-cells and then one 2-cell. Here we
can see one of the most compelling reasons for expanding our view from simplicial
complexes to more general cell complexes. Every simplicial decomposition of the
2-torus has at least 7 vertices, 21 edges and 14 triangles.
It may seem that quite a bit has been lost in the transition from simplicial
complexes to general cell complexes. After all, a simplicial complex is completely
described by a finite amount of combinatorial data. On the other hand, the con-
struction of a cell decomposition requires the choice of a number of continuous
maps. However, if one is only concerned with the homotopy type of the resulting
cell complex, then things begin to look a bit more manageable. Namely, the homo-
topy type of X ∪f σ depends only on the homotopy type of X and the homotopy
class of f .
Theorem 1. Let h : X → X denote a homotopy equivalence, σ a cell, and f1 :
σ̇ → X, f2 : σ̇ → X two continuous maps. If h ◦ f1 is homotopic to f2 , then
X ∪f1 σ and X ∪f2 σ are homotopy equivalent.
(See Theorem 2.3 on page 120 of [68].) An important special case is when h is the
identity map. We state this case separately for future reference.
Corollary 2. Let X be a topological space, σ a cell, and f1 , f2 : σ̇ → X two
continuous maps. If f1 and f2 are homotopic, then X ∪f1 σ and X ∪f2 σ are
homotopy equivalent.
Therefore, the homotopy type of a cell complex is determined by the homotopy
classes of the attaching maps. Since homotopy clases are discrete objects, we have
now recaptured a bit of the combinatorial atmosphere that we seemingly lost when
generalizing from simplicial complexes to cell complexes.
Let us now present some examples.
1) Suppose X is a topological space which has a cell decomposition consisting
of exactly one 0-cell and one d-cell. Then X has a cell decomposition ∅ ⊂ X0 ⊂
X1 = X. The space X0 must be the 0-cell, and X = X1 is the result of attaching
the d-cell to X0 . Since X0 consists of a single point, the only possible attaching
map is the constant map. Thus X is constructed from taking a closed d-ball and
140 R. FORMAN, COMB. DIFFERENTIAL TOPOLOGY AND GEOMETRY
identifying all of the points on its boundary. One can easily see that this implies
that the resulting space is a d-sphere.
2) Suppose X is a topological space which has a cell decomposition consisting of
exactly one 0-cell and n d-cells. Then X has a cell decomposition as in (1) such that
X0 is the 0-cell, and for each i = 1, 2, . . . , n the space Xi is the result of attaching
a d-cell to X(i−1) . From the previous example, we know that X1 is a d-sphere.
The space X2 is constructed by attaching a d-cell to X1 . The attaching map is a
continuous map from a (d − 1)-sphere to X1 . Every map of the (d − 1)-sphere into
X1 is homotopic to a constant map (since π(d−1) (X1 ) ∼ = π(d−1) (S d ) ∼
= 0). If the
attaching map is actually a constant map, then it is easy to see that the space X2
is the wedge of two d-spheres, denoted by S d ∨ S d . (The wedge of a collection of
topological spaces is the space resulting from choosing a point in each space, taking
the disjoint union of the spaces, and identifying all of the chosen points.) Since the
attaching map must be homotopic to a constant map, Corollary 2 implies that X2
is homotopy equivalent to a wedge of two d-spheres.
When constructing X3 by attaching a d-cell to X2 , the relevant information is a
map from S d−1 to X2 , and the homotopy type of the resulting space is determined
by the homotopy class of this map. All such maps are homotopic to a constant
map (since πd−1 (X2 ) ∼ = πd−1 (S d ∨ S d ) ∼
= 0). Since X2 is homotopy equivalent to
a wedge of two d-spheres, and the attaching map is homotopic to a constant map,
it follows from Theorem 1 that X3 is homotopy equivalent to the space that would
result from attaching a d-cell to S d ∨ S d via a constant map, i.e. X3 is homotopy
equivalent to a wedge of three d-spheres.
Continuing in this fashion, we can see that X must be homotopy equivalent to
a wedge of n d-spheres.
The reader should not get the impression that the homotopy type of a cell com-
plex is determined by the number of cells of each dimension. This is true only for
very few spaces (and the reader might enjoy coming up with some other examples).
The fact that wedges of spheres can, in fact, be identified by this numerical data
partly explains why the main theorem of many papers in combinatorial topology
is that a certain simplicial complex is homotopy equivalent to a wedge of spheres.
Namely such complexes are the easiest to recognize. However, that does not ex-
plain why so many simplicial complexes that arise in combinatorics are homotopy
equivalent to a wedge of spheres. I have often wondered if perhaps there is some
deeper explanation for this.
3) Suppose that X is a cell complex which has a cell decomposition consisting
of exactly one 0-cell, one 1-cell and one 2-cell. Let us consider a cell decomposition
for X with these cells: ∅ ⊂ X0 ⊂ X1 ⊂ X2 = X. We know that X0 is the 0-cell.
Suppose that X1 is the result of attaching the 1-cell to X0 . Then X1 must be a
circle, and X2 arises from attaching a 2-cell to X1 . The attaching map is a map
from the boundary of the 2-cell, i.e. a circle, to X1 which is also a circle. Up to
homotopy, such a map is determined by its winding number, which can be taken
to be a nonnegative integer. If the winding number is 0, then without altering the
homotopy type of X we may assume that the attaching map is a constant map,
which yields that X ∼ S 1 ∨ S 2 (where ∼ denotes homotopy equivalence). If the
winding number is 1 then without altering the homotopy type of X we may assume
that the attaching map is a homeomorphism, in which case X is a 2-dimensional
disc. If the winding number is 2, then without altering the homotopy type of X
LECTURE 1. DISCRETE MORSE THEORY 141
we may assume that the attaching map is a standard degree 2 mapping (i.e. that
wraps one circle around the other twice, with no backtracking). The reader should
convince him/herself that the result in this case is that X is the 2-dimensional
projective space P2 . In fact, each winding number results in a homotopically distinct
space. These spaces can be distinguished by their homology, since H1 (X, Z) for the
space X resulting from an attaching map with winding number n is isomorphic to
Z/nZ.
It seems that we are not quite done with this example, because we assumed
that the 1-cell was attached before the 2-cell, and we must consider the alternative
order, in which X1 is the result of attaching a 2-cell to X0 . In this case, X1 is a
2-sphere, and X = X2 is the result of attaching a 1-cell to X1 . The attaching map
is a map of S 0 into S 2 . Since S 2 is connected (i.e. π0 (S 2 ) = 0) all such maps are
homotopic to a constant map. Taking the attaching map to be a constant map
yields that X = S 1 ∨ S 2 . Thus adding the cells in this order merely resulted in
fewer possibilities for the homotopy type of X. This is a general phenomenon.
Generalizing the argument we just presented, using the fact that πi (S d ) = 0 for
i < d, yields the following statement.
Proposition 3. Let
(2) ∅ ⊂ X0 ⊂ X1 ⊂ · · · ⊂ Xn = X
be a cell decomposition of a finite cell complex X. Then X is homotopy equivalent to
a finite cell decomposition with precisely the same number of cells of each dimension
as in (2), and with the cells attached so that their dimensions form a nondecreasing
sequence.
A CW complex is one that can be constructed in this fashion. In fact, even
more is required.
Definition 4. A CW complex is a cell complex with the property that the boundary
of each cell is mapped into the union of the cells of lower dimension.
In some sense, this is a merely technical requirement, as every cell complex
is homotopy equivalent to a CW complex. However, there are certain advantages
to working with CW complexes, and all of the cell complexes which arise in these
notes will be CW complexes.
I first learned of simplicial complexes in a course on algebraic topology. They
were introduced as a category of topological spaces for which it was rather easy to
define homology and cohomology, i.e. in terms of the simplical chain- and cochain-
complexes. One might be concerned that in the transition from simplicial complexes
to cell complexes we have lost this ability to easily compute these topological in-
variants. In fact, much of this computability remains. Let X be a cell complex
with a fixed cell decomposition. Suppose that in this decomposition X is con-
structed from exactly cd cells of dimension d for each d = 0, 1, 2, . . . , n = dim(K),
and let Cd (X, Z) denote the space Zcd (more precisely, Cd (X, Z) denotes the free
abelian group generated by the d-cells of X, each endowed with an orientation).
The following is one of the fundamental results in the theory of cell complexes.
Theorem 5. There are boundary maps ∂d : Cd (X, Z) → Cd−1 (X, Z), for each d,
so that
∂d−1 ◦ ∂d = 0
142 R. FORMAN, COMB. DIFFERENTIAL TOPOLOGY AND GEOMETRY
As the name “Weak Morse Inequalities” implies, this theorem can be strength-
ened. The following inequalities, known as the “Strong Morse Inequalities”, also
follow from standard linear algebra.
Theorem 7 (The Strong Morse Inequalities). With all notation as in Theorem 6,
for each d = 0, 1, 2, . . .
As the names imply, Theorem 7 does directly imply Theorem 6, as one can see by
comparing Strong Morse Inequalities for consecutive values of d, and using the fact
that bi = 0 for i larger than the dimension of K.
We mentioned earlier that a great benefit of passing from simplicial complexes
to the more general cell complexes is that one often can use many fewer cells. Let us
take another look at this phenomenon in light of the Morse inequalities. Consider
the case where X is a two-dimensional torus, so that with respect to any coefficient
field b0 = 1, b1 = 2, b2 = 1. From the weak Morse inequalities, we have that for any
cell decomposition,
c0 ≥ b 0 = 1
c1 ≥ b 1 = 2
c2 ≥ b2 = 1.
A simplicial decomposition is a special case of a CW decomposition, so these in-
equalities are satisfied when cd denotes the number of d-simplices in a fixed sim-
plicial decomposition. However, every simplicial decomposition has at least seven
0-simplices, twenty-one 1-simplices and fourteen 2-simplices, so these inequalities
are far from equality. It is generally the case that for a simplicial decomposition
these inequalities are very far from optimal, and hence are generally of little in-
terest. On the other hand, earlier we demonstrated a CW decomposition of the
two-torus with exactly one 0-cell, two 1-cells and one 2-cell. The inequalities tell
us, in particular, that one cannot build the torus using fewer cells.
α0
α1
α2 α3 α4 α5
Figure 4. A V -path.
β
α
K1 K2
123
1
12 13 23
1 2 3
2 3
empty
1 123
12 13 23
12 3
2 3
empty
3 3
2 1 2 1
t
1 3 2 1 e 3 2
(i) (ii)
6. Sphere Theorems
As mentioned in our discussion at the end of Section 5, one can sometimes use
discrete Morse theory to make statements about more than just the homotopy type
of the simplicial complex. One can sometimes classify the complex up to homeo-
morphism or combinatorial equivalence. In this section we give some examples of
such arguments. An interesting application of these ideas is presented in the next
section. So far, we have not placed any restrictions on the simplicial complexes
under consideration. The main idea of this section is that if our simplicial complex
has some additional structure, then one may be able to strengthen the conclusion.
This idea rests on some very deep work of J. H. C. Whitehead [95].
A simplicial complex K is a combinatorial d-ball if K and the standard d-
simplex σd have isomorphic subdivisions. A simplicial complex K is a combinatorial
(d − 1)-sphere if K and σ̇d have isomorphic subdivisions (where σ̇d denotes the
boundary of σd with its induced simplicial structure). A simplicial complex K is
a combinatorial d-manifold with boundary if the link of every vertex is either a
combinatorial (d − 1)-sphere or a combinatorial (d − 1)-ball. The following is a
special case of the powerful main theorem of [95].
Theorem 14. Let K be a combinatorial d-manifold with boundary which simpli-
cially collapses to a vertex. (That is, K can be a reduced to a vertex by a sequence
of elementary simplicial collapses.) Then K is a combinatorial d-ball.
With this theorem, and its generalizations, one can sometimes strengthen the con-
clusion of Theorem 11 beyond homotopy equivalence. We present just one example.
Theorem 15. Let X be a combinatorial d-manifold with a discrete gradient vector
field with exactly two critical simplices. Then X is a combinatorial d-sphere.
The proof is quite simple (given Theorem 14). The statement is trivial for
d = 0, so we assume that d ≥ 1. Suppose that X is a combinatorial d-manifold
with a discrete gradient vector field V with exactly two critical simplices. Let x0
be a vertex of X. If x0 is not critical, then {x0 < e} is an element of V , for some
edge e. Let x1 be the other endpoint of e. Then x0 , e, x1 is a V-path. If x1 is
not critical, we can follow the V -path to the next vertex x2 , etc. Since there are
only a finite number of vertices, and there are no loops, we must eventually reach
a critical vertex. We can run this argument in reverse for d-simplices. That is, if
α0 is a d-simplex, and α0 is not critical, then {β < α0 } is an element of V for some
(d−1)-simplex β. Let α1 denote the other d-simplex incident to β. Then α1 , β, α0 is
a V -path, and we can follow this path backwards until reaching a critical d-simplex.
Thus, there must be precisely one critical vertex x, and one critical d-simplex α.
Then X − α is a combinatorial d-manifold with boundary with a discrete gradient
vector field with only a single critical simplex, namely the vertex x. It follows
that X − α collapses to x. Whitehead’s theorem now implies that X − α is a
combinatorial d-ball, which implies that X is a combinatorial d-sphere.
((x0 x1 x2 )x3 )
(x0 x1 x2 )
x0 x1 x2 x3 x4
parenthesis between x1 and x2 . Pair such an expression with the one resulting from
adding a pair of parentheses around x2 and x3 if possible. Continue this process as
long as possible. When it has terminated, the only expressions that have not been
paired up with any other expression are s∗ and the one that has a left parenthesis
between every consecutive pair x1 and xi+1 for i = 0, 1, . . . , n−1, i.e. the expression
t∗ = (x0 (x1 (x2 (. . . (xn−2 (xn−1 xn )))) . . . ). Note that t∗ is an (n − 2)-simplex of the
complex Mn .
This completes our construction of the vector field V . All that needs to be
checked is there are no closed V -paths. Denote by Vk the discrete vector field that
has been constructed after the k th step in the construction, i.e. after considera-
tion of the pair xk−1 , xk . It is simple to check that V1 has no closed orbits. Let
(p) (p+1) (p)
s 0 , t0 , s1 denote a V -path. This requires that s0 and t0 be paired in V . Sup-
pose that s0 and t0 are paired in Vk The reader can check that this implies that
either s1 is the head of an arrow in Vk (and hence the V -path cannot be continued)
or s1 is paired in Vk−1 . Thus, by induction, there can be no closed V -paths.
Let K be a simplicial complex, and let x and y be two points not in K. Then the
suspension of K is defined to be the join of K and the set {x, y}. More geometrically,
embed K in some Rd , and embed Rd in Rd+1 by adding a final coordinate. Let
x be the point (0, . . . , 0, 1) and y the point (0, . . . , 0, −1). Then the suspension of
K is the union of all of the closed line segments connecting x to a point in K and
all of the closed line segments connecting the point y to a point in K. This space
comes with a natural simplicial decomposition induced from that of K.
Let S be a simplex, and M a nonempty proper subcomplex of S. There are
two interesting topological spaces to consider in this setting. One is M itself, and
the other is S/M , the result of identifying all of the points in M to a single point.
While S/M is not a simplicial complex, it does have a canonical cell decomposition
giving S/M the structure of a CW complex. Moreover, if α < β are two faces of S
which are not in M, and α∗ and β ∗ are their images in S/M, then α∗ < β ∗ , and
moreover, α∗ is a regular face of β ∗ .
In fact, the two spaces M and S/M are closely related, and one can deduce
essentially the entire topological structure of either one from a knowledge of the
other. More precisely, we have the following statement.
Theorem 19. S/M is homotopy equivalent to the suspension of M .
Of particular interest to us is the following result.
p+1 (S/M, Z) ∼
Corollary 20. For any p, H =H p (M, Z).
These results are not hard to prove using standard methods, but we present a
discrete Morse theory proof of Corollary 20, as the technique (more than the result)
will prove useful later (see the next section). In fact, a more careful analysis of this
proof allows one to deduce Theorem 19, but we will leave that to the reader. Our
approach is to simultaneously construct gradient vector fields U and V on M and
S/M , respectively. Let v be any vertex of M . If α is a nonempty simplex of M
which does not contain v and which has the property that v ∗ α is also in M , then
153
154 R. FORMAN, COMB. DIFFERENTIAL TOPOLOGY AND GEOMETRY
homology of S/M , with the isomporphism shifting all degrees up by 1. This suffices
to prove Theorem 20. A more careful consideration of the implications of Theorem
22 yields Theorem 19.
1 2
connected connected
component component
Figure 10. Graphs which are critical for V12 have two components.
Similarly, if vertex 3 is in G2 and G does not contain the edge (2, 3), then pair
G with G + (2, 3). Let V3 denote the resulting discrete vector field.
The unpaired graphs in V3 are g and those that either contain the edge (1,3)
and have the property that G − (1, 3) is the union of three connected components,
one containing vertex 1, one containing vertex 2, and one containing vertex 3, or
contain the edge (2,3) and have the property that G − (2, 3) is the union of three
connected components, one containing vertex 1, one containing vertex 2, and one
containing vertex 3. We illustrate these graphs in Figure 11. The circles in this
figure indicate connected subgraphs.
Now consider the location of the vertex label 4, and pair any graph G which
is unpaired in V3 with G + (1, 4), G + (2, 4), or G + (3, 4) if possible (at most one
of these graphs is unpaired in V3 ). Call the resulting discrete vector field V4 . We
continue in this fashion, considering in turn the vertices label 5, 6, . . . , n. Let Vi
denote the discrete vector field that has been constructed after the consideration
of vertex i, and V = Vn the final discrete vector field. When we are done the
only unpaired graphs in V will be g and those graphs that are the union of two
connected trees, one containing the vertex 1 and one containing the vertex 2. In
addition, both trees have the property that the vertex labels are increasing along
every ray starting from the vertex 1 or the vertex 2. There are precisely (n−1)! such
graphs, and they each have n − 2 edges, and hence correspond to an (n − 3)-simplex
in Δn .
It remains to see that the discrete vector field V is a gradient vector field,
i.e. that there are no closed V -paths. We first check that V12 is a gradient vector
(p) (p+1) (p)
field. Let γ = α0 , β0 , α1 denote a V12 -path. Then α0 must be the “tail of an
arrow”, i.e. the smaller graph of some pair in V12 , with β0 being the head of the
arrow, i.e. β0 = α0 + (1, 2). The simplex α1 is a codimension-one face of β0 other
than α0 . Thus, α1 corresponds to a graph of the form α0 + (1, 2) − e, where e is an
edge of α0 other than (1,2). Since α1 contains the edge (1, 2) it is the “head of an
arrow” in V12 , i.e. the larger graph of some pair in V12 , which implies that γ cannot
be continued to a longer V12 -path. This certainly implies that there are no closed
V12 -paths.
1 2 1 2
3 3
connected connected
component component
Figure 11. The two types of graphs which are critical for V3 .
158 R. FORMAN, COMB. DIFFERENTIAL TOPOLOGY AND GEOMETRY
1 1
··· ···
··· ···
(i) v3 v4 v5 vn 2 (ii) v3 v4 v5 vn 2
Figure 12. (i) Critical Graphs in Δ2n . (ii) Critical Graphs in Nn2 .
The same sort of argument will work for V . Recall that V is constructed in
stages, by first considering the edge (1, 2) and then the vertices 3, 4, 5, . . . in order.
Let γ = α0 , β0 , α1 denote a V -path. In particular, α0 and β0 must be paired in V .
The reader can check that if α0 and β0 are first paired in Vi , i ≥ 3, then either α1
is the head of an arrow in Vi , in which case the V -path cannot be continued, or α1
is paired in Vi−1 . It follows by induction that there can be no closed V -paths.
In summary, V is a discrete gradient vector field on Nn with exactly one un-
paired vertex, and (n−1)! unpaired (n−3)-simplices. We can now apply Theorem 11
to conclude
Theorem 23 ([93]). The complex Δn of disconnected graphs on n vertices is ho-
motopy equivalent to the wedge of (n − 1)! spheres of dimension (n − 3).
Now let G3 denote the induced maximal 2-connected subgraph which is adjacent
to G1 in G(2), and let v(G) be the vertex G1 ∩ G3 . It is clear that v(G) = 2.
Moreover, if v(G) = 1 then vertex 1 would be a cut vertex of G+(1, 2), contradicting
the assumption that G + (1, 2) is 2-connected. Therefore v(G) ∈ / {1, 2}. Suppose
G ∈ M1 and (v(G), 2) ∈ / G. It is easy to see that v(G) is a cut vertex of G+(v(G), 2),
and hence G + (v(G), 2) is not 2-connected. Moreover, [G + (v(G), 2)] + (1, 2) =
[G + (1, 2)] + (v(G), 2) is 2-connected (since [G + (1, 2)] is), so G + (v(G), 2) ∈ M1 .
Now define
Let M2 contain the graphs which are not paired in V1 or V2 . Then M2 consists
of those graphs G in M1 which contain (v(G), 2), and which have the property that
G − (v(G), 2) ∈/ M1 . First note that since (v(G), 2) ∈ G, v(G) and 2 are contained
in an induced 2-connected subgraph, which implies that v(G) ∈ G2 , and hence G1
and G2 are connected in G(2). From the previous lemma, we learn that G(2) must
consist of only the two vertices G1 and G2 and the edge between them. The only
way G − (v(G), 2) could fail to be in M1 is if G − (v(G), 2) failed to be connected.
This can happen only if G2 − (v(G), 2) is not connected. However, since G2 is 2-
connected, this can happen only if G2 consists entirely of the vertices 2 and v(G) and
the edge between them. Thus, the graphs G in M2 are precisely those that can be
constructed by taking a 2-connected graph G1 on the vertex set {1, 2, . . . , n} − {2},
adding the vertex 2, and adding the edge (i, 2) for some i ∈ / {1, 2} (in which case
v(G) = i).
Let M2 (i), i = 3, 4, . . . , n, denote those graphs G in M2 with v(G) = i. Then
M2 is the disjoint union of the M2 (i)’s. Each M2 (i) can be canonically identified
with the complex Γ of 2-connected graphs on the n − 1 vertices {1, 3, 4, . . . , n}.
By induction, there is a gradient vector field on Γ with precisely (n − 3)! critical
simplices of dimension 2(n − 1) − 5. = 2n − 7. Using the identification, we get a
gradient vector field V3 (i) on M2 (i) with (n − 3)! critical simplices of dimension
2n − 6. Let
V = V1 ∪ V2 ∪ (∪ni=3 V3 (i)).
Since there are n − 2 such M2 (i)’s, the total number of unmatched simplices in V
is (n − 2)(n − 3)! = (n − 2)!, each of dimension 2n − 6. The theorem now follows
once we know that V is a gradient vector field.
Lemma 28. The vector field V constructed above is a gradient vector field.
The proof is left as a (rather non-trivial) exercise.
It is, in fact, quite easy to identify more explicitly the critical simplices in the
above gradient vector field. To find the critical graphs in M2 (i), i = 3, 4, . . . , n,
we take the critical graphs in the complex of 2-connected graphs on the vertex set
{1, 3, 4, . . . , n} with respect to some optimal gradient vector field add the vertex 2
and the edge (i, 2) for some i = 3, 4, . . . , n. Fixing i, identify {1, 3, 4, . . . , n} with
{1, 2, . . . , n−1} via a correspondence that identifies 1 with 1, and identifies i with 2.
By induction, there is a gradient vector field on the 2-connected graphs on the vertex
set {1, 2, . . . , n − 1} whose critical simplices have the form shown in Figure 12(ii).
Using the identification, we get a gradient vector field on 2-connected graphs on the
LECTURE 2. DISCRETE MORSE THEORY, CONTINUED 161
··· ···
··· ···
v3 v4 vi−1 vi+1 vn i
Figure 13. Critical 2-connected graphs on the vertex set {1, 2, 3, . . . , n}.
vertex set {1, 3, 4, . . . , n} whose critical simplices are of the form shown in Figure 13
(where v3 , v4 , . . . , vi−1 , vi+1 , . . . , vn is any permutation of 3, 4, . . . , i−1, i+1, . . ., n).
Adding a vertex 2 to each such graph, and adding an edge between vertex i and
vertex 2 yields the desired collection of graphs shown in Figure 12(i). Corollary 26
now follows from Theorem 22.
Since homotopy equivalent spaces have isomorphic homology, the following theorem
follows from Theorems 11 and 5.
Theorem 31. There are boundary maps ∂˜d : Md → Md−1 , for each d, so that
∂˜d−1 ◦ ∂˜d = 0
and such that the resulting differential complex
(3) ∂˜ ∂˜
0 −−−−→ Mn −−−n−→ . . . −−−1−→ M0 −−−−→ 0
calculates the homology of K. That is, if we define
˜
˜ = Ker(∂d )
Hd (M, ∂)
Im(∂˜d+1 )
then for each d
˜ ∼
Hd (M, ∂) = Hd (K, Z).
In fact, this statement is equivalent to the Strong Morse inequalities (see Exer-
cise 1 of Lecture 1). The main goal of this section is to present an explicit formula
for the boundary operator ∂. ˜ This requires a closer look at the notion of a gradient
path. Let β be a critical (p + 1) simplex, and and α a critical p-simplex. Then it
is easy to check that any gradient-path from β to α has the form
(p+1) (p) (p+1) (p) (p)
β = β0 , α1 , β1 , α2 , . . . , βr(p+1) , αr+1 = α
such that for each i = 0, 1, 2, . . . , r , {αi+1 < βi+1 } ∈ V, and αi+1 < βi , but
{αi+1 < βi } ∈ / V. In Figure 14 we show a single gradient path from the boundary of
a critical 2-simplex β to a critical edge α, where the arrows pointing from an edge
to a 2-cell indicate the gradient vector field V .
Given a gradient path as shown in Figure 14, an orientation on β induces an
orientation on α. We will not state the precise definition here. The idea is that
one “slides” the orientation from β along the gradient path to α. For example, for
the gradient path shown in Figure 14, the indicated orientation on β induces the
indicated orientation on α.
We are now ready to state the desired formula.
Theorem 32. Choose an orientation for each simplex. Then for any critical (p+1)-
simplex β set
(4) =
∂β cα,β α
critical α(p)
where
cα,β = m(γ)
γ∈Γ(β,α)
where Γ(β, α) is the set of gradient paths which go from β to α. The multiplicity
m(γ) of any gradient path γ is equal to ±1, depending on whether, given γ, the
orientation on β induces the chosen orientation on α, or the opposite orientation.
With this differential, the complex (3) computes the homology of K.
We refer to the complex (3) with the differential (4) as the Morse complex (it
goes by many different names in the literature). An extensive study of the Morse
complex in the smooth category appears in [78]. In is section, we have focused
our attention on simplicial complexes. However, it is worth noting that this entire
LECTURE 2. DISCRETE MORSE THEORY, CONTINUED 163
β α
discussion applies, without any change, to any regular CW-complex, and, after
some refinement of the notion of the multiplicity m(γ), to all CW complexes. See
[32] for details.
We only have time to present the main ideas the proof of Theorem 32. For
the details, consult Sections 7 and 8 of [32]. The key ingredient in the proof is the
notion of a (discrete time) flow associated to a discrete vector field V . In the case
of smooth manifolds, the gradient vector field defines a dynamical system, namely
the flow along the vector field. Viewing the Morse function from the point of view
of this dynamical system leads to important new insights [83]. The same is true in
the combinatorial category.
Up to this point in the notes, we have been thinking of V as a collection of pairs
of simplices. Now it is better to think of V as a map of oriented simplices. Namely,
choose an orientation for each simplex of M . If {β (p) < α(p+1) } is an element of
V , then we set V (β) = −iα where i = ±1 is the incidence number of β and α (i.e
i = 1 if the orientations agree, and −1 otherwise). Set V (β (p) ) = 0 if there is no
such α(p+1) , i.e. if β is not the tail of any arrow in V . Now extend V linearly to a
map
V : Cp (M, Z) → Cp+1 (M, Z),
and do this for each p.
The flow Φ along the gradient vector field V is a map
Φ : Cp (M, Z) → Cp (M, Z),
for each p, defined by the formula
Φ = 1 + ∂V + V ∂.
See Figure 15 for the flow of an oriented edge e. In this figure, we indicate the
orientation of e, and just enough of the vector field V in order to determine Φ(e).
We observe that the map Φ commutes with the boundary operator. The other
main fact is that for a finite simplicial complex, the map Φ stabilizes in finite time.
That is, there is an N such that ΦN = ΦN +1 = ΦN +2 = . . . (it is only here that it
is necessary that the vector field V be a gradient vector field), and we denote this
map by Φ∞ .
Now let us return to the analysis of the Morse complex. Let
∂ ∂
C∗ : 0 −−−−→ Cn (K, Z) −−−n−→ . . . −−−1−→ C0 (K, Z) −−−−→ 0
denote the usual simplicial chain complex of K. Let CpΦ (K, Z) ⊂ Cp (K, Z) denote
the subspace of Φ-invariant chains (i.e. the chains c such that Φ(c) = c). Then,
since Φ commutes with the boundary operator ∂, the boundary map takes CpΦ (K, Z)
Φ
to Cp−1 (K, Z). Now consider the complex of Φ-invariant chains.
∂ ∂
C∗Φ : 0 −−−−→ CnΦ (K, Z) −−−n−→ . . . −−−1−→ C0Φ (K, Z) −−−−→ 0.
164 R. FORMAN, COMB. DIFFERENTIAL TOPOLOGY AND GEOMETRY
+ +
Φ(e)
The first step is to see that this complex has the same homology as C∗ . There
are obvious maps between the two complexes, since C∗Φ injects into C∗ , and Φ∞
maps C∗ onto C∗Φ . The composition yields the identity map on C∗Φ . Thus, it
is sufficient to show that the map Φ∞ : C∗ → C∗ induces an isomorphism on
homology. For this, it is sufficient to find a homotopy operator. That is, an operator
L : C∗ (K, F) → C∗−1 (K, F) with the property that Φ∞ − 1 = L∂ + ∂L. If Φ∞ = Φ,
then one could take L = V . The general case of Φ∞ = ΦN is similar.
To make the transition to critical simplices, one can establish that
Φ∞ : Mp → Cp (K, Z)
is an isomorphism for each p, with inverse the restriction map r : Cp (K, Z) → Mp .
Theorem 32 now follows if we take ∂ = r ◦ ∂ ◦ Φ∞ . One must then calculate that
this is precisely the operator defined in the statement of the theorem.
A different proof of Theorem 32 is suggested in the exercises.
Example 33. We end this section with a demonstration of how the ideas of this
section may be applied to the example of the real projective plane P2 as illustrated
in Figure 8(ii). We saw in Section 11 how discrete Morse Theory can help us see
that P2 has a CW decomposition with exactly one 0-cell, one 1-cell and one 2-
cell. Here we will see how Morse theory can distinguish between the spaces which
have such a CW decomposition. Let us now calculate the boundary map in the
Morse complex corresponding to the gradient vector field illustrated in Figure 8(ii).
Choose an orientation for the edge e. To calculate ∂(e), we must count all of the
gradient paths from e to v. There are precisely two such paths, since the unique
gradient path beginning at each endpoint of e leads to v. (The gradient path
beginning at vertex 1 is the trivial path of 0 steps.) Since the orientation of e
induces a + on one endpoint of e, and a − orientation on the other, adding these
two paths with their corresponding signs leads us to the formula that ∂(e) = 0.
Now choose an orientation for t. It can be seen from Figure 8(ii) that there are
precisely two gradient paths from t to e, and both induce the same orientation on
LECTURE 2. DISCRETE MORSE THEORY, CONTINUED 165
β α
smooth. Moreover, one must check that the new vector field is a gradient vector
field, so that, in particular, modifying the vectors did not result in the creation of
a closed orbit. This is an example of the sort of complications which arise in the
smooth setting, but which do not make an appearance in the discrete theory.
This theorem was recently put to very good use in [7], in which discrete Morse
theory is used to determine the homotopy type of some simplicial complexes arising
in the study of partitions. It is fascinating, and quite pleasing, to see the same
idea play a central role in two subjects, the Poincaré conjecture and the study of
partitions, which seem to have so little to do with one another. In [50], Hersh
generalizes this cancellation technique and investigates, among other ideas, when
families of pairs of critical simplices can be canceled simultaneously. The main
theorem of this section is also used extensively in [54] as a basic computational
tool for searching for optimal gradient vector fields. To see other computational
approaches to finding optimal gradient vector fields, the reader can take a look at
[62], [63], [64].
1 − v2 − v3 − · · · − vn
where {v2 , v3 , . . . , vn } = {2, 3, . . . , n}. We observe that there are precisely
(n−1)! of these graphs, and each has (n−1) edges. Your job is to construct
a gradient vector field on the simplicial complex of connected graphs on
LECTURE 2. DISCRETE MORSE THEORY, CONTINUED 167
n vertices for which the critical graphs are precisely the graphs in Pn .
(In Vassiliev’s original work on this complex, this is the form in which he
presented the answer.)
(2) Let G be any graph, and let P be any monotone decreasing graph property.
Then we can consider the simplicial complex of all spanning subgraphs of
G that satisfy P.
In the lecture we only considered the case where G is a complete
graph. For other graphs G, these complexes are quite interesting and
largely unexplored.
(a) Examine this in the case where P is the property of being discon-
nected. What is the homotopy type of the resulting complex? That
is, given a graph G, what is the homotopy type of the simplicial
complex of disconnected spanning subgraphs of G?
(b) Pick your favorite monotone graph property and your favorite graph
and examine the resulting simplicial complex.
(3) Let M be a simplicial complex with a gradient vector field V. Prove that
the homology of the Morse complex (as defined in this lecture) is isomor-
phic to the homology of M by following these steps:
(a) Suppose that V is the empty gradient vector field. Then the Morse
complex is just the standard simplicial chain complex of M.
(b) Now prove that the homology of the Morse complex of V does not
change if one pair is removed from V (i.e. if one arrow is erased). Do
this by showing that if
d d d d
M : 0 → Mn → Mn−1 → Mn−2 → · · · → M0 → 0
and
d d d d
M : 0 → Mn → Mn−1
→ Mn−2 → c . . . → M0 → 0
are the Morse complexes corresponding to gradient vector fields V
and V on M which differ by a single arrow, then there is a map
Φ : Mi → Mi which induces an isomorphism on homology. Try to
construct the map Φ as explicitly as possible.
Together (a) and (b) prove the desired result.
(4) In the exercises to Lecture 1 we proved that every triangulated surface has
a perfect gradient vector field. Consider the Morse complex corresponding
to such a vector field. Prove that all of the differentials vanish (that is,
each differential is the zero map). Can you understand this directly from
the definition of the differential – that is by counting gradient paths?
LECTURE 3
Discrete Morse Theory and Evasiveness
c
no yes
a b
no yes no yes
b b a a
no yes no yes no yes no yes
Let us begin with a simple example of the sort of thing we wish to study.
Suppose there are three yes/no questions that we can easily ask. We label these
questions {a, b, c}.
Assumption 1: We suppose that these questions have the property that their
answers are independent of the order in which they are asked. (We will make this
assumption for the rest of these notes.)
Then there are eight possible outcomes resulting from asking these three questions.
We label these outcomes by listing the questions that yield the answer “yes” for
that outcome. The possibilities are: [ ], [a], [b], [c], [ab], [ac], [bc], [abc].
Assumption 2: We assume that every set of answers is possible.
That is, one can easily imagine a set of questions with the property that questions
b and c can not both be answered “yes”, but we will not consider this possibility
in these lectures. We make this assumption only for reasons of simplicity. The
general situation is considered in [41]. Suppose that the following four outcomes
are good: [a],[b],[c],[ab], and the remaining outcomes are bad. By asking these three
questions, our goal is to determine whether the outcome is good or bad. We can,
of course, accomplish this goal by asking all three questions. We are considered
to have won this game if we achieve the goal before we ask the third question. A
winning strategy, then, is one which guarantees that no matter what the outcome is,
we can determine whether or not it is good or bad before asking the third question.
For example, consider the search algorithm shown in Figure 17, in which case
we have listed the question to be asked next, given the answers to the previous
questions. For example, we ask question c first, and if we get the answer “yes” we
ask question b, but if we get the answer “no”, we ask the question a. We observe that,
asking questions in the indicated order, if the outcome is in the set {[ ], [b], [c], [ac]},
then we must ask the third question. Outcomes which require us to ask the third
question are called evaders of the search algorithm, so the algorithm has 4 evaders.
In fact, this is the best one can do. The following proposition is fairly easy to check
by straightforward means.
Proposition 35. Every search algorithm for the problem of determining member-
ship in the set of good outcomes {[a], [b], [c], [ab]} has at least 4 evaders. The number
of evaders which are good equals the number of evaders which are bad, and hence
there must be at least two of each.
If we assume that each outcome is equally likely, then this proposition implies
that no matter which search algorithm we choose, we will have to ask the third
question at least half of the time. Note that this theorem does not say that every
search algorithm has exactly 4 evaders, and it is rather easy to find search algorithms
LECTURE 3. DISCRETE MORSE THEORY AND EVASIVENESS 171
ab ac
abc
b bc c
with more than 4 evaders. If every search algorithm has some evaders, so that we
have no winning strategy, then we say that the problem is evasive.
It is probably not at all clear to the reader what this topic is doing in a series
of lectures on discrete Morse theory, but we will show that in fact these topics are
intimately related. In particular, we will show that algebraic topology gives a way
of understanding why some problems of this form are easy, and others are hard.
First we observe that the problem can easily be stated in a more topological way.
Consider the 2-dimensional simplex S with vertices labeled {a, b, c}. Then the faces
of S can be indentified with the subsets of {a, b, c}, and hence with the 8 possible
outcomes (see Figure 18). Then the good and bad outcomes partition the faces of
S into 2 sets. In this setting we are given a partition of the set of faces, the outcome
is a face σ of the simplex, and our goal is to determine which block of the partition
contains σ. We are permitted to ask questions of the form “Is vertex v in σ?”.
In this way, we can convert binary search problems (which satisfy Assumption
1) into the language of simplices. If we also require Assumption 2, then the sort
of search problems we are considering lead to problems of the following form. Let
S be an n-dimensional simplex, with vertices v0 , v1 , . . . , vn , F the set of faces of S,
and
P : F = P1 P2 . . . Pk
a partition of F, which is known to you. Let σ be a face of S which is not known
to you. Your goal is to determine which block of the partition P contains σ. In
particular, you need not determine the face σ. You are permitted to ask questions
of the form “Is vi in σ?”. You may use the answers to the questions you have
already asked in determining which vertex to ask about next. Of course, you can
determine which block contains σ by asking n + 1 questions, since by asking about
all n + 1 vertices you can completely determine σ. You win this game if you answer
the given question after asking fewer than n + 1 questions.
Say that P is nonevasive if there is a winning strategy for this game, i.e there
is a search algorithm that determines which block contains σ in fewer than n + 1
questions, no matter what σ is. Say P is evasive otherwise.
One of the main issues we will have to deal with is that a block Pi of the
partition need not be a subcomplex or have any other nice structure. Hence, the
notion of the homology of such a set is problematic. Let P be any set of faces
of a simplex S, and let F be a field. One of the main contributions of this and
the following sections is a definition of the F-Betti numbers of P. More precisely,
for each i = −1, 0, 1, . . . , we will define Bi (P, F), the ith Betti number of P with
respect to the field F. We will also define the even and odd Betti numbers, denoted
Be (P, F) and Bo (P, F), respectively, and the total Betti number B(P, F). For ease of
notation, we will assume that the field F is fixed, and refer to Bi (P ), Be (P ), Bo (P )
172 R. FORMAN, COMB. DIFFERENTIAL TOPOLOGY AND GEOMETRY
and B(P ). We will present the precise definition of these numbers in the next
section. The basic idea is that the Betti number Bi (P ) is defined by restricting
the chain complex of S over the field F to the faces in P . The result need not be
a complex, since the composition of its consecutive differentials might be nonzero,
but the dimension of its ith “homology” can still be defined as the dimension of its
ith kernel minus the dimension of its (i + 1)st image (if that is nonnegative, and 0
otherwise); see Proposition 47. At this point, we will state the main properties of
these Betti numbers.
Theorem 36. For any set of faces P
(i) Be (P ) ≥ Bi (P ),
i even
Bo (P ) ≥ Bi (P ),
i odd
B(P ) = Be (P ) + Bo (P ).
(ii) Bi (P ) = 0 for i larger thanthe dimension of P .
(iii) Be (P ) − Bo (P ) = χ(P ) = i (−1)i #{i-simplices in P }
Our notion of a Betti number is equal to a standard notion in a number of settings.
Theorem 37.
(1) If P is a subcomplex of S, and the empty set (considered as a face of S)
is an element of P , then for each i
i (P, F).
Bi (P ) = dim H
where the tilde denotes reduced homology. Moreover,
even (P, F)
Be (P ) = dim H
odd (P, F)
Bo (P ) = dim H
B(P ) = dim H ∗ (P, F).
(2) If P is a subcomplex of S, except that the empty set is not element of P ,
then for each i
Bi (P ) = dim Hi (P, F).
Moreover,
Be (P ) = dim Heven (P, F)
Bo (P ) = dim Hodd (P, F)
B(P ) = dim H∗ (P, F).
(3) Let P denote the closure of P (i.e. the set consisting of the faces of P
along with all of their faces), and let Ṗ = P − P. If Ṗ is a subcomplex of
S (which contains the empty set) then for each i
Bi (P ) = dim Hi (P , P, F).
Moreover
Be (P ) = dim Heven (P , P, F)
Bo (P ) = dim Hodd (P , P, F)
B(P ) = dim H∗ (P , P, F).
Assuming these results for now, as well as the still undefined notion of Betti
number, we present the main theorem of this section.
LECTURE 3. DISCRETE MORSE THEORY AND EVASIVENESS 173
Theorem 38. With all notation as above, for any search algorithm A the number
of evaders of A which lie in any block Pj of the partition P is at least B(Pj ). Hence
k
the total number of evaders is at least j=1 B(Pj ).
In fact, we can make this statement much more precise. Define the dimension
of an evader to be the dimension of the face of S to which it corresponds. That is,
if σ is any possible outcome, dim(σ) is
(the number of questions answered “yes” if the outcome is σ) − 1.
Theorem 39. With all notation as above, for any search algorithm A the number
of evaders of A of dimension i which lie in any block Pj of the partition P is at
least Bi (Pj ). The number of even-dimensional evaders which lie in block Pj is at
least Be (Pj ), and the number of odd-dimensional evaders which lie in block Pj is at
least Bo (Pj ) .
Before discussing the proof of this result, we would like to point out that Kahn,
Saks and Sturtevant [55] first observed the relationship between evasiveness and
algebraic topology. In their setting, the partition consists of precisely two blocks,
P : S = P1 P2 , in which P1 is a subcomplex. They proved the following theorem.
Theorem 40. If H̃∗ (P1 ) = 0, where H̃∗ (P1 ) denotes the reduced homology of P1 ,
then P is evasive.
In fact, they proved something stronger, and we will come back to this point
later. In [39] we used discrete Morse theory to make some of their results more
quantitative along the lines of Theorems 38 and 39. The generalization in this
section to more than two blocks is relatively minor. The extension to more general
sets of faces is the major value of this newer work.
We illustrate the previous theorems by returning to the example introduced
at the beginning of this section. Let P1 denote the set {[a], [b], [c], [ab]} of good
outcomes, and let P2 denote the complement, the set of bad outcomes. We observe
that P1 is a simplicial complex which does not contain the empty face. Hence by
Theorem 37, B(P1 ) is equal to the dimension of the (unreduced) homology of P1 ,
which is 2. By Theorem 38, we learn that for any search algorithm, the number of
evaders which lie in P1 is at least 2. We observe that P2 does not satisfy any of
the hypotheses presented in Theorem 37, so one can not deduce its Betti numbers
from that result. However, as the reader can check (after we present the definition
of Betti numbers in the next section), its total Betti number is also 2.
The link between evasiveness and algebraic topology is provided by discrete
Morse theory. Morse theory comes to the fore when one observes that a search
algorithm induces a discrete vector field on S. For example, the search algorithm
shown in Figure 17 induces the vector field
V = { {[ ] < [b]}, {[a] < [a, b]}, {[c] < [a, c]}, {[b, c] < [a, b, c]} }
That is, V consists of those pairs of faces of S which are not distinguished by the
search algorithm until the last question. There is slight subtlety here in that a search
algorithm pairs a vertex with the empty face [ ], while in our original definition, it
was not permitted to pair a simplex with [ ]. Thus, to get a true discrete vector
field, we must remove this pair from V . (It is precisely this subtle point that results
in the reduced homology of K being the relevant measure of topological complexity
174 R. FORMAN, COMB. DIFFERENTIAL TOPOLOGY AND GEOMETRY
ab ac
abc
b c
bc
in Theorem 37(1), rather than the unreduced homology.) However, for simplicity,
from now on we will simply ignore this technical point.
Theorem 41. For any search algorithm, let V denote the vector field consisting of
pairs of nonempty faces of S which are not distinguished by the search algorithm
until the last question. Then V is a gradient vector field.
We will postpone the proof of this result until the end of this section.
For now, suppose that block Pj of the partition P is a subcomplex (containing
the empty face). We will complete the proof in this setting before discussing the
general case. Now restrict V to Pj by taking only those pairs in V such that both
simplices are in Pj , and denote the resulting vector field by Vj . In our simple
example, this results in the vector field
V1 = {{[a] < [a, b]}}.
From the previous theorem, V has no closed orbits. Any discrete vector field
consisting of a subset of the pairs of V has fewer paths, and hence also has no
closed orbits. Therefore, Vj is a gradient vector field on Pj . Note that V pairs
every face of S with another face, and hence there are no critical simplices except
for the vertex which is paired with the empty set. Thus, ignoring that special vertex
for the moment, the critical simplices of Vj are precisely the simplices of Pj which
are paired in V with a face of S which is not in Pj . These are precisely the simplices
of Pj which are the evaders of the search algorithm.
The Morse inequalities of Theorem 13 (i) immediately imply the following re-
sult.
Corollary 42. If the block Pj of the partition P is a subcomplex (containing the
∗ (Pj ).
empty face of S) then the number of evaders in Pj is at least dim H
(We must use reduced homology here because of the minor issue surrounding the
vertex paired with the empty set.) This yields Theorem 37 (in the case of a sim-
plicial complex containing the empty face).
Suppose that P is nonevasive. Then there is some search algorithm which has
no evaders. From our above discussion we have seen that this implies that Pj has a
gradient vector field with no critical simplices. Actually, this is not quite true. The
gradient vector field must have a critical vertex – the vertex that is paired with the
empty face. These ideas lead to the following strengthening of Theorem 40.
Theorem 43. If P is nonevasive, and if the block Pj of the partition is a subcom-
plex, then Pj collapses to a vertex.
This theorem appears in [55], the paper that first established, and used to very
good effect, a close relationship between evasiveness and topology. The interested
LECTURE 3. DISCRETE MORSE THEORY AND EVASIVENESS 175
reader can consult [36] for some additional refinements of this theorem. This topic
has been the subject of much study, and the reader can find more information about
the connection between evasiveness and topology in the references [11], [56], [76],
[77], and [94] .
We now present a proof of Theorem 41. Let S denote an n-simplex, and fix a
search algorithm. Associate to each p-simplex α of S the sequence of integers
n(α) = n0 (α) < n1 (α) < · · · < np (α)
where, for each i, question number ni (α) is answered “yes” if σ = α, and these are
the only questions answered “yes”.
Let V be the vector field induced by the search algorithm and
α1 , α2
be a V -path. Then either (i) α1 is a face of α2 and {α1 < α2 } ∈ V, or (ii) α2 is a
face of α1 and {α1 < α2 } ∈ / V. Let us consider case (ii) first. In this case, α2 has
one fewer vertex than α1 , and the vertex is not the subject of the (n + 1)st question.
Suppose the the vertex is the subject of the ni (α1 )st question. Then this question
is answered “yes” for α1 , but “no” for α2 . This implies that
n(α2 ) = n0 (α1 ) < n1 (α1 ) < · · · < ni−1 (α1 ) < ni (α2 ) < · · ·
for some i < n + 1, and such that ni (α2 ) > ni (α1 ). Thus n(α2 ) > n(α1 ) in the
lexicographic order.
We now consider case (i), in which {α1 < α2 } ∈ V, and continue the V -path one
more step to α1 , α2 , α3 . Then α1 and α2 are not distinguished until the (n + 1)st
question. Thus,
n(α2 ) = n0 (α1 ) < n1 (α1 ) < · · · < np (α1 ) < n + 1.
We now observe that the vertices of α3 are a subset of the vertices of α2 . Suppose
the vertex of α2 which is not in α3 is the vertex tested in question ni (α2 ). Then
we must have i = n + 1. This demonstrates that
n(α3 ) = n0 (α1 ) < n1 (α1 ) < · · · < ni−1 (α1 ) < ni (α3 ) < · · ·
for some i < n + 1, and such that ni (α3 ) > ni (α1 ). Thus n(α3 ) > n(α1 ) in the
lexicographic order, which is sufficient to prove that there are no closed orbits.
In [53], Jonsson investigates further the question of which gradient vector fields
arise from decision trees. Anyone interested in this topic should also consult [84].
If ∂p ◦ ∂p+1 = 0 for each p, then we say that V is a differential complex and that
the ∂i ’s form a differential. We recall that one defines the homology of a differential
complex by the formula
Ker ∂p
(5) Hp (V) := .
Im ∂p+1
For each p, choose a subspace Xp which is mapped isomorphically onto Im ∂p . Then
we have that
Vp = Xp ⊕ Ker ∂p .
In the case of a differential complex, Im ∂p+1 ⊂ Ker ∂p we can find a Zp ⊂ Vp so
that
Ker ∂p = Im ∂p+1 ⊕ Zp ,
which implies that
Vp = Xp ⊕ Im ∂p+1 ⊕ Zp ,
and the reader can easily check that Zp ∼ = Hp (V).
We now return to the general case of a nondifferential complex. That is, we
no longer assume that ∂p ◦ ∂p+1 = 0. We will use the construction of the previous
paragraph to define the homology of such a complex.
Definition 44. A homological decomposition D of the complex S is a decomposi-
tion
Vi = Xi ⊕ Yi ⊕ Zi ,
for each i, with the property that for each i, ∂i maps Xi isomorphically onto Yi−1 .
By the notation Vi = Xi ⊕ Yi ⊕ Zi we mean that Xi , Yi and Zi are linear subspaces
of Vi , such that their pairwise intersections are {0}, and they sum to give all of Vi .
Homological decompositions always exist, since one can take Xi = 0, Yi = 0, and
Zi = Vi , for each i.
For any homological decomposition D of V, and any i, let Bi (V, D) denote the
dimension of Zi . We also define the even Betti number of D
Be (V, D) := Bi (V, D),
i even
the odd Betti number of D
Bo (V, D) = Bi (V, D),
i odd
and the total Betti number of D
B(V, D) = Be (V, D) + Bo (V, D) = Bi (V, D).
i
We now define the Betti numbers of S by
Bi (V) := min Bi (V, D),
D
Be (V) := min Be (V, D),
D
Bo (V) := min Bo (V, D),
D
and
B(V) := min B(V, D).
D
We observe the following facts.
LECTURE 3. DISCRETE MORSE THEORY AND EVASIVENESS 177
Example 46. A simple example will serve to show that the inequalities in part (i)
of the proposition can be strict when V is not a differential complex. Consider the
complex V with V0 = V1 = V2 = F, and Vi = 0 for i = −1 and i > 2. Suppose that
∂1 and ∂0 are both the identity map.
∂ ∂
V : 0 −−−−→ F −−−1−→ F −−−0−→ F −−−−→ 0
Let D1 denote the homological decomposition
∂ ∂
0 −−−−→ F ⊕ 0 ⊕ 0 −−−1−→ 0 ⊕ F ⊕ 0 −−−0−→ 0 ⊕ 0 ⊕ F −−−−→ 0.
We have that B1 (V, D1 ) = B2 (V, D1 ) = 0, while B0 (V, D1 ) = 1, which implies that
B1 (V) = B2 (V) = 0, and B0 (V) ≤ 1.
Let D2 denote the homological decomposition
∂ ∂
0 −−−−→ 0 ⊕ 0 ⊕ F −−−1−→ F ⊕ 0 ⊕ 0 −−−0−→ 0 ⊕ F ⊕ 0 −−−−→ 0.
In this case we see that B0 (V, D2 ) = B1 (V, D2 ) = 0, while B2 (V, D2 ) = 1, which
implies that B0 (V) = B1 (V) = 0, and B2 (V) ≤ 1.
Thus we learn that Bi (V) = 0 for every i. On the other hand Bo (V, D1 ) =
Bo (V, D2 ) = 0, which implies that Bo (V) = 0. We note that Be (V, D1 ) =
Be (V, D2 ) = 1, and, in fact, once can easily see that Be (V) = 1.
ab ac
abc
b bc c
space. We define the basic combinatorial notions just as for a simplicial complex.
A face a of S is said to be a maximal element of K if a is in K, and a is not a
proper subset of any element in K. If a is a maximal element of K, we say that b is
a free face of a in K if: b is in K, b is a codimension-one face of a, and a is the only
element of K which properly contains b. Let K be a simplex space, a a maximal
face of K, and b is a free face of a in K. The act of replacing K by K − {a, b} is
called a simplicial collapse. Say that K is collapsible if one can transform K into
the empty simplex space by a sequence of simplical collapses.
Let K be a simplex space, and a a maximal element of K. We will call the act
of replacing K by K − a a simplical removal.
We will use the term elementary simplicial reduction to refer to either a simpli-
cal collapse or a simplicial removal. A complete reduction of K is any sequence of
elementary reductions that transforms K into the empty simplex space. In partic-
ular, K is collapsible if and only if there is a complete reduction consisting solely
of simplicial collapses.
Lemma 50. Let K be a simplex space.
(i) If K = K − α for some maximal d-simplex α, then
B(K ) ≥ B(K) − 1.
(ii) If K = K − (Int(α) ∪ Int(β)) is the result of a simplicial collapse, where α
is a maximal d-simplex and β is a free face of α, then
B(K ) ≥ B(K).
Together, parts (i) and (ii) imply the following theorem.
Theorem 51. Let K be a simplex space. In any complete reduction of K, the
number of simplices which are taken out by a simplicial removal is at least B(K).
Corollary 52. Let K be any simplex space, and V any gradient vector field on K.
Then the number of critical cells of V is at least B(K).
Theorem 50 can be made more precise to include an understanding of how the
individual Bd (K) can change under simplicial collapse and simplicial removal that
is sufficient to imply Theorem 39.
Example 53. We end this section with an example to illustrate that, unlike in the
case of a simplicial complex, a simplicial collapse can increase the Betti numbers of a
simplex space. Let S denote the two-dimensional simplex with vertices label a, b, c.
Let K denote the simplex space consisting of the four faces [a], [a, b], [b, c], [a, b, c].
Then K is collapsible, since one can remove [b, c] and [a, b, c] by one simplicial
180 R. FORMAN, COMB. DIFFERENTIAL TOPOLOGY AND GEOMETRY
collapse, and the remaining two faces with a second simplicial collapse. Thus, all
Betti numbers of K are zero. One the other hand, beginning with K, one can
also remove the faces [a, b] and [a, b, c] by a simplicial collapse, resulting in the
simplex space K consisting of the faces [a] and [b, c]. One can easily check that
B0 (K ) = B1 (K ) = Be (K ) = Bo (K ) = 1.
a a
ab ac
ab ac
abc abc
b bc c b bc c
K K
Figure 21. The simplex space K with vanishing Betti numbers collapses to
K , which has nonzero Betti numbers.
1. Introduction
These notes are intended to be an introduction to the Charney-Davis conjectures
and some of their combinatorial implications. My aim is to provide a stimulating
advertisement for a circle of ideas that is the subject of some fascinating recent
work, most of which creates more questions than answers, and which has shed new
light on some of the central questions in geometric combinatorics. The subject is
a beautiful one, borrowing techniques and ideas from geometry, topology, analysis,
algebra, algebraic geometry and combinatorics. My goal in these lectures is to
present the topological and geometric context of these conjectures (as presented
e.g. in [15],[22]), along with the most recent combinatorial understanding of them
(as in Gal [43] and Brändén[12].) These notes will have been successful if some
readers are inspired to consult these original sources, and to begin thinking about
these conjectures.
The Charney-Davis conjectures, concerned with the relationship between geom-
etry and topology, find their origins, as do most such questions, in the Gauss-Bonnet
Theorem. Recall that the Gauss-Bonnet theorem states that if M is a compact sur-
face with a Riemannian metric, then
1
χ(M ) = K darea
2π M
where K denotes the Gauss curvature of M . It follows that if K ≤ 0 everywhere,
then χ(M ) ≤ 0.
Hopf conjectured the following generalization.
Conjecture 54. If M is a compact Riemannian manifold of dimension 2n and the
sectional curvature of M is ≤ 0 then (−1)n χ(M ) ≥ 0.
[Recall that if M is an odd-dimensional manifold, then χ(M ) = 0.] This is
not a suitable place for a primer in differential geometry, so we hope it will suffice
to say that the condition that the sectional curvature is nonpositive means that
every two-dimensional “orthogonal slice” of M is a surface of nonpositive Gauss
curvature. This conjecture may seem a bit surprising, and perhaps unintuitive,
at first glance. However, some general considerations point in this direction. Most
notably, one has the following observations.
181
182 R. FORMAN, COMB. DIFFERENTIAL TOPOLOGY AND GEOMETRY
Proposition 55.
(1) Let M1 and M2 be compact manifolds, then
χ(M1 × M2 ) = χ(M1 )χ(M2 ).
(2) If M1 and M2 are Riemannian manifolds with nonpositive sectional cur-
vature, and M1 × M2 is endowed with the product Riemannian metric,
then M1 × M2 has nonpositive sectional curvature.
Thus, if M1 and M2 are nonpositively curved Riemannian manifolds for which
the conclusion of Hopf’s conjecture holds, then the same is true for M1 ×M2 . In par-
ticular, Hopf’s conjecture holds for any product of arbitrarily many nonpositively
curved surfaces.
Allendoerfer, Fenchel and Weil ([1],[29], [2]), and later Chern ([17]), proved
a higher dimensional version of the Gauss-Bonnet theorem, which, for a compact
Riemannian manifold, has the general form
χ(M ) = R dvol,
M
where R is a function of the curvature of M and is usually called the Chern-
Gauss-Bonnet integrand. Chern [18] gives a proof (attributed to Milnor) that in
dimension 4, if the sectional curvature is ≤ 0 everywhere, then R ≥ 0. In particular:
Corollary 56. If M is a compact Riemannian 4-manifold with sectional curvature
≤ 0, then χ(M ) ≥ 0.
However, Geroch [44] proved that this approach is insufficient to settle Hopf’s
conjecture in higher dimensions.
Theorem 57. In even dimensions ≥ 6, there exist Riemannian metrics with sec-
tional curvature ≤ 0 such that the Chern-Gauss-Bonnet integrand achieves both
signs.
So, in higher dimensions another approach is necessary. Before discussing al-
ternate approaches, and partial results, we will take a detour to discuss some gen-
eralizations and extensions of Hopf’s conjecture. From now on, when we say that
a Riemannian manifold M has nonpositive curvature, we mean that all sectional
curvatures are ≤ 0.
It is a theorem of Cartan and Hadamard that if M n has nonpositive curvature,
then M , the universal cover of M , is diffeomorphic to Rn . A manifold is said to
be aspherical if its universal cover is contractible. Thurston generalized Hopf’s
conjecture to the following
Conjecture 58. Let M 2n be a smooth, compact, aspherical manifold. Then
(−1)n χ(M ) ≥ 0.
This is quite interesting, as the hypothesis has changed from a geometric condition
to one that is purely topological. Our interests, however, lie in a different direction.
Riemannian curvature is expressed in terms of 2nd derivatives of the metric. Thus,
Hopf’s conjecture, as it is usually understood, is a statement about manifolds which
are at least twice differentiable. However, Alexandrov showed how one could speak
of nonpositive curvature for continuous, but nonsmooth, manifolds. Let M be a
complete Riemannian manifold. The Hopf-Rinow theorem states that for any two
points p and q in M , there exists a minimal geodesic from p to q (i.e. a curve γ
LECTURE 4. THE CHARNEY-DAVIS CONJECTURES 183
where the sum is over all faces f which contain v, and angle(f, v) ∈ [0, 1] denotes
the normalized interior angle of f at v, i.e. the usual angle (in radians) divided by
2π. Then one can check in a straightforward manner the following very classical
formula
(7) χ(M ) = k(v).
v
The relationship between the previous discussion and the current topic is provided
by the following lemma.
Lemma 65. A PE surface M is nonpositively curved if and only if k(v) ≤ 0 for
each vertex v.
The Charney-Davis conjecture, in the case of PE surfaces, follows immediately.
where {α(i) > v} denotes the set of i-dimensional cells of M which contain v, and
[v, α] denotes the normalized exterior angle of α at v. That is, [v, α] is the fraction
of the sphere consisting of outward pointing normals to supporting hyperplanes of
α at v. Equivalently, [v, α] is the fraction of linear functions on α which achieve
their maximum at v. Banchoff proved the following generalization of (7).
Theorem 66. If M is a polyhedron, then
(9) χ(M ) = k(v).
v
Recall that the local approach that was sufficient to prove Hopf’s theorem
in dimensions 2 and 4 is not sufficient in higher directions. Charney and Davis,
perhaps somewhat surprisingly, conjecture that the corresponding local approach
to their conjecture works in all dimensions.
Conjecture 67. Let M be a PE manifold of dimension 2n. If M is nonpositively
curved, then for every vertex of M
(−1)n k(v) ≥ 0.
The function k(v) can, in a straightforward way, be written in terms of the link
of v with its natural geometry as a complex of spherical cells. Let k denote this
function, so that
k(v) = k(link(v)).
The next step is to determine which simplicial complexes can arise as links of
vertices in nonpositively curved PE manifolds. Roughly speaking, a Riemannian
manifold has nonpositive curvature if and only if the boundary of each small metric
ball is larger, in some sense, than the corresponding Euclidean sphere. Something
similar is true for PE manifolds. That is, a PE manifold is nonpositively curved
if the link is larger, in a precise sense, than a standard sphere of radius 1. More
precisely, say that a complex L of spherical cells is large if for every pair of points
x and y in L, with dist(x, y) < π, there is a unique geodesic connecting them. The
following is a theorem of Gromov [48].
186 R. FORMAN, COMB. DIFFERENTIAL TOPOLOGY AND GEOMETRY
where fi (L) denotes the number of i-simplices in L. Gromov showed that there is
a simple combinatorial test for whether such a link is large.
Definition 70. Say that a simplicial complex L is flag if every clique spans a
simplex. That is, is v1 , v2 , . . . , vk are vertices in L, and they are all pairwise
adjacent, then they span a simplex.
Theorem 71. A cubical PE manifold is nonpositively curved if and only if the link
of every vertex is flag.
Thus, in this case, Conjecture 69 implies the following statement.
Conjecture 72. Let L be a simplicial complex which is homeomorphic to a sphere
of dimension 2n − 1. If L is flag, then (−1)n
k(L) ≥ 0, where
k(L) is given by the
formula (10
This conjecture is very combinatorial in nature, but still has one topological,
noncombinatorial, ingredient, namely the hypothesis that L be homeomorphic to a
sphere. There is a natural generalization of triangulated spheres which has a more
combinatorial flavor. A Gorenstein* complex (or a generalized homology sphere)
is a simplicial complex with the property that, for every p ≥ 0, the link of every
p-simplex has the homology of an (n − p − 1)-sphere. If L is a simplicial complex
which is homeomorphic to an n-sphere, or, more generally, any homology n-sphere,
then L is Gorenstein*. Thus, to place these ideas in a more combinatorial setting,
it is natural to consider the following generalization of Conjecture 72.
LECTURE 4. THE CHARNEY-DAVIS CONJECTURES 187
Let X be a finite CW complex, and let C p (X) denote the space of real-valued
p-cochains on X. Let
dp : C p (X) → C p+1 (X)
denote the usual coboundary operator. Then d2 = 0, and the singular cohomology
of X, H ∗ (X, R), is isomorphic to the cohomology of the cochain complex
d d
C ∗ (X) : 0 −−−−→ C 0 (X) −−−0−→ C 1 (X) −−−1−→ C 2 (X) −−−−→ · · ·
That is
Ker dp
H p (X, R) ∼
= .
Im dp−1
Now endow each C p (X) with a (positive definite) inner product by declaring the
canonical basis to be orthonormal. More explicitly, for each p, let Sp (X) denote
the set of p-cells in X. Choose an orientation for each element in Sp (X). Then for
α and β in C p (X) set
(11) α, β = α(y)β(y).
y∈Sp (x)
Cochains in the kernel of p are called harmonic. We will denote the space of
harmonic p-cochains in X by Hp (X).
So far, in this section, we have been considering the case of a finite CW complex.
How do things change in the case of an infinite complex? Of particular interest to
us is the case of the universal cover of a finite complex. With that in mind, let Y
denote an infinite CW complex that is the covering space of some finite complex.
Let us take a look at Hodge theory on Y . Let
d d
C ∗ (Y ) : 0 −−−−→ C 0 (Y ) −−−0−→ C 1 (Y ) −−−1−→ C 2 (Y ) −−−−→ · · ·
denote the cochain complex of Y . Hodge theory requires inner products. We quickly
realize that the standard formula (11) does not yield a well-defined inner product
in the infinite setting. There are various possible ways to proceed. However, if one
desires to work with Hilbert spaces, there is a natural choice. Let C2p (Y ) denote
the L2 p -cochains on Y . That is, if Sp (Y ) denotes the set of p-cells in Y , each
endowed with an orientation, then
C2p (Y ) = {α ∈ C p (Y ) s.t. (α(y))2 < ∞}.
y∈Sp (Y )
The next step is to replace the standard cochain complex on Y by the complex of
L2 cochains. To do this, one requires the following lemma.
Lemma 77. dp (C2p (Y )) ⊂ C2p+1 (Y ).
The proof is left as an exercise. (See Exercise 1 at the end of this lecture.)
Now consider the L2 cochain complex
d d
C2∗ (Y ) : 0 −−−−→ C20 (Y ) −−−0−→ C21 (Y ) −−−1−→ C22 (Y ) −−−−→ · · · .
One might wish to proceed by defining the L2 -cohomology of Y by the usual formula
Ker(dp : C2p (Y ) → C2p+1 (Y ))
Im(dp−1 : C2p−1 (Y ) → C2p (Y ))
This is certainly possible (this is called the unreduced L2 cohomology). However,
it does lead to certain difficulties, since Im dp−1 need not be a closed subspace of
Ker dp , and hence the quotient need not inherit the structure of a Hilbert space.
With that in mind, we define the L2 cohomology of Y to be
Ker(dp )
H2p (Y ) :=
Im(dp−1 )
where Im(dp−1 ) indicates that we take the closure of Im(dp−1 ) in Ker(dp ). (This
quotient is sometimes called the reduced L2 -cohomology.)
Following the same procedure as before, we can construct the adjoint operator
d∗ C2p (Y ) → C2p−1 (Y ) (see exercise 1 at the end of this section), and the Laplace
operator
p2 : C2p (Y ) −→ C2p (Y ).
Let H2p (Y ) denote the kernel of the operator p2 . The proof of Theorem 76, applied
in this setting, yields the following result.
Theorem 78. H2p (Y ) ∼ p
= H2 (Y ).
Now let X be a finite CW complex. We define the pth Betti number of X to
be the dimension of H p (X), and denote this number by bp (X). From Theorem 76
we know that
bp (X) = dim Hp (X).
Let π p : C p (x) → Hp (x) denote the orthogonal projection. Then we can also write
bp (X) = trace(π p ).
A useful way to calculate the trace of an operator is to express the operator as
a matrix with respect to some basis, and then to take the sum of the diagonal
elements of the matrix. Let us carry out this procedure here, and represent π p as
a matrix with respect to the standard basis for the cochain space. Let α1 , . . . , αk
denote an orthogonal basis for Hp (X) (so that k = bp (X)). Then we can write
k
πp = αi ⊗ α∗i
i=1
where α∗i : C (X) → R is the map that takes β to β, αi . The function
p
K p : Sp (X) × Sp (X) −→ R
192 R. FORMAN, COMB. DIFFERENTIAL TOPOLOGY AND GEOMETRY
given by
k
K p (x, y) = αi (x)αi (y)
i=1
is a matrix for the operator π p , in the sense that for any β ∈ C p (X), and any
x ∈ Sp (X),
⎛ ⎞
k
k
[π p (β)](x) = αi (x)β, αi = αi (x) ⎝ αi (y)β(y)⎠
i=1 i=1 y∈Sp (X)
k
= αi (x)αi (y) β(y) = K p (x, y)β(y).
y i=1 y
It follows that
bp (X) = trace(π p ) = K p (x, x)
x∈Sp (X)
(This identity can easily be proved directly, without the preceeding discussion.)
The function
k
K p (x, x) = α2i (x)
i=1
is sometimes called the pth local Betti number (as its integral gives the pth Betti
number). We chose an orthonormal basis for the space of harmonic cochains on
order to define this function, but one can easily check that it is independent of the
choices.
Now let us consider again the case of an infinite CW complex Y which has an
action by a group G, such that Y /G is finite. In this case, if H2p (Y ) = 0, then it is
necessarily infinite-dimensional. Still, much of the previous discussion makes sense
in this setting. That is, one can define the orthogonal projection
π p : C2p (Y ) → H2p (Y ).
While the trace of this operator is not defined, we can still construct a kernel
K p (x, y), x, y ∈ Sp (Y ), given by
K p (x, y) = αi (x)αi (y)
i
where {αi } is an orthonormal basis for Hp (Y ). Just as for the finite complex, we
can restrict this operator to the diagonal and consider the pth local Betti number
K (x, x) = i α2i (x), x ∈ Sp (Y ). Summing these entries, however, yields an infinite
p
result. At this point, we use the extra information we have, namely the fact that
everything is invariant under the action of the group G. Let Sp∗ (Y ) ⊂ Sp (Y ) denote
a set of p-cells of Y containing exactly one p-cell from each G-orbit in Sp (Y ). Then
Sp∗ (Y ) is a finite set, and the values K p (x, x) for x ∈ Sp∗ (Y ) determine K p (x, x) for
all x.
With that in mind, define the G-trace of π p , denoted by τG (π p ) to be the result
of summing K p (x, x) over x ∈ Sp∗ (Y ). That is
τG (π p ) = K p (x, x) ∈ [0, ∞).
x∈Sp∗ (Y )
LECTURE 5. FROM ANALYSIS TO COMBINATORICS 193
Now let us restrict attention to the case in which X is the universal cover of
a finite CW complex X, and we take the group G to be the fundamental group of
X, acting freely on X in the usual way. In this setting, Dodziuk proved that the
2
L -Betti numbers of X computed combinatorially from a cell decomposition are
equal to those calculated from the Riemannian Laplacian, and that these numbers
are homotopy invariants of X [23]. For our purposes, the main property of the
L2 -Betti numbers of X is the following result of Atiyah [4].
Theorem 79. Let X be the universal cover of a finite CW complex X, and take
the group G to be the fundamental group of X. Then
(12) χ(X) =
(−1)i bpG (X).
i
2
Thus, L -Betti numbers are another tool at our disposal for investigating the Euler
characteristic. It may not be clear how one could use this new information to inves-
tigate the Hopf-Charney-Davis conjectures, but a link is provided by the following
beautiful conjecture of Singer.
the universal
Conjecture 80. Let X be a compact aspherical n-manifold, and X
= 0.
cover of X. Then for all p = n/2, H2p (X)
Applying Theorem 79, we see that Singer’s conjecture immediately implies the
Hopf conjecture (as generalized by Thurston, Conjecture 58). While the Hopf con-
jecture is trivial for odd dimensional manifolds, Singer’s conjecture is not. Singer’s
conjecture can quite easily be shown to be true for surfaces (it follows from the
fact that there are no L2 harmonic functions on a complete Riemannian mani-
fold of infinite volume). It has also been shown to hold for 3-manifolds for which
Thurston’s geometrization conjecture is true [66], locally symmetric spaces [24],
negatively curved Kähler manifolds [49], manifolds with sufficiently pinched neg-
ative curvature [26], aspherical manifolds whose fundamental group contains an
infinite amenable normal subgroup [16], and manifolds which fiber over S 1 [67].
It is quite natural to guess that Singer’s conjecture also holds for suitable
piecewise Euclidean manifolds. The following conjecture, along with a series of
related conjectures, appears in Section 8 of [22].
Conjecture 81. Let X be a compact nonpositively curved P E manifold of dimen-
be the universal cover of X. Then for all p = n/2, Hp (X)
sion n, and let X = 0.
2
This conjecture implies the Charney-Davis conjecture 64. In [22], Davis and
Okun used this circle of ideas to establish Conjecture 72 for 3-dimensional flag
simplicial spheres (and somewhat more generally).
Theorem 82. Let L be any flag simplicial 3-sphere, then
k(L) ≥ 0, where
k is as
in (10).
194 R. FORMAN, COMB. DIFFERENTIAL TOPOLOGY AND GEOMETRY
Very roughly speaking, for any flag simplicial 3-sphere, a special nonpositively
curved 4-dimensional cubical PE manifold M is constructed (using the structure of
right-angled Coxeter groups) which has the properties that the link of every vertex
is identified with L, and Singer’s conjecture can be shown to hold for M . This is
a wonderful result, requiring a lot of hard work, and Davis and Okun introduce
some powerful new ideas into the subject. The reader is strongly encouraged to
consult their paper. Following the discussion in Section 1, Theorem 82 implies the
following general result.
Theorem 83. If X is any finite nonpositively curved cubical PE manifold of di-
mension 4, then χ(X) ≥ 0.
In this section we focus attention on Conjectures 72 and 73, and show that they
reside quite naturally in the well-developed circle of ideas surrounding the inves-
tigation of f -vectors of simplicial complexes. In (10) the formula for the relevant
function k(L) is given in terms of the f -numbers of L. In a number of settings,
especially those related to commutative algebra and toric varieties, it has proved
very useful to study certain special linear combinations of the f -numbers, called the
h-numbers. In [15] Charney and Davis observe that Conjectures 72 and 73 can be
restated in a very nice way in terms of the h-numbers. For any finite n-dimensional
simplicial complex K, define the f -polynomial of K to be the generating function
n+1
of the f -numbers. More explicitly, set f (K, t) = i=0 fi−1 (K)ti ,where we define
n+1
f−1 to be 1. Define the h-polynomial of K, h(K, t) = i=0 hi (K)ti , by the formula
t
(13) h(K, t) = (1 − t)n+1 f (K, ).
t−1
(We will often write h(t) for h(K, t) if it will not cause any confusion.) It follows
immediately from (10) and (13) that for any n-dimensional simplicial complex K,
h(K, −1) = 2n+1
k(K).
Hence, we can now restate Conjecture 73 as
Conjecture 84. If K is any simplicial Gorenstein* (2n − 1)-complex which is flag,
then (−1)n h(K, −1) ≥ 0.
In this form, the conjecture can more easily be compared to other conjectures
and results concerning the h-vectors of simplicial spheres and related spaces. One
advantage of the h-polynomial is that it is quite easy to state the Dehn-Somerville
relations. Say that K is Eulerian if the link of every i-simplex, i ≥ −1, has the
same Euler characteristic as a sphere of dimension n − i − 1, so that, in particular,
every Gorenstein* complex, and thus every triangulated sphere, is Eulerian. If K
is Eulerian, then
(14) h(t) = tn+1 h(t−1 ),
(equivalently, hi = hn+1−i for each i).
An n-dimensional simplicial complex is said to be Cohen-Macaulay if for every i
the link of every i-simplex has nonzero reduced homology only in dimension n−i−1.
So, for example, every Gorenstein* complex, and thus every triangulated sphere,
LECTURE 5. FROM ANALYSIS TO COMBINATORICS 195
These are the most general linear inequalities satisfied by the f -vectors of sim-
plicial polytopes. One of the central open problems in the study of f -vectors is
to determine precisely for which simplicial complexes the conclusion of the GLBT
holds. For example, does it hold for all Gorenstein* complexes (this is sometimes
called the Generalized Lower Bound Conjecture), or all triangulated spheres, or
all PL spheres? Independently, Kalai and Stanley have shown that the conclusion
holds for the boundary complex of any triangulated (n + 1)-ball which appears as a
subcomplex of a simplicial (n + 1)-polytope, but it is not clear which spheres arise
in this way. This unimodality property has some relation to the Charney-Davis
conjecture.
Theorem 88. Let K be a Gorenstein* complex of dimension 2n − 1. Suppose that
h(K, t) only real roots. Then the following two conclusions hold.
(1) the h-numbers of K are unimodal, i.e. the conclusion of the GLBT holds;
(2) (−1)n h(K, −1) ≥ 0.
While we have stated this result in terms of h-polynomials of simplicial complexes,
this theorem is really just a statement about polynomials with real coefficients
satisfying a symmetry relation as in (14). Part two of this theorem is due to Charney
and Davis (see Lemma 7.5 of [15]). In fact, they prove the stronger statement that
the conclusion holds as long as h(K, t) has no nonreal roots of modulus 1. The first
part of this theorem is due to Isaac Newton!
With this theorem in mind, it is natural to make the following Real Root Con-
jecture (apparently due originally to Januzkiewicz, see [20]).
Conjecture 89. For any Gorenstein* complex K which is flag, h(K, t) has only
real roots.
In [75], Reiner and Welker consider these questions for the order complex of a
graded poset P . This special case of the real root conjecture was formulated earlier,
and is known as the Neggers-Stanley conjecture. Without proving the Neggers-
Stanley conjecture, they are able to prove the implications of this conjecture. More
precisely, they construct a simplicial polytope with the same h-polynomial as the
order complex of P , and thus the unimodality of the h numbers of the order complex
follows from Theorem 87. By other means (using the results of [61]) they establish
the Charney-Davis conjecture for KP for graded posets of width 2.
More recently, Brändén [12] has proved the Charney-Davis conjecture for KP ,
as well as the unimodality of the h-numbers, for any graded poset P . That is, he
establishes both conclusions of Theorem 88 for such complexes. He does not do this
by proving that the h-polynomial has real roots, however. Let us take a moment to
discuss Brändén’s approach, an approach that was also presented, independently,
in the recent work of Gal [43]. We know from the Dehn-Sommerville relations (14)
that the h-polynomial of any Eulerian complex of dimension 2n − 1 satisfies the
symmetry
pi (t) = ti (1 + t)2n−2i , i = 0, 1, 2, . . . , n
form a basis for the vector space of polynomials of degree 2n with this symmetry.
Thus, for any Eulerian complex K of dimension 2n we can write
n
(15) h(K, t) = ai (K)pi (t).
i=0
and the ai (K)’s are uniquely determined by this identity. We can make two simple
observations. First, we see that
1 = h(K, 0) = a0 (K).
Second, we see that
Proof. Let γ(K, t) denote the generating function of the ai ’s. That is
n
γ(K, t) = ai (K)ti .
i=0
where :p
C2p (Y)→ C2p (Y
) is the Laplace operator on Y . Show that
(a) I(0) is the left hand side of (12) and limt→∞ I(t) is the right hand
side of (12).
(b) d/dt I(t) = 0 for all t ∈ [0, ∞).
(4) Show that if Y is an infinite polyhedron, then H20 (Y ) = 0.
(5) Let Y denote the real line given the structure of an infinite polyhedron
by placing a vertex at each integer point. What is the (reduced) L2 -
cohomology of Y ? What is the unreduced L2 -cohomology of Y ?
Exercises for Section 2 of Lecture 5.
(1) The best exercise is to calculate f (t), h(t), and γ(t) for your favorite
Eulerian complexes. Start with simple complexes, and then keep going.
(2) If you have never done this before: Prove the Dehn-Somerville relations
(14) for any Eulerian complex.
(3) Prove identity (17).
(4) Find explicit formulas for the first few coefficients of γ(K, t) in terms of
the f -vector of K.
(5) Show that the coefficient of t in γ(K, t) is always ≥ 0 for a Gorenstein*
complex that is flag.
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hamiltonian graphs, J. Combin. Theory, Ser. A., 104 (2003), pp. 169–199.
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bin., 12 (2005), pp. 211–239.
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Workshop on Graphs and Combinatorial Optimization (electronic), Electron.
Notes Discrete Math., 17. Elsevier, Amsterdam, (2004), pp. 191–195.
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Combinatorica, 4 (1984), pp. 297–306.
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conjecture, J. Combin. Theory, Ser. B, 28 (1980), pp. 85–95.
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of the Amer. Math. Soc., 128 (2000), pp. 2253–2259.
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resonances, Israel J. of Math., 132 (2002), 189–206.
204 R. FORMAN, COMB. DIFFERENTIAL TOPOLOGY AND GEOMETRY
Introduction
The aim of these lectures was to give an overview of some combinatorial, symmetric-
function theoretic, and representation-theoretic developments during the last sev-
eral years in the theory of Hall-Littlewood and Macdonald polynomials. The
motivating problem for all these developments was Macdonald’s 1988 positivity
conjecture [20, 21]. The positivity conjecture asserts that certain polynomials
have non-negative integer coefficients, and so it naturally raised the question of
how to understand Macdonald polynomials combinatorially. This question remains
open, even after the proof of the positivity conjecture in [16], using methods from
algebraic geometry. The latest developments, which will be discussed at the end
of these notes, for the first time promise progress on the combinatorial side of the
problem.
The lectures start with basics and proceed towards a discussion of the most
recent combinatorial advances. Along the way, I have taken as my central topic
the q and q, t-analogs of classical combinatorial themes such as Catalan numbers,
enumeration of trees and parking functions, and Lagrange inversion. The surprising
connection between these themes and the theory of Macdonald polynomials was one
of the most beautiful discoveries to emerge from work on the positivity conjecture.
This topic also serves nicely to motivate the combinatorial conjectures discussed in
the final lecture.
The subject as a whole has grown far beyond what can be covered in a series
of introductory lectures. Omitted entirely are the algebraic geometrical aspects
[2, 14, 16, 17]. Also omitted is a treatment of the full list of other quantities, not
quite so immediately connected with classical enumeration, which are also expressed
by formulas involving Macdonald polynomials, and are known or conjectured to be
Schur-positive, for which combinatorial interpretations are still sought [1]. Yet
another direction not touched on here is the link with representation theory of
1 Dept.
of Mathematics, University of California, Berkeley, CA.
E-mail address: mhaiman@math.berkeley.edu, awoo@math.berkeley.edu.
Work supported in part by NSF grant DMS-0301072 (M.H.).
c
2007 American Mathematical Society
209
210 HAIMAN AND WOO, GEOMETRY OF q AND q, t-ANALOGS
Cherednik algebras and their degenerations [4, 5, 10, 11]. A more advanced but
less up-to-date survey of some of these topics can be found in [18].
My heartfelt thanks go to Alexander Woo, who conducted discussion and ex-
ercise sessions associated with the lectures and did most of the work in preparing
these notes. In the process he greatly improved the exposition, worked out missing
details, and took pains to clarify those points which proved most troublesome for
students in the discussion sections. Credit for whatever good qualities the following
notes may possess is mostly due to him.
–M.H.
LECTURE 1
Kostka Numbers and q-Analogs
211
212 HAIMAN AND WOO, GEOMETRY OF q AND q, t-ANALOGS
3 5
2 2 4 4
1 1 1 2 3
where the sum is taken over all partitions μ, or equivalently all partitions μ of
size |λ|.
1.3. Sn Representations
Let V be a finite dimensional Sn representation, that is, a finite dimensional C-
vector space with a linear action by Sn . For any partition μ of n, there is the Young
subgroup Sμ = Sμ1 × Sμ2 × · · ·× Sμk ⊆ Sn , where the Sμ1 factor permutes the first
μ1 letters, the Sμ2 factor permutes the μ1 + 1-th through μ1 + μ2 -th letters, and so
on. Now let V Sμ denote the subspace of V fixed by every element of Sμ . Then there
LECTURE 1. KOSTKA NUMBERS AND q-ANALOGS 213
This can be proven by identifying C·Wμ with the induced representation C ↑SSnμ
and using Frobenius reciprocity.
214 HAIMAN AND WOO, GEOMETRY OF q AND q, t-ANALOGS
The most basic examples are λ = (k), in which case tr(Symk V, g(x)) = hk (x),
k
and λ = (1k ) for k ≤ n, for which tr( V, g(x)) = ek (x).
LECTURE 1. KOSTKA NUMBERS AND q-ANALOGS 215
with the map π being the projection onto the first factor.
Now we show that Z is smooth. Let ψ : Z → G/B be the projection onto
the second factor, and let E• be the standard flag, that is, the flag with Ei =
C · {e1 , . . . , ei }, where {e1 , . . . , en } is the standard basis of Cn . The fiber ψ −1 (E
•)
is given by ψ −1 (E• ) = {(X, E• ) : X is upper triangular}, so ψ −1 (E• ) is a n2 -
dimensionalvector space. Moreover, for any flag F• , F• = gE• for some g ∈ G, and
−1 −1
ψn (F• ) = (gXg , F• ) : X is upper triangular , also a vector space of dimension
2 . This makes Z into a vector bundle over G/B; since G/B is smooth, Z must
also be smooth. (Technically we also need to check that Z is locally trivial over
G/B, but this is also easy to check using the group action.)
The map π is projective because G/B is a projective variety. Also, for any X
whose Jordan form has only one Jordan block, π −1 (X) consists of a single flag, so,
as these matrices X form an open dense subset of N , π is birational.
Now let G act on N by conjugation; that is, g · X := gXg −1 for g ∈ G and
X ∈ N . Let μ be a partition. Let Mμ be the nilpotent Jordan matrix with Jordan
blocks of size μ1 , μ2 , · · · , μk , and Oμ = GLn · Mμ . These orbits cover all of N , since
every matrix has a Jordan form and we can conjugate by permutation matrices
to rearrange the Jordan blocks so that their sizes are in non-increasing order. We
have a corresponding action on Z by g · (X, F• ) := (gXg −1 , gF• ), so the fibers of π
over points in the same G orbit are isomorphic. Let Yμ = π −1 (P ) for some point
P ∈ Oμ . (We will only be interested in isomorphism invariants of Yμ , so the choice
of point is irrelevant.)
216 HAIMAN AND WOO, GEOMETRY OF q AND q, t-ANALOGS
For example, for μ = (n), Y(n) is a single point, as already stated above. At
the other extreme, when X is the zero matrix, (X, F• ) ∈ Z for every F• , so for
μ = (1n ), Y(1n ) ∼
= G/B.
The following theorem allows what we will consider the definition of Kλμ (q).
Theorem 3.
(1) The natural map H ∗ (G/B, C) → H ∗ (Yμ , C) is surjective.
(2) There are geometrically defined Sn actions on H ∗ (G/B, C) and H ∗ (Yμ , C)
such that the above map is Sn -equivariant.
(3) H ∗ (Yμ , C) ∼
= C · Wμ ∼
⊕K
= λ Vλ λμ as Sn -representations.
λμ (q) by
Now we define K
λμ (q) =
K
(i)
Kλμ q i ,
i
(i)
where Kλμ is defined by
⊕Kλμ
(i)
H 2i (Yμ , C) ∼
= Sn Vλ .
1.6. Exercises
(1) Prove Lemma 1.
(2) Define hμ (x) := hμ1 (x)hμ2 (x) · · · hμl(μ) (x). Show that FC·Wμ (x) = hμ (x).
Deduce that hμ (x) = λ Kλμ sλ .
(3) Find a basis and weight space decomposition of V λ (the GLn representa-
tion) for λ = (2, 1k−2 ).
(4) Let V = Cn = C·{e1 , . . . , en } be the defining representation of Sn , that is,
with the action w · ei = ew(i) . Decompose V into irreducibles and FV (x)
into Schur functions, corresponding to your decomposition of V .
LECTURE 2
Catalan Numbers, Trees, Lagrange Inversion, and their
q-Analogs
217
218 HAIMAN AND WOO, GEOMETRY OF q AND q, t-ANALOGS
paths from (0, n) to (n, 0), or equivalently the number of partitions inside δ(n). We
have C0 = C1 = 1, C2 = 2, and C3 = 5, as demonstrated by the Figure 2.
As is frequently useful in combinatorics, we can try to calculate or get a formula
for Cn by using a generating
function. In this case, this means a power series
C(x) defined by C(x) := n Cn xn .
Given a proper parentheses string, the initial “(” matches with some “)”, and
between those parentheses is a proper parenthesization of some length k, while after
the specified “)” is another proper parentheses string of length n − 1 − k. In other
words, a non-empty proper parentheses string looks like (A)B, where A and B are
respectively parentheses strings of length k and n − 1 − k. Therefore, the Catalan
n−1
numbers satisfy the recurrence Cn = k=0 Ck Cn−1−k . In terms of the generating
function, we have
C(x) = 1 + xC(x)2 .
We can get a formula for Cn by solving for C(x) and using the binomial theorem,
but we will instead get one by using Lagrange inversion later in this lecture. For
now, note that our equation can be rewritten as
xC(x)(1 − xC(x)) = x,
or equivalently F1 (x) ◦ (xC(x)) = x, where ◦ denotes functional composition and
F1 (x) = x(1 − x). In other words, F1 (x) and xC(x) are compositional inverses.
10 5 8
11
1 2 7
6 4 3 9
relate the generating series for the number of rooted trees and the number of rooted
forests, so
that, if hn = tn+1 /(n + 1) is the number of rooted labelled forests and
H(x) = n hn xn /n!, we have H(x) = eT (x) . Therefore,
tn+1 xn xn T (x)
eT (x) = = tn+1 = .
n
n + 1 n! n
(n + 1)! x
n = 7, k=3
qk
q (2)−|ν|
k
111
000
000
111
λn−k+1 = k + 1
ν 000
111
111
000
000
11100
11
00
11
q(
n−1−k
2 )−|ρ|
00000
11111
ρ
It turns out to be slightly easier to solve for hn , the number of rooted forests.
If we let E(x) = ex , then x/E(x) = F2 (x), and
x
◦ (xH(x)) = x.
E(x)
Once again applying Lagrange inversion,
hn 1 1 (n + 1)n
= [xn ] e(n+1)x = ,
n! n+1 n+1 n!
so hn = (n + 1)n−1 , and tn = nn−1 .
2.4. q-Analogs
The Catalan numbers have two q-analogs, but we will only be concerned with the
n
one originally defined by Carlitz and Riordan [3], namely Cn (q) = λ⊆δ(n) q ( 2 )−|λ| .
This q-analog satisfies a recurrence as follows. We can separate all partitions λ ⊆
(k) (k)
δ(n) into classes Cn for 0 ≤ k < n − 1 by putting λ in Cn if k is the smallest
number such that λn−k−1 = k + 1 = δ(n)n−k−1 , and k = n − 1 if no such number
exists. For example, as illustrated in Figure 4, the partition (6, 4, 4, 1) ⊆ δ(7)
(3) (k)
belongs in C7 . Now, for λ ∈ Cn , let ν be the partition defined by νi = λn−k−1+i ,
and let ρ be the partition defined by ρi = λi − k − 1, for i, 1 ≤ i ≤ n − k − 1, as
illustrated in Figure 4. Notice that ν ⊆ δk , and ρ ⊆ δn−1−k . Furthermore,
n k n−1−k
− |λ| = (k + − |ν|) + ( − |ρ|),
2 2 2
so we have the recurrence
n−1
Cn (q) = q k Ck (q)Cn−1−k (q).
k=0
6
4
2
3
5
1
Figure 5. Tableau associated with the parking function f (2) = f (4) = f (6) =
1, f (3) = 3, f (1) = f (5) = 4
for us to define this q-analog later using parking functions. A parking function
is a function f : {1, . . . , n} → {1, . . . , n} such that #f −1 ({1, . . . , k}) ≥ k for all
k ∈ {1, . . . , n}. (The name comes from the following description. Suppose we have
n parking spaces on a one way street, labelled in order, and n cars. The cars arrive
at the street in order, and each car k immediately goes to its preferred parking
space f (k). If it is already filled by a previous car, then it keeps going and parks in
the first empty space. The condition above is then satisfied if and only if every car
successfully finds a parking space without having to enter the street a second time.)
Denote the set of parking functions for n cars by PF(n) The symmetric group Sn
acts on PF(n) by w · f = f ◦ w−1 for w ∈ Sn and f ∈ PF(n).
We can represent a parking function by a tableaux of skew shape (λ + (1n ))/λ
for some partition λ, that is a filling of the boxes in λ + (1n ) but not in λ, strictly
increasing in columns and weakly increasing in rows (although in this case there
are no relevant rows) as usual. Let f be f sorted into non-increasing order; in other
words, we want f = w ·f for some w such that f(i) ≥ f(i+1) for all i, 1 ≤ i ≤ n−1.
Now we specify λ by requiring λi = f(i)−1. Note that the requirement that f (or f)
be a parking function is equivalent to requiring that λ ⊆ δ(n). Now the j-th column
in (λ + (1n ))/λ will have f −1 (j) many open boxes to fill, and we fill them with the
elements of f −1 (j) in increasing order. Figure 5 shows the tableau associated with
the parking function f (2) = f (4) = f (6) = 1, f (3) = 3, f (1) = f (5) = 4. The
n
content of this tableaux
is
always (1 ).
Note that 2 − |λ| = i=1 i − ni=1 f (i), and we will denote this quantity as
n n
wt(f ). (This quantity is sometimes known as the “frustration factor” of the parking
function since it counts the sum total of how far drivers park from their preferred
space.) Let Pn (q) := f ∈PF(n) q wt(f ) . Counting parking functions according to the
partition representing them, we get that
n n
Pn (q) = q ( 2 )−|λ| ,
α0 , α1 , · · · , αn−1
λ⊆δ(n)
Then apply the equation with Φ(x) = F (q −2 x)F (q −1 x), and the last equation, to
get
(F (q −2 x)F (q −1 x)F (x)) ◦q G(x) = x3 .
By induction, we have
(F (q −(n−1) x) · · · F (q −1 x)F (x)) ◦q G(x) = xn .
Therefore, for any power series Ψ(x) = n ψn xn ,
(Ψ(x) ◦q−1 F (x)) ◦q G(x) = ψn (F (q −(n−1) x) · · · F (qx)F (x)) ◦q G(x)
n
= ψn xn
n
= Ψ(x),
as desired.
For usual functional composition, it turned out that it was easier to get the
explicit Lagrange inversion formula for the modified form
x
◦ xK(x) = x,
E(x)
or equivalently,
K(x) = E(x) ◦ xK(x),
was easier to solve for the coefficients. (The equivalence is obvious once one stops
using the ◦ notation.) Similarly, for q-composition, it is easier to state the q-
Lagrange inversion formula for the following forms, whose equivalence is left as a
(not so trivial) exercise.
Proposition 2.
x
◦q xK(qx) = x
E(x)
if and only if
K(x) = E(x) ◦q xK(x).
Now we are ready state the q-Lagrange inversion formula. It will not have a
simple algebraic form, but will instead be a combinatorial sum that relates to the
q-analogs described in section 2.4.
Theorem 6. Let E(x) = n en xn and K(x) = n kn (q)xn be power series, with
e0 = k0 (q) = 1. Then
x
◦q (xK(qx)) = x
E(x)
if and only if
n
kn (q) = q ( 2 )−|λ| eα0 (λ) eα1 (λ) · · · eαn−1 (λ) ,
λ∈δ(n)
n−1
where αi (λ) is the number of parts of λ having size i, and α0 = n − i=1 αi . (For
example, if n = 4 and λ = (3, 1, 1), then α1 = 2, α0 = α3 = 1, and α2 = 0.)
224 HAIMAN AND WOO, GEOMETRY OF q AND q, t-ANALOGS
Proof. By Proposition 2,
x
◦q xK(qx) = x
E(x)
if and only if
kn (q) = [xn ] 1 + er q r−1 xK(q r−1 x) · · · qxK(qx)xK(x)
r>0
r
n
= [x ] er q (2) xr K(q r−1 x) · · · K(qx)K(x)
r>0
r
= er q (2) xn−r K(q r−1 x) · · · K(qx)K(x)
r>0
r
= er q (2) [xm
i ] K(q
r−i
x)
r>0 m1 +···+mr =n−r i
r
= er q (2) q (r−i)mi kmi (q)
r>0 m1 +···+mr =n−r i
= er q i (mi +1)(r−i) kmi (q)
r>0 m1 +···+mr =n−r i
It is clear that this recurrence has a unique solution (given the initial condition
k0 (q) = 1), so we need to show that
n
kn (q) = q ( 2 )−|λ| eα0 (λ) eα1 (λ) · · · eαn−1 (λ)
λ∈δ(n)
n = 14
q (m1 +1)(r−1)
r=6
q(
m1
2 )−|ν (1) |
q (m3 +1)(r−3)
m1
q ( 2 )−|ν |
11
00
m3 (3)
ν (1)
(m2 = 0)
11
00 q (m4 +1)(r−4)
111
000
111
000
m3
q (m5 +1)(r−5)
m4
(m5 = 0)
111
000
m6
ν (3)
Figure 6. The q-Lagrange inversion recurrence illustrated for λ = (13, 10, 7, 7, 6, 2, 2, 1).
2.6. Exercises
(1) Prove Lemma 2.
(2) Prove Proposition 2.
(3) Use Theorem 6 to prove Theorem 4 by setting q = 1. (Hint: First
show that, if (α0 , . . . , αn ) ∈ N satisfy α0 + · · · + αn = n, the sequence
(α0 , . . . , αn ) has a unique rotation (β0 , . . . , βn ) such that there is a parti-
tion λ ⊆ δ(n) with αi (λ) = βi for all i.)
(4) Prove directly that there are (n + 1)n−1 parking functions on {1, . . . , n}.
(5) Let Sn act on PF(n) as previously stated, and view C · PF(n) as an Sn
representation graded by wt(f ). Show that C · PF(n) is a direct sum of
induced representations C ↑SSnμ (which are respectively isomorphic to the
representations C · Wμ introduced in Lecture 1) in which the generating
function for the multiplicity of C ↑SSnμ in the graded degrees is equal to the
coefficient of eμ1 · · · eμk in kn (q).
LECTURE 3
Macdonald Polynomials
The Macdonald polynomials are a basis for the ring of symmetric functions over
the base field Q(q, t). This basis has a number of useful and interesting properties,
but, unfortunately, the polynomials are difficult to write out explicitly; indeed we
will only have space to give an abstract definition and a number of their important
properties, mostly without proof. These statements will require some notation and
machinery, as well as motivation, from the general theory of symmetric functions,
which we will now proceed to explain in the first part of this lecture.
Throughout this lecture, Λk denotes the ring of symmetric functions over the
base field (or occasionally base ring) k.
Proposition 3. Two bases {uλ } and {vλ } are dual (with respect to the Hall inner
product) if and only if Ω[XY ] = λ uλ [X]vλ [Y ].
LECTURE 3. MACDONALD POLYNOMIALS 229
Proof. First note that Ω[XY ] = i,j 1/(1 − xi yj ) = i Ω[xi Y ] = i n xni hn [Y ] =
λ mλ [X]hλ [Y ], and, by symmetry, Ω[XY ] = λ hλ [X]mλ [Y ]. (This is known as
the first Cauchy formula.)
Let ·, ·x denote the Hall inner product with respect to the x variables only.
Then mλ [X], Ω[XY ]x = mλ [X], hλ [X]mλ [Y ]x = mλ [Y ], so linearity implies
If Ω[XY ] = λ uλ [X]vλ [Y ], then {uλ } and {vλ }
f [X], Ω[XY ]x = f [Y ] for all f .
are dual bases, because vλ [X], λ uλ [X]vλ [Y ]x = vλ [Y ]. Since the Hall inner
product is non-degenerate, the only way to have vλ [X], g[XY ]x = vλ [X] for all λ
is to have g = Ω, which proves the reverse direction.
It can be shown, for example
by using the Robinson-Schensted-Knuth corre-
spondence, that Ω[XY ] = λ sλ [X]sλ [Y ], so the Schur functions are an orthonor-
mal basis for ΛQ under this inner product. Therefore, in terms of the representation
theory of Sn , we therefore have that dim(HomSn (V, W )) = FV , FW for any two
representations V and W .
Finally, note that ω is an isometry with respect to the Hall inner product. In
other words, ωf, ωg = f, g for any symmetric functions f and g.
μ (x; q, t) charac-
Theorem-Definition 2. The ring ΛQ(q,t) has a unique basis H
terized by
230 HAIMAN AND WOO, GEOMETRY OF q AND q, t-ANALOGS
moves to the right and powers of t increase as one moves up.) Therefore, eigenfunc-
tions for Δ must satisfy (1). Furthermore, Δ and Bλ are symmetric with respect to
simultaneously exchanging q and t and exchanging μ and μ , so these eigenfunctions
must satisfy (2). Condition (3) is just a scalar normalization factor. Note that, in
particular,
ΔH μ (x; q, t) = Bμ (q, t)H μ (x; q, t).
Some properties of Macdonald polynomials are easy to see from the definition
and theorem. First, H μ (x; 0, t) = H μ (x; t), and K λμ (0, t) = K λμ (t). In other
words, setting q = 0 in a Macdonald polynomial recovers the corresponding Hall-
Littlewood polynomial. Also, the definition looks the same when we both swap q
and t and swap μ and μ , so by uniqueness, H μ (x; q, t) = H μ (x; t, q). In particular,
if μ = μ , then H μ (x; q, t) is symmetric under switching q and t.
From the definition it is possible to compute H (n) (x; q, t). Every partition
n
dominates (1 ) = (n) , so the second condition is vacuous. The first condition
states that H (n) [X(1 − q); q, t] = f hn (x) for some f ∈ Q(q, t), or, equivalently, that
H (n) (x; q, t) = f hn [X/(1 − q); q, t] for some f . Now we use the third condition to
solve for f ; namely f = H (n) [1; q, t]/hn [1/(1 − q)]. Note that
1
hn [1/(1−q)] = hn (1, q, q 2 , . . .) = q |λ| = q |λ| = .
(1 − q)(1 − q 2 ) · · · (1 − q n )
l(λ)≤n λ1 ≤n
Therefore,
(n) (x; q, t) = (1 − q) · · · (1 − q n )hn X
H .
1−q
Next we compute H μ (x; q, 1) for all μ. First, note that Δ |t=1 is a derivation
on ΛQ(q) ; that is, for any f, g ∈ ΛQ(q) , Δ(f g) |t=1 = f (Δ(g) |t=1 ) + (Δ(f ) |t=1 )g.
Since Δ |t=1 is linear on ΛQ(q) , this statement can be proven by showing that it
holds when f = pμ and g = pν , and this is left as an exercise. Now note that
(n) (x; q, t) = H
H (n) (x; q, 1), so we have that
2
t
1
t
+
0
t
0 1 2
q q q
we get that
μ (x; 1, 1) =
H (μ ) (x; 1, 1) = h(1|μ| ) (x).
H i
i
Note in particular this does not depend on the partition μ as long as |μ| = n.
00
11
00
11
00
11
00
11
l(c)
0000000
1111111
00
11
0000000
1111111
c
a(c)
We can use this formula to derive K λμ when λ a hook shape, that is, if λ = (n−r, 1r )
for some r. Specifically,
K (n−r,1r ),μ = er [Bμ − 1].
Finally, we describe a q, t-analog of the Hall inner product and give a corre-
sponding Cauchy formula for Macdonald polynomials. Define
where the inner product on the right is the usual Hall inner product (with respect
to the x variables). Then H μ (x; q, t)∗ = H
λ (x; q, t), H λ [X(1−q); q, t], ω H μ [X(1−
t); q, t], and, expanding both parts of the inner product in terms of the orthonormal
basis of Schur functions, we see that H μ (x; q, t)∗ = 0 iff {ν : ν ≥
λ (x; q, t), H
λ} ∩ {ν : ν ≥ μ } = ∅ iff λ ≤ μ. By symmetry of the inner product (which follows
from ω being an isometry and Π(1−q) and Π1/(1−q) being adjoint), we also have
λ ≥ μ, so H μ (x; q, t)∗ = 0 if λ = μ.
λ (x; q, t), H
Let c be a cell in the diagram of some partition λ. The arm and leg of c,
respectively denoted a(c) and l(c), are the number of boxes strictly to the right of,
and respectively the number of boxes strictly above, the box c in the diagram of λ,
as illustrated in Figure 2. It turns out that
H μ (x; q, t)∗ = tn(μ) q n(μ )
μ (x; q, t), H (1 − tl(c)+1 q −a(c) )(1 − t−l(c) q a(c)+1 ).
c∈μ
Therefore, we have
t−n(μ) q −n(μ ) H
μ [X(1 − q); q, t]ω H μ [Y (1 − t); q, t]
Ω[XY ] = −a(c)
,
μ c∈μ (1 − t
l(c)+1 q )(1 − t−l(c) q a(c)+1 )
or, after substituting X/(1 − q) for X and −Y /(1 − t) for Y , taking the degree n
piece, and multiplying both sides by (−1)n ,
μ (x; q, t)H
μ (y; q, t)
XY t−n(μ) q −n(μ ) H
en = .
(1 − q)(1 − t) c∈μ (1 − t
l(c)+1 q −a(c) )(1 − t−l(c) q a(c)+1 )
|μ|=n
234 HAIMAN AND WOO, GEOMETRY OF q AND q, t-ANALOGS
3.7. Exercises
(1) Let X = x1 + x2 + · · · and Y = y1 + y2 + · · · . Express en [X − Y ] in terms
of symmetric functions separately in the x and y variables.
(2) Show that the graded Frobenius series of C[x1 , · · · , xn ] as an Sn repre-
C[x1 ,··· ,xn ]d (where C[x1 , · · · , xn ]d denotes the
d
sentation, that is, d t F
polynomials of degree d), is hn [X/(1 − t)].
(3) Prove Proposition 5.
(4) Show that Δ |t=1 is a derivation on ΛQ(q) by showing that
Δ(pμ pν ) |t=1 = Δ(pμ ) |t=1 pν + pμ Δ(pν ) |t=1 .
(5) Show that
μ (x; q, t) = tn(μ) q n(μ ) H
ωH
μ (x; q −1 , t−1 ).
(You will need to use the Macdonald specialization formula.)
λμ (q, t) = er [Bμ −
(6) Use the Macdonald specialization formula to show that K
1] when λ = (n − r, 1r ).
(7) (a) Prove that for any expression A
en [(1 − u)A]
|u=1 = (−1)n−1 pn [A].
1−u
(b) For the Macdonald operator Δ, show that
X en [X]
Δ (−1)n−1 pn = .
(1 − q)(1 − t) (1 − q)(1 − t)
(c) Let Πμ (q, t) = (i,j)∈μ\(0,0) (1 − q j ti ). Now use parts (a) and (b), the
Macdonald specialization, and the Cauchy formula to prove that
t−n(μ) q −n(μ ) (1 − q)(1 − t)Πμ (q, t)Bμ (q, t)H μ (x; q, t)
en (x) = −a(c) −l(c)
.
c∈μ (1 − t )(1 − t
l(c)+1 q q a(c)+1 )
|μ=n|
LECTURE 4
Connecting Macdonald Polynomials and q-Lagrange
Inversion; (q, t)-Analogs
235
236 HAIMAN AND WOO, GEOMETRY OF q AND q, t-ANALOGS
where the third equality comes from the Macdonald specialization formula as dis-
cussed in Lecture 3. Therefore,
tn(μ) q n(μ ) (1 − q)(1 − t)Πμ (q, t)Bμ (q, t)
∇en , en = l(c)+1 q −a(c) )(1 − t−l(c) q a(c)+1 )
.
|μ=n| c∈μ (1 − t
Define Cn (q, t) to be this rational function ∇en , en . It turns out that Cn (q, t)
is a polynomial with positive integer coefficients, and that Cn (q, 1) = Cn (q), the
q-analog of the Catalan numbers discussed in Lecture 2. Furthermore, Cn (q, t)
is symmetric under exchanging q and t; that is, Cn (q, t) = Cn (t, q). For example,
C3 (q, t) = q 3 +q 2 t+qt+qt2 +t3 , and specializing to t = 1 gives C3 (q) = q 3 +q 2 +2q+1
which is what we had earlier. Therefore, it makes sense to think of Cn (q, t) as a
(q, t)-analog of the Catalan numbers.
Now
notice that Cn (q) = kn (q)|ek →1 , as we saw at the end of Lecture 2. Since
hn = |μ|=n mμ and {hμ } and {mμ } are dual bases, hμ , hn = 1 for all μ, and con-
sequently, since ω is an isometry with respect to the Hall inner product, eμ , en = 1
for all μ. Therefore, if we pretend that the ek in kn (q) actually stand for elementary
symmetric functions, then Cn (q) = kn (q), en .
Comparing the equations Cn (q) = kn (q), en and Cn (q, t) = ∇en , en hints
at a possible connection between kn (q) and ∇en . It turns out that there is indeed
a connection given by the following theorem, which we will spend most of the
remainder of this lecture proving.
Theorem 7. Interpreting the ek in kn (q) as elementary symmetric functions, we
have that
∇en |t=1 = kn (q).
Before we go into the proof, let us mention two corollaries giving (q, t)-analogs
of our main examples from Lecture 2. The first corollary
follows from the discussion
n
above. To prove the second, recall that h(1n ) = |μ|=n μ1 ,...,μ mμ , so eμ , e(1n ) =
n l
μ1 ,...,μl .
n 1
= g n z n q −( 2 )
n
E[z(1 − q )X/(1 − q −1 )]
−n
n E[zq −n X/(1 − q −1 )]
= g n z n q −( 2 ) .
n
E[zX/(1 − q −1 )]
Hence
n q −n zX zX
(1) g n z n q −( 2 ) E = zE .
n
1 − q −1 1 − q −1
n
For any series Ψ(z) = n Ψn z n , define ∨Ψ(z) = n Ψn q ( 2 ) z n = Ψ(z) ◦q z.
Now we need a lemma about the behavior of ∨.
Lemma 3. We have the identities
n
(1) ∨(z n q −( 2 ) Ψ(q −n z)) = z n∨Ψ(qz)
(2) ∨(zΨ(z)) = z ∨Ψ(qz).
Proof. By linearity, it suffices to prove this for Ψ(z) = z r (for all r) in both cases.
We see that
n n+r n
∨
(z n q −( 2 ) (q −n z)r ) = q ( 2 ) q −( 2 )−nr z n+r
r
= q (2) z n+r
= z n∨z r .
Also,
r+1
∨
(z r+1 ) = q( 2 ) z r+1
r
= zq (2) q r z r
= z ∨((qz)r ).
Apply the operator ∨ to both sides of equation 1. Using the first part of the
lemma on the left hand side and the second part on the right hand side, we get
n∨ zX ∨ qzX
gn z E −1
=z E .
n
1 − q 1 − q −1
Hence,
z ∨E qzX/(1 − q −1 )
n
zK(qz) = G(z) = gn z = ∨ ,
n
E [zX/(1 − q −1 )]
238 HAIMAN AND WOO, GEOMETRY OF q AND q, t-ANALOGS
Equivalently, we could also define fμ as the dual basis to {eμ } under the Hall inner
product, or by letting fμ := ωmμ . Secondly, we introduce a fictitious alphabet A
such that
n
hn [A] := q ( 2 ) hn [X/(1 − q)].
Now we produce the following identity to simplify our expression for K(z):
−1
∨ q zX ∨ −zX
E −1
= E
1−q 1−q
∨
= (−1)n ωen [X/(1 − q)]z n
n
∨
= hn [X/(1 − q)](−z)n
n
n
= q ( 2 ) hn [X/(1 − q)](−z)n
n
= hn [A](−z)n = en [−zA] = E[−zA] = 1/E[zA].
n n n
i i
LECTURE 4. MACDONALD POLYNOMIALS AND LAGRANGE INVERSION 239
3
t
2
t
1
t
+
0
t
0 1 2 3
q q q q
Figure 1. ∇e3
as desired.
Notice that P3 (q, t) and C3 (q, t) are both polynomials in q and t with pos-
itive integer coefficients. This in turn follows from the coefficients of sλ in the
Schur function expansion of ∇e3 all being polynomials with positive integer coef-
ficients. This and further calculations suggest that ∇en , sμ should always be a
polynomial with positive integer coefficients. One can hope to prove this positiv-
ity in two ways. First, one can hope that ∇en has a combinatorial interpretation
under which one can calculate ∇en , sλ by counting some set of objects (associ-
ated with the partition λ) with appropriate weights. More precisely, there should
be combinatorially defined sets Sλ and functions qwt, twt : Sλ → N such that
∇en , sλ = s∈Sλ q qwt(s) ttwt(s) . Secondly, one can hope ∇en has a representation
theoretic interpretation by which ∇en is the bi-graded Frobenius characteristic
(q,t)
FVn for some naturally defined family of bi-graded Sn representations Vn .
Since the Macdonald polynomials H μ (x; q, t) are also Schur-positive, that is,
have only positive integer polynomial coefficients in their Schur function expansions,
there should also be similar interpretations of the Macdonald polynomials.
At present, there are known interpretations of the Macdonald polynomials and
of ∇en in terms of Sn -representation theory. Both ∇en and H μ turn out to be
the Frobenius characteristics of certain finite dimensional quotients of the rings
C[x1 , . . . , xn , y1 , . . . , yn ] which we will describe in the last lecture. Although these
quotient rings can be defined in an elementary way, the existing proofs of these
theorems require some fairly sophisticated algebraic geometry involving the Hilbert
scheme of points in the plane [16, 17].
As for combinatorial interpretations, those relating to ∇en are known and
proved only for Cn (q, t) and some related specializations. Some recent conjectures
have, however, shed further light on this subject. These will be the main topic of
the final lecture.
4.4. Exercises
(n) , and that therefore Pn (q, 0) = H
(1) Prove that ∇en |t=0 = H (n) , e(1n ) =
qk −1
[n]q !, where by definition [n]q ! = [n]q [n − 1]q · · · [1]q and [k]q = q−1 .
LECTURE 5
Positivity and Combinatorics?
μ (x; q, t)
5.1. Representation Theory of H
Recall the Frobenius characteristic of an Sn representation V is defined as
FV (x) = (dimV Sμ )mμ (x).
|μ|=n
By construction, FV (x; q, t), sλ ∈ N[q, t] for every λ. Therefore, one method
for showing that a symmetric function f ∈ ΛQ(q,t) has the property that f, sλ ∈
N[q, t] for every λ is to show that f = FV (x; q, t) for some bi-graded representation
V.
In this section we will construct this representation V for f = H μ (x; q, t),
which shows that Kλμ ∈ N[q, t]. In the next section we will do the same for
f = ∇en . Although we will be able to explicitly describe these representations, the
proof that they have the right Frobenius characteristic involves fairly sophisticated
algebraic geometry involving the Hilbert scheme of points in the plane, and would
require another entire series of lectures to present. No elementary proof that these
representations have the right Frobenius characteristic is known.
Given a partition μ with |μ| = n, let {(p1 , q1 ), . . . , (pn , qn )} be the coordinates
of the boxes in its diagram. Now define
p q
Δμ (x1 , x2 , . . . , xn , y1 , y2 , . . . , yn ) = det xi j yi j .
241
242 HAIMAN AND WOO, GEOMETRY OF q AND q, t-ANALOGS
(Note that the convention is for powers of x to increase along the vertical axis in the
partition diagram and for powers of y to increase along the horizontal axis, contrary
to the usual expectation for Cartesian coordinates. Our peculiar convention has
become established in the literature because the rings Rμ we will soon define were
first studied in the case of Hall-Littlewood polynomials, and these are conventionally
written in terms of t and the x-variables, setting q and the y-variables to 0.)
Now let S denote the ring C[x1 , . . . , xn , y1 , . . . , yn ], bi-graded so that its (i, j)-
th graded piece consists of polynomials homogeneous of degree j in the x variables
and degree i in the y variables. (In the lectures and in a number of places in the
literature, Q is used instead of C here. This is an irrelevant difference since the
representation theory of Sn is exactly the same over the two fields. We have reverted
to using C since that is more consistent with earlier lectures and the general study
of representation theory.) Now for each partition μ with |μ| = n, define an ideal Jμ
of S by % &
∂ ∂
Jμ = f : f ( , )Δμ (x, y) = 0 .
∂x ∂y
In other words, Jμ consists of all polynomials that, when considered as partial
differentiation operators, annihilate Δμ . Now let Rμ = S/Jμ .
The simplest example is μ = (1n ). As mentioned before, Δ(1n ) is the classical
Vandermonde determinant, and J(1n ) = y1 , . . . , yn , e1 (x), . . . , en (x). Therefore,
R(1n ) = C[x]/C[x]S+n ,
or, in words, the polynomial ring in the x variables modulo the ideal generated by
all homogeneous non-constant symmetric functions. This ring is known as the ring
of covariants, and it is a classical theorem that R(1n ) ∼
= S n C · Sn ∼
n
= C ↑S1 , and,
furthermore, that
(1n ) (x; q, t).
FR(1n ) (x; q, t) = (1 − t)(1 − t2 ) · · · (1 − tn )hn [X/(1 − t)] = H
Generalizing this, we have the following theorem.
Theorem 8 ([16]). There holds the identity
μ (x; q, t).
FRμ (x; q, t) = H
Now let Rn = C/Jn , the coinvariant ring for the diagonal action. We have the
following theorem.
Theorem 9 ([17]). There holds the identity
FRn (x; q, t) = ∇en .
Corollary 3.
(1) ∇en ∈ N[q, t] · {sλ : |λ| = n}.
(2) Cn (q, t) = ∇en , en = i,j dim(Rn )i,j ti q j ∈ N[q, t].
(3) Pn (q, t) = ∇en , e(1n ) = i,j dim(Rn )i,j ti q j ∈ N[q, t].
Proof. (1) holds because the Frobenius series of any (positively graded) Sn -module
is in N[q, t] · {sλ : |λ| = n}. Since f, en picks out the coefficient of en = s(1n ) in the
expansion of f in the Schur function basis, if f = FV for some Sn representation
V , f, en gives the multiplicity of the sign representation in V . Since the sign
representation is 1-dimensional, (2) follows. Finally, for any Sn representation
V , FV , e(1n ) = FV , h(1n ) , which is the coefficient of m(1n ) in the monomial
expansion of FV . By definition, this is the dimension of the subspace of V fixed by
the trivial group, which is all of V , giving (3).
More pictorially, a cell c attacks c if either c and c are on the same diagonal
with c to the left of c , or c is one diagonal above and strictly to the right of c .
( and c attacks c }. Figure 1 shows that
Now simply let twtn (λ) = #{(c, c )|c, c ∈ λ
twt6 ((4, 4, 2)) = 9.
Now we have the following theorem.
Theorem 10 ([6, 13]). There holds the identity
Cn (q, t) = q qwtn (λ) ttwtn (λ) .
λ∈δ(n)
This conjecture, if true, would have the following corollary; recall that a tableau
of skew shape λ( and content (1n ) corresponds directly to a parking function.
It is also mysterious why this should be symmetric under switching q and t, and
what connection these combinatorics may have with the ring Rn described above.
It can at least be shown that insofar as Cn (q, t) is concerned, Conjecture 1
agrees with Theorem 10. First of all, in keeping with how ω usually acts on objects
indexed by tableaux, it can be shown that
ωDλ (x; t) = ttwt(T ) xT ,
T ∈SSYT− (λ)
where SSYT− (λ) ( denotes the set of imaginary tableaux T of shape λ, ( whose
“imaginary” entries increase weakly along columns and strictly along rows (the
requirement on rows is irrelevant in the case of the shapes λ ( occurring in the above
formula). For imaginary tableaux, twt(T ) is redefined to allow a contribution from a
pair of cells (c, c ) if c attacks c and T (c) ≥ T (c ) (instead of requiring T (c) > T (c )).
For each λ ⊆ δ(n), there is a unique imaginary tableau T of shape λ ( with
all
entries being 1, and for this imaginary tableau, twt(T ) = twt n (λ). Therefore,
λ∈δ(n) q qwtn (λ) ωDλ (x; t), hn = Cn (q, t). Since λ⊆δ(n) q qwtn (λ) Dλ (x; t), en =
λ⊆δ(n) q qwtn (λ) ωDλ (x; t), hn , the theorem for Cn (q, t) agrees with the conjec-
ture.
μ (x; q, t)
5.4. Combinatorics of H
This topic was addressed, not in these lectures, but in a satellite lecture by Jim
Haglund. We will comment briefly on the latest developments. Haglund conjec-
tured, and discussed in his lecture, a combinatorial formula analogous to Conjec-
ture 1 for the monomial expansion of H ( μ (x; q, t). Like Conjecture 1, Haglund’s
formula can be expressed as a q-weighted sum of LLT polynomials in the parame-
ter t, which shows in particular that it is in fact a symmetric function. (This also
shows, subject to a general Schur-positivity conjecture for LLT polynomials, that
Haglund’s formula is Schur-positive. The special case of the LLT positivity conjec-
ture required for Schur-positivity of the formula in Conjecture 1 is known to hold.)
Between the the PCMI meeting and the preparation of the final version of these
notes, Haglund’s conjecture has been proven by Haglund, Haiman and Loehr, who
verify directly that Haglund’s formula satisfies the defining axioms for Macdonald
polynomials in Theorem-Definition 2. For details, see [12].
5.5. Exercises
Show that Conjecture 1 gives the correct predictions for the following.
(1) ∇en |t=1 = kn (q)
(1n ) (x; q, t)
(2) ∇en |q=0 = (1 − t)(1 − t2 ) · · · (1 − tn )hn [X/(1 − t)] = H
(3) ∇en |t=0 (This one is trickier.)
246 HAIMAN AND WOO, GEOMETRY OF q AND q, t-ANALOGS
Proving that the conjecture gives the correct prediction for ∇en |q=1 is an open
problem. Using the first exercise, this is presumably a special case for showing
combinatorially that the conjecture gives a function symmetric under switching q
and t.
BIBLIOGRAPHY
Dmitry N. Kozlov
IAS/Park City Mathematics Series
Volume 14, 2004
Preamble
Combinatorics, in particular graph theory, has a rich history of being a domain
of successful applications of tools from other areas of mathematics, including topo-
logical methods. Here, we survey the study of the Hom -complexes, and the ways
these can be used to obtain lower bounds for the chromatic numbers of graphs, pre-
sented in a recent series of papers [BK03a, BK03b, BK04, CK04a, CK04b,
Ko04, Ko05b].
The structural theory is developed and put in the historical context, culminat-
ing in the proof of the Lovász Conjecture, which can be stated as follows:
For a graph G, such that the complex Hom (C2r+1 , G) is k-connected
for some r, k ∈ Z, r ≥ 1, k ≥ −1, we have χ(G) ≥ k + 4.
Beyond the, more customary in this area, cohomology groups, the algebro-
topological concepts involved are spectral sequences and Stiefel-Whitney charac-
teristic classes. Complete proofs are included for all the new results appearing in
this survey for the first time.
1 Institute
of Theoretical Computer Science / Department of Mathematics, Eidgenössische Tech-
nische Hochschule - Zürich, CH-8006 Zürich, Switzerland.
E-mail address: dkozlov@inf.ethz.ch.
The author would like to thank the Swiss National Science Foundation and Mathematical Science
Research Institute, Berkeley for the generous support.
c
2007 American Mathematical Society
251
LECTURE 1
Introduction
Figure 1.1.1. A graph with chromatic number 4, which does not contain K4
as an induced subgraph.
Theorem 1.1.3 (The Four-Color Theorem). (Appel & Haken, [AH89]; revised
proof by Robertson, Sanders, Seymour & Thomas, [RSST]).
Every planar graph is four colorable.
where the infimum is taken over all pairs (n, k) such that there exists a graph ho-
momorphism from G to Kn,k .
Here, the state graphs are the Kneser graphs, {Kn,k }n≥2k , and the chosen
valuation on this family is Kn,k → n/k.
where the infimum is taken over all pairs (n, k) such that there exists a graph ho-
momorphism from G to Rn,k .
Here, the state graphs are {Rn,k }n≥2k , and the chosen valuation on this family
is again Rn,k → n/k.
258 D. N. KOZLOV, MORPHISM COMPLEXES, AND STIEFEL-WHITNEY CLASSES
111
000
1 2
000
111
2,3 3 2 3
1
1
3 2 1 3
000
111
000
111
1,2 3 3
2,3 2,3
000
111
00000
11111
1
3 3
3 1 2 3 2
00000
11111
000001111
111110000 3 2
2 1
3 3
2
0000
1111
Hom (L2 = K2 , K3 ) 1,3 1,3 Hom (L3 , K3 )
Definition 2.1.5. Hom (T, G) is the subcomplex of C(T, G) defined by the following
condition: σ = x∈V (T ) σx ∈ Hom (T, G) if and only if for any x, y ∈ V (T ), if
(x, y) ∈ E(T ), then (σx , σy ) is a complete bipartite subgraph of G.
2 1
2 1 2 13
23 1
000
111
1 2
000
111
2 13
111
000000
111
3 1 13 23 1 2
1 23
0000
1111 000
111000
111 000
111
000
111 000
111
12 1111
0000 000
111
000
111000
111 000
111
13 2
000
111
23
0000
1111 000
111000
111 000
111 000
111 2 1
3 12 1111
0000 000
111000
111
000
111 000
111 000
111
0000
1111
00
11 000
111000
111 000
111 000
111
000
111
000
111
0000
1111
00
11 000
111 000
111
000
111
3 3
12 3 1111
0000 000 111
111
00
11
0000
1111 000
000
111
00000
11111
00
11
3 2 1111
0000 000
111
000 2 3
111
0000
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00
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000
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00
11
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11111 000000
111111
0000
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111
00
11
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11111 000000
111111 12
13 2
00000
11111
00000
11111 000000
111111
00000
11111
00000
11111 1 3
2 13 1 2 00000
11111
1 2 1 23
3 12
1 2 23 12
23 13
2 1 23 1
3 3
Hom (L4 , K3 )
(3) In the literature there are several different notations for the set of all graph
homomorphisms from a graph T to the graph G. Since an untangling of the de-
finitions shows that this set is precisely the set of vertices of Hom (T, G), i.e., its
0-skeleton, it feels natural to denote it by Hom 0 (T, G).
(4) On the intuitive level, one can think of each η : V (T ) → 2V (G) \ {∅}, satisfying
the conditions of the Definition 2.1.5, as associating non-empty lists of vertices of
G to vertices of T with the condition on this collection of lists being that any choice
of one vertex from each list will yield a graph homomorphism from T to G.
(5) The standard way to turn a polyhedral complex into a simplicial one is to take
the barycentric subdivision. This is readily done by taking the face poset and then
taking its nerve (order complex). So, here, if we consider the partially ordered set
F (Hom (T, G)) of all η as in Definition 2.1.5, with the partial order defined by η̃ ≤ η
if and only if η̃(v) ⊆ η(v), for all v ∈ V (T ), then we get that the order complex
Δ(F (Hom (T, G))) is a barycentric subdivision of Hom (T, G). A cell τ of Hom (T, G)
corresponds to the union of all the simplices of Δ(F (Hom (T, G))) labeled by the
chains with the maximal element τ .
Some examples are shown on Figures 2.1.2, 2.1.3, 2.1.4, and 2.1.5. On these figures
we used the following notations: Ln denotes an n-string, i.e., a tree with n vertices
and no branching points, Cm denotes a cycle with m vertices, i.e., V (Cm ) = Zm ,
E(Cm ) = {(x, x + 1), (x + 1, x) | x ∈ Zm }.
2 1 2 1
1 2 1 3
2
111
000
13
000
111
000
111
23 1
000
111
000
111 000
111
000
111 000
111
000
111
13 2 000
111
000
111 000
11100
11
1 23
000
111 3 1 00
11
00
11
3 111
000
12
000
111 00
11 3 12
000 1 2 11
111 00
00
11
000
111 00
11
12 111
000
00000
11111 00 12
11
3
000
111
00000
11111
00000
11111 3
00000
11111
00000
11111
00000
11111
00000
11111
1 23 13 2
23 1 2 13
Hom (C4 , K3 )
1 1
3 2 2 3
2 3 1 3 2 1
3 3 2 2
1 1
2 1 3 1
3 2 2 3
2 1 3 1
Hom (C5 , K3 )
know the set A, we can only make the statements of the type the labels of these two
edges are the same/different.
Assume we have S ⊆ M , such that S can be written as a direct product
S = S1 × S2 × · · · × St . Assume furthermore that the subgraph of Γ induced by S
is precisely the 1-skeleton of the corresponding cell.
First, consider 3 elements a, b, c ∈ S1 × · · · × St , which have the same indices
in all Si ’s except for exactly one. Then, by our assumption on S, the subgraph of
Γ induced on the vertices a, b, and c, is a triangle. Clearly, if 3 changes of a value
of a function result in the same function, then the changes were done in the same
element of A, i.e., λ(a, b) = λ(a, c) = λ(b, c).
Next, consider 4 elements a, b, c, d ∈ S1 × · · · × St , such that pairs of vertices
(a, b), (b, c), (c, d), and (a, d), have the same indices in all Si ’s except for exactly
one. Assume further that this index is not the same for (a, b) and (b, c): say a and
b differ in S1 , and b and c differ in S2 .
According to our assumption on S, (a, b), (b, c), (c, d), and (a, d) are edges of Γ.
If λ(a, b) = λ(b, c), then Γ contains the edge (a, c), and λ(a, b) = λ(a, c), which
contradicts our choice of S.
If, on the other hand, λ(a, b) = λ(b, c), then, since changes of functions along
the paths a → b → c and a → d → c should give the same answer, we are left with
the only possibility: namely λ(a, b) = λ(c, d), and λ(b, c) = λ(a, d).
Let a ∈ S1 ×· · ·×St , a = (a1 , . . . , at ). By our first argument, if b ∈ S1 ×· · ·×St ,
b = (a1 , . . . , ãi , . . . , at ), then λ(a, b) does not depend on ai and ãi . Furthermore, let
c, d ∈ S1 ×· · ·×St , d = (a1 , . . . , ãj , . . . , at ), c = (a1 , . . . , ãi , . . . , ãj , . . . , at ), for i = j.
By our second argument, applied to a, b, c, d, we get that λ(a, b) = λ(c, d). If iterated
for various j, this implies that λ(a, b) does not depend on a1 , . . . , ai−1 , ai+1 , . . . , at
either; thus it may depend only on the index i.
Finally, this label should be different for different i’s, as otherwise, by the same
argument as above, we would get more edges in the subgraph of Γ induced by S,
than what we allowed by our assumptions.
Summarizing, we have shown that the cell σ = x∈A σx ∈ C(A, B) belongs to
Hom M (A, B) if and only if the 1-skeleton of σ is a subgraph of Γ. This implies that
Hom M (A, B) is uniquely determined by its 1-skeleton.
Intuitively, one can interpret Proposition 2.2.2 as saying that, with respect to its
1-skeleton, Hom M (A, B) is the polyhedral analog of the flag complex construction.
For a directed graph G, let u(G) be the undirected graph obtained from G by
forgetting the directions, and identifying the multiple edges. We remark that for
any two directed graphs G and H, the complexes Hom (G, H) and Hom (u(G), u(H))
are isomorphic, if E(H) is Z2 -invariant.
G= 1111111
0000000
000
111
0000000
1111111
11
000
111
0000000
1111111
000
111
0000000
11111110000
1111
000
111
0000000
1111111
21 13
00000
11111
0000000
11111110000
1111
1
00000
11111
00000
11111
00000001111
11111110000
H= 12
00000
11111
00000
11111
2 3
00000
11111
22 0000
1111 32 33
(3) Let A and B be the vertex sets of simplicial complexes Δ1 and Δ2 , and let M
be the set of simplicial maps from Δ1 to Δ2 , then Hom M (A, B) is the analog of
Hom (T, G) for simplicial complexes.
(4) Recall that a hypergraph with the vertex set V is a subset H ⊆ 2V . Let A
and B be the vertex sets of hypergraphs H1 and H2 . There are various choices
for when to call a map ϕ : A → B a hypergraph homomorphism. Two possibilities
which we mention here are: one could require that ϕ(H1 ) ⊆ H2 , or one could ask
that for any H1 ∈ H1 , there exists H2 ∈ H2 , such that ϕ(H1 ) ⊆ H2 . The example
(3) is a special case of both. Either way, the corresponding complex Hom M (A, B)
provides us with an analog of Hom (T, G) for hypergraphs.
(5) Let A and B be the vertex sets of posets P and Q, and let M be the set of
order-preserving maps from P to Q, then Hom M (A, B) is the analog of Hom (T, G)
for posets.
consists of all k-subsets of [n], and the set of hyperedges consists of all r-tuples of
disjoint k-subsets.
Using the introduced notations, we can now formulate the generalization of
Theorem 2.3.1.
Theorem 2.3.7 (Alon-Frankl-Lovász, [AFL86]).
For arbitrary positive integers n, k, r, such that r ≥ 2, and n ≥ rk, we have
r n − rk + r
χ(Kn,k )= .
r−1
Theorem 2.3.7 can be proved using the generalization of the Borsuk-Ulam the-
orem from [BSS81].
2.3.2.3. Further References.
There has been a substantial body of further important work, which, due to space
constraints, we do not pursue in detail in this survey, some of the references are
[Dol88, Kr92, Kr00, Ma04, MZ04, Sar90, Zie02].
There have also been multiple constructions, such as box complexes, designed to
generalize the original Lovász neighbourhood complexes. However, as later research
showed, the bounds obtained in that way were essentially convertible, since the Z2 -
homotopy types of these complexes were very closely related, either by simply being
the same, or by means of one being the suspension of another, or something close
to that. This means that all these constructions are avatars of the same object, as
explained in [Ziv04].
2.4.2. Products
For any three graphs G, H, and K, we have the following homotopy equivalence,
see [Ba05]:
(2.4.2) Hom (G, H × K) Hom (G, H) × Hom (G, K).
In fact, the formula (2.4.2) can be strengthened to state that the left hand
side is simple homotopy equivalent (in the sense of Whitehead, see [Co73]) to the
right hand side. Since this simple homotopy equivalence result is new, we include
a complete argument, as promised in the abstract.
270 D. N. KOZLOV, MORPHISM COMPLEXES, AND STIEFEL-WHITNEY CLASSES
Consider the following three maps 2pH : 2V (H)×V (K) → 2V (H) , 2pK :
2V (H)×V (K)
→ 2V (K) , and c : 2V (H) × 2V (K) → 2V (H)×V (K) , where 2pH and
2 pK
are induced by the standard projection maps pH : V (H) × V (K) → V (H)
and pK : V (H) × V (K) → V (K), and c is given by c(A, B) = A × B.
We let ψ : 2V (H)×V (K) → 2V (H)×V (K) denote the composition map ψ(S) =
c(2pH (S), 2pK (S)) = 2pH (S) × 2pK (S).
Given a cell of Hom (G, H × K) indexed by η : V (G) → 2V (H)×V (K) \ {∅}, one
can see that the composition function ψ ◦ η : V (G) → 2V (H)×V (K) \ {∅} will also
index a cell. Indeed, for any (x, y) ∈ E(G) we know that (η(x), η(y)) is a complete
bipartite subgraph of H × K, which is the same as to say that, for any α ∈ η(x),
and β ∈ η(y), we have (pH (α), pH (β)) ∈ E(H), and (pK (α), pK (β)) ∈ E(K).
If we now choose α̃ ∈ ψ(η(x)), and β̃ ∈ ψ(η(y)), we have pH (α̃) = pH (α), for
some α ∈ η(x), and pH (β̃) = pH (β), for some β ∈ η(y), hence verifying that
(pH (α̃), pH (β̃)) ∈ E(H). The fact that (pK (α̃), pK (β̃)) ∈ E(K) can be proved
analogously.
This means that we have a map ϕ : F (Hom (G, H × K)) → F (Hom (G, H × K)).
It is easy to see that ϕ is order-preserving and ascending (meaning ϕ(x) ≥ x,
for any x ∈ F(Hom (G, H × K))). It follows from [Ko05a, Theorem 3.1] that
Δ(F (Hom (G, H × K))) = Bd (Hom (G, H × K)) collapses onto Δ(im ϕ).
On the other hand, F (Hom (G, H)) × F(Hom (G, K)) ∼ = im ϕ with the iso-
morphism given by the map (η1 , η2 ) → η, where η(x) = η1 (x) × η2 (x), for
any x ∈ V (G). Thus we conclude that Bd (Hom (G, H × K)) collapses onto
Δ(F (Hom (G, H)) × F(Hom (G, K))) ∼ = Δ(F (Hom (G, H))) × Δ(F (Hom (G, K))) =
Bd (Hom (G, H)) × Bd (Hom (G, K)) ∼ = Hom (G, H) × Hom (G, K), and our argument is
now complete.
For the analog of the formula (2.4.2), where the direct product is taken on the
left, we need the following additional standard notion.
Definition 2.4.1. For two graphs H and K, the power graph K H is defined by
• V (K H ) is the set of all set maps f : V (H) → V (K);
• (f, g) ∈ E(K H ), for f, g : V (H) → V (K), if and only if, whenever
(v, w) ∈ E(H), we also have (f (v), g(w)) ∈ E(K).
It is easy to see that the power graph notion is introduced precisely so that for
any triple of graphs the following adjunction relation holds:
(2.4.3) Hom 0 (G × H, K) = Hom 0 (G, K H ).
In our topological situation the formula (2.4.3) generalizes up to homotopy.
More precisely, we have the following homotopy equivalence, see [Ba05],
(2.4.4) Hom (G × H, K) Hom (G, K H ).
The formula (2.4.4) can as well be strengthened to yield a simple homotopy
equivalence. Below we include a complete argument.
H H
Define a map ψ : 2V (K ) → 2V (K ) , ψ : Ω → ψ(Ω), as follows: g ∈ ψ(Ω) if and
only if g(x) ∈ {f (x) | f ∈ Ω}, for all x ∈ V (H). In other words, we use the collection
of functions Ω to specify the sets of values, which functions from ψ(Ω) are allowed
to take. Clearly, we have ψ(Ω) ⊇ Ω. Take a cell of Hom (G, K H ), η : V (G) →
H H
2V (K ) \ {∅}, and consider the composition map ψ ◦ η : V (G) → 2V (K ) \ {∅}.
LECTURE 2. THE FUNCTOR Hom (−, −) 271
Since η is a cell, we know that if (x, y) ∈ E(G), and α ∈ η(x), β ∈ η(y), then
(α, β) ∈ E(K H ), i.e., whenever (v, w) ∈ E(H), we have (α(v), β(w)) ∈ E(K).
Choose α̃ ∈ ψ(η(x)), and β̃ ∈ ψ(η(y)). To check that (α̃, β̃) ∈ E(K H ), we need
to check that for any (v, w) ∈ E(H), we have (α̃(v), β̃(w)) ∈ E(K). However, by
the definition of ψ, we know that α̃(v) = α(v), for some α ∈ η(x), and β̃(w) = β(w),
for some β ∈ η(y). It follows that (α̃(v), β̃(w)) = (α(v), β(w)) ∈ E(K), and hence
ψ ◦ η is again a cell.
As a consequence, the composition gives us an order-preserving ascending map
ϕ : F (Hom (G, K H )) → F (Hom (G, K H )). The image of this map is isomorphic to
F (Hom (G × H, K)). The isomorphism map takes the poset element η : V (G) ×
H
V (H) → 2V (K) \ {∅} to the poset element η̃ : V (G) → 2V (K ) \ {∅} defined by
for all x ∈ V (G). By [Ko05a, Theorem 3.1], we conclude that the complex
Δ(F (Hom (G, K H ))) = Bd (Hom (G, K H )) collapses onto its subcomplex Δ(im ϕ) =
Bd (Hom (G × H, K)).
We obtain an interesting special case of the formula (2.4.4) when substituting
G = K1o (which means a graph with one looped vertex). Since K1o × H = H, for
any graph H, we conclude that Hom (H, K) Hom (K1o , K H ) for any two graphs H
and K. As seen directly, for an arbitrary graph G, Hom (K1o , G) is the clique complex
of the looped part of G, i.e., of the subgraph induced by the set of vertices which
have loops. In particular, the complex Hom (K1o , G) is simplicial. On the other hand,
a vertex f ∈ V (K H ) has a loop if and only if f is a graph homomorphism. We can
therefore conclude that for arbitrary graphs H and K the complex Hom (H, K) is
homotopy equivalent to the clique complex of the subgraph of K H , induced by the
set of all graph homomorphisms from H to K.
defined above. Hence, we can conclude that this f comes from a cellular map from
Hom (T, G) to Hom (T, K), which we denote by ϕT .
Moreover, a detailed pointwise analysis of the polyhedral structure of Hom (T, G)
shows that cells (direct products of simplices) map surjectively to other cells, and
that this map is a product map induced by the corresponding maps on the simplices.
Therefore, ϕT is a polyhedral map.
The situation is slightly more complicated if one considers the functoriality in
the first argument. Let us choose some proper graph homomorphism ψ from T
to G, and let K be some graph. Again, by using composition we can define a poset
map g : F (Hom (G, K)) → F (Hom (T, K)), namely, for η : V (G) → 2V (K) \ {∅},
and v ∈ V (T ), we have g(η)(v) = η(ψ(v)). This map is well-defined, since, first
if v, w ∈ V (T ), and (v, w) ∈ E(T ), then (ψ(v), ψ(w)) ∈ E(G), and therefore, for
any x ∈ η(ψ(v)), and y ∈ η(ψ(w)), we have (x, y) ∈ E(K), and second, by the
properness assumption, v∈V (T ) (|η(ψ(v))| − 1) < ∞. Furthermore, this map is
order-preserving: if τ ≥ η, i.e., if τ (w) ⊇ η(w), for any w ∈ V (T ), then g(τ )(w) =
τ (ψ(w)) ⊇ η(ψ(w)) = g(η)(w).
Intuitively, one can think of the map g as the pullback map. It is important to
remark that, if ψ is not injective, it may happen that dim g(η) > dim η.
For an arbitrary regular CW complex X, let Bd (X) denote the barycentric
subdivision of X. Since g is an order-preserving map, the induced map
Δ(g) : Bd (Hom (G, K)) → Bd (Hom (T, K))
is simplicial and gives the corresponding map of topological spaces, which we de-
note ψK . However, g does not always come from a cellular map. In fact, one can
check that there exists a cellular map ψK : Hom (G, K) → Hom (T, K), such that
F (ψK ) = g, if ψ is injective on the vertices of T .
In any case, we see that Hom (T, −) is a covariant functor from Graphs to
Top, while Hom (−, K) is a contravariant functor from Graphsp to Top; here Top
denotes the category whose objects are topological spaces, and whose morphisms
are all continuous maps.
and hence, since for any posets P1 and P2 , the simplicial complex Δ(P1 × P2 ) is
homeomorphic to the polyhedral complex Δ(P1 ) × Δ(P2 ) (in fact it is its subdivi-
sion), we have a corresponding topological map
Hom (T, G) × Hom (G, K) −→ Hom (T, K).
2.4.6. Universality
In topological combinatorics it happens very often that the family of the stud-
ied objects is universal with respect to the invariants which one is interested in
computing. This is also the case not only for the Hom -complexes, but even for
the Hom (K2 , −)-complexes. The following result is due to Csorba, [Cs04a], and,
independently, to Živaljević, [Ziv04].
Theorem 2.4.2 ([Cs04a, Ziv04]).
For each finite, free Z2 -complex X, there exists a graph G, such that Hom (K2 , G) is
Z2 -homotopy equivalent to X.
We note that Theorem 2.4.2 can be verified by combining [Ziv04, Theorem 32],
with a remark in the beginning of [Ziv04, Section 7].
2.5. Folds
2.5.1. Sequences of Collapses Induced by Folds
Hom -complexes behave well with respect to the following standard operation from
graph theory.
Definition 2.5.1. For a graph G and v ∈ V (G), G − v is called a fold of G if
there exists u ∈ V (G), u = v, such that N (u) ⊇ N (v).
Let G − v be a fold of G. We let i : G − v → G denote the inclusion ho-
momorphism, and let f : G → G − v denote the folding homomorphism defined
by
u, for x = v;
f (x) =
x, for x = v.
274 D. N. KOZLOV, MORPHISM COMPLEXES, AND STIEFEL-WHITNEY CLASSES
Figure 2.5.1 shows an example of the collapsing sequence appearing in the proof
of Theorem 2.5.2 (1).
Note that Theorem 2.5.2 cannot be generalized to encompass arbitrary graph
homomorphisms φ of G onto H, where H is a subgraph of G, and φ is identity on
H. For example, Hom (C6 , K3 ) Hom (K2 , K3 ), see Figures 2.5.2 and 2.5.3, despite
of the existence of the sequence of graph homomorphisms K2 → C6 → K2 which
compose to give an identity.
We remark that, for the sake of transparency, the striped rectangles are shown
on Figure 2.5.2 only around one of the 6 joining vertices, and only two out of the
three. The big connected component corresponds to the graph homomorphisms
ϕ : C6 → K3 having the winding number 0. The isolated points correspond to the
6 possible tight windings of C6 around K3 . Observe also that the cubes are solid.
2.5.2. Applications
When G is a graph, and H is an induced subgraph of G, we say that G reduces to
H, if there exists a sequence of folds leading from G to H.
LECTURE 2. THE FUNCTOR Hom (−, −) 275
1 23
1 2 23 1 12 3
23 13 1 23 3 12
1 2 12 3
1 23 2 13
2 1 13 2
111
000
00
11
13 2 000
111
00
11
000
111
2 13
00
11
000
111
1 2
00
11
00
11
1 2
2 1 00
11 3 3
1 2 2 1
23 1
13 2
1 23
2 13 3 12
23 1
13 2 12 3
3 12
Hom (C6 , K3 )
Figure 2.5.2. The figure depicts the polyhedral complex of all 3-colorings of
a 6-cycle.
The Theorem 2.5.2 can be used to obtain complete understanding of the ho-
motopy type of the Hom -complexes for certain specific families of graphs.
Proposition 2.5.4. If T is a tree with at least one edge, and G an arbitrary graph,
then Hom (T, G) is homotopy equivalent to N (G). As a consequence, if F is a forest,
and T1 , . . . , Tk are all its connected components consisting of at least 2 vertices, then
k
Hom (F, G) i=1 N (G).
An even more special case was important in [BK03b, BK04] for the proof of
Lovász Conjecture.
Corollary 2.5.5 ([BK03b, Proposition 5.4]).
If T is a finite tree with at least one edge, then the map iKn : Hom (T, Kn ) →
Hom (K2 , Kn ) induced by any inclusion i : K2 → T is a homotopy equivalence, in
particular Hom (T, Kn ) S n−2 .
If F is a finite forest, and T1 , . . . , Tk are all its connected components consisting of
k
at least 2 vertices, then Hom (F, Kn ) i=1 S n−2 .
In this case, Corollary 2.5.3 can be applied to describe the Z2 -homotopy type as
well. First, some new notations: let San , resp. Stn , denote the n-dimensional sphere,
equipped with an antipodal, resp. trivial Z2 -action, where n is a nonnegative integer,
or infinity.
276 D. N. KOZLOV, MORPHISM COMPLEXES, AND STIEFEL-WHITNEY CLASSES
1 23
1 2
23 1
v= 2 1 13 2
1 23
111111
000000
2 1
1 2
11
000
1
00000
11111
0
1 000000
111111
00000
11111
00
110
1
00000
11111
1 23
00
110
1
00000
11111
00
110
1
00000
11111
00
110
1
1 23
00000
11111
00
110
1
00000
11111
2 1
00
110
1
00000
11111
1 2
00
110
1
00000
11111
00
11
13 2
00
110
1
00000
11111
0
1
23 13
111111
000000
0
1 000000
111111
000000
111111
1 2
0
1
0
1 000000
111111
13 2
000000
111111
2 13
13 2
Figure 2.5.3. The figure on the left shows the neighbourhood of the vertex
v of Hom (C6 , K3 ). The figure on the right shows the link of this vertex.
ϕ (ϕ/Γ)∗
X Y H ∗ (X/Γ) H ∗ (Y /Γ)
v
v◦ϕ ((v ◦ ϕ)/Γ)∗ (v/Γ)∗
EΓ H ∗ (BΓ)
In other words the characteristic classes associated to a finite group action are
natural, or, as one sometimes says, functorial.
We refer the reader to the wonderful book of tom Dieck, [tD87], for further de-
tails on equivariant maps and associated bundles. We also recommend the classical
book of Milnor&Stasheff, [MS74], as an excellent source for the theory of charac-
teristic classes of vector bundles. The generalities on bundles, including principal
bundles, can be found in [Ste51].
where the sum is taken over representatives of Z2 -orbits of multicolored edges, one
representative per orbit.
To describe the powers of the Stiefel-Whitney classes, 1k (X), we need to recall
how the cohomology multiplication is done simplicially. In fact, to evaluate 1k (X)
on a k-simplex (v0 , v1 , . . . , vk ), we need to evaluate 1 (X) on each of the edges
(vi , vi+1 ), for i = 0, . . . , k − 1, and then multiply the results. Thus, the only
k-simplices, on which the power 1k (X) evaluates nontrivially, are those whose
ordered set of vertices has alternating elements from A and from B. We call these
simplices multicolored. We summarize
(3.2.2) 1k (X) = τσ ,
multicolored σ
14 1
1 4 12 2 3 2 35
34
1 234
1 1
123 4 25 3 24 3
13 24
1 15
2 34 2 3
and
m−1
m+k+1 m
(3.3.4) f (m, n) = (−1) kn ,
k+1
k=1
for n ≥ m ≥ 1.
In particular, for small values of m, we obtain the following explicit formulae:
f (2, n) = 1, for n ≥ 2, f (3, n) = 2n − 3, for n ≥ 3, f (4, n) = 3n − 4 · 2n + 6, for
n ≥ 4, f (5, n) = 4n − 5 · 3n + 10 · 2n − 10, for n ≥ 5.
for Hom . So the only difference is that η(x) is allowed to be an empty set, for
x ∈ V (T ).
2
1 3 = ×
Λ K2 × Λ
111
000
000
111
000
111
000
111
000
111
000
111
111111
000000
000000
111111
0000000
1111111
000
111
000
111
000
111
0000000
1111111
000
111
Λ+ Hom (K , Λ)
+ 2 Hom (K2 , Λ+ )
2
1 2
111111111
000000000
0000
1111
00000
11111
1,3 2 1,2,3
0000
1111
0000
1111
00000
11111
1
00000
11111
0000
1111
00000
11111
0000
1111
0000
1111
3
00000
11111
00000
11111
00000
11111
00000
11111
00000
11111
00000
11111
1,2,3
+ 2
00000
11111
Hom (K , Λ) 11111
00000 00000
11111
11111
00000
2
Δ[2]
The above facts mix well with the cellular structure. First, for a cell complex X,
the cellular cohomology groups of X are by definition isomorphic to the cohomology
groups of the associated cochain complex C ∗ (X). Second, for two cell complexes
X and Y , a cellular map ϕ : X → Y induces a cochain complex map between
associated cochain complexes (but in the opposite direction!), and hence a map
between corresponding cohomology groups.
4.2.2. Filtrations
In concrete situations it can be difficult to compute the cohomology groups H i (C)
without auxiliary constructions. The idea behind spectral sequences is to break
up this large task into smaller subtasks, with the formal machinery to help the
bookkeeping. This ”break up” is usually phrased in terms of a filtration.
A cochain subcomplex of C is a sequence
i−2 i−1 i i+1
C = · · · −→ C
i−1 d−→ C
i −→ i+1 −→
d d d
C ...,
where C i is an R-submodule of C i , and the differentials are restrictions of those
In this situation, one can form the quotient
in C. We shall simply write C ⊇ C.
cochain complex
di−2 di−1 di di+1
C/C = · · · −→ C i−1 /C
i−1 −→ C i /C
i −→ C i+1 /C
i+1 −→ . . . .
The cohomology groups of this complex, H ∗ (C/C), are usually denoted H ∗ (C, C),
and are called the relative cohomology groups.
If X is a cell complex, and Y its cell subcomplex, then the cellular cochain
complex of Y is a cochain subcomplex of the cellular cochain complex of X. The
corresponding cohomology groups of the quotient cochain complex are precisely the
relative cohomology groups of the pair of topological spaces (X, Y ).
Definition 4.2.1. A (finite) filtration on a cochain complex C is a nested sequence
of cochain complexes
di−2 di−1 di di+1
Cj = · · · −→ Cji−1 −→ Cji −→ Cji+1 −→ . . . ,
for j = 0, 1, 2, . . . , t, such that C = Ct ⊇ Ct−1 ⊇ · · · ⊇ C0 (that is why we suppressed
the lower index in the differential).
In general, infinite filtrations can be considered, but in this article we limit our
considerations to the finite ones. Given a filtration C = Ct ⊇ Ct−1 ⊇ · · · ⊇ C0 , we
set C−1 = 0, for the convenience of notations.
There are many standard filtrations of cochain complexes. For example, if
a pure cochain complex is bounded, say C i = 0, for i < 0, or i > t, then, the
standard skeleton filtration is defined as follows:
i C i , if i ≤ j;
Cj =
0, otherwise.
This filtration is not very interesting though, since computing the cohomology
groups with its help is canonically equivalent with computing the cohomology
groups from the cochain complex directly.
For a cell complex X, a classical way to define a filtration on its cellular cochain
complex, is to choose a cell filtration on X, i.e., a sequence of cell subcomplexes
X = Xt ⊇ Xt−1 ⊇ · · · ⊇ X0 (again for the convenience of notations, we set
LECTURE 4. THE SPECTRAL SEQUENCE APPROACH 291
X−1 = ∅). As mentioned above, the corresponding cellular cochain complexes form
a sequence of nested subcomplexes.
If the cell complex X is finite dimensional, then, taking Xi to be the i-th
skeleton of X, we recover the standard skeleton filtration on C ∗ (X), which explains
the name of this filtration.
A much more interesting situation is the following.
Definition 4.2.2. Assume that we have a cell map ϕ : X → Y and a filtration
Y = Yt ⊇ Yt−1 ⊇ · · · ⊇ Y0 . Define a filtration on X as follows: Xi := ϕ−1 (Yi ),
for i = 0, . . . , t. This filtration on X is called the pullback of the filtration on Y
along ϕ.
In the case when the filtration on Y is simply the skeleton filtration, the cor-
responding pullback filtration on X is called the Serre filtration. We use the same
name for the corresponding filtration on the cellular cochain complex of X.
However, if R is an arbitrary ring (for example R = Z), then one may need to solve
a number of extension problems before obtaining the final answer. This has to do
with the fact, that in a short exact sequence of R-modules
α β
0 −→ A −→ B −→ C −→ 0,
B does not necessarily split as a direct sum of the submodule A and the quotient
module C. This is not even true for R = Z, a classical example is to take A = B =
Z, C = Z2 , to take α : x → 2x to be the doubling map (injective), and to take
β : x → x mod 2 to be the parity map (surjective).
Let us now describe more precisely how the tableaux En∗,∗ and the differentials
dn are constructed. As auxiliary modules, set
Znp,q := Cpp+q ∩ d−1 (Cp+n
p+q+1
),
292 D. N. KOZLOV, MORPHISM COMPLEXES, AND STIEFEL-WHITNEY CLASSES
where d−1 denotes the inverse of the differential d, i.e., Znp,q consists of all elements
p+q+1
of Cpp+q whose boundary is in Cp+n ; and set
p+q−1
Bnp,q := Cpp+q ∩ d (Cp−n ),
i.e., Bnp,q consists of all elements of Cpp+q which constitute the image of d from
p+q−1
Cp−n . These are the settings for n ≥ 0. Finally, for n = −1, we use the following
convention:
p,q p,q p+q−1
Z−1 := Cpp+q , and B−1 := d(Cp+1 ).
With these notations, we set
p+1,q−1 p,q
(4.2.2) Enp,q := Znp,q /(Zn−1 + Bn−1 ),
for all 0 ≤ n ≤ ∞.
It is an easy check, which we leave to the reader, that d(Znp,q ) ⊆ Znp+n,q−n+1 ,
p+1,q−1 p,q p+n+1,q−n p+n,q−n+1
and that d(Zn−1 + Bn−1 ) ⊆ Zn−1 + Bn−1 . It follows that, via the
quotient maps, the differential d induces a map from Enp,q to Enp+n,q−n+1 , which we
choose to call dp,q
n (or just dn , if it is clear what the coefficients p and q are).
One can view the tableau (En∗,∗ , dn ) as a collection of (nearly) diagonal cochain
complexes. This allows one to compute the cohomology groups, just like for the
usual cochain complexes, by setting
(4.2.3) p,q
En+1 = H p,q (En∗,∗ , dn ).
Please note, that the equation (4.2.3) is not trivial, and needs a proof. It can be
deduced directly from the equation (4.2.2), see e.g., [McC01].
Let us start with unwinding these definitions for n = 0. It follows from our
conventions for n = −1, that
Moreover, one can show that dp,q 1 : E1p,q −→ E1p+1,q is the connecting homo-
morphism ∂ : H p+q
(Cp , Cp+1 ) −→ H p+q+1
(Cp+1 , Cp+2 ) in the long exact sequence
of the triple (Cp , Cp+1 , Cp+2 ).
Unless some additional specific information is available, it is hard to say what
happens in the tableaux for n ≥ 2. The important thing is that with the setup
above, the spectral sequence runs its course and eventually converges (modulo
the extension difficulties outlined above) to the cohomology groups of the origi-
nal cochain complex.
LECTURE 4. THE SPECTRAL SEQUENCE APPROACH 293
∂ ∗ is the restriction of the differential in C ∗ (Hom+ (T, G); R), and Hom+ (T, G) de-
(q)
notes the q-th skeleton of Hom+ (T, G). Phrased verbally: F p,q is generated by all
elementary cochains corresponding to q-dimensional cells, which are supported in
at least p + 1 vertices of T . Note, that this restriction defines a filtration, since the
differential does not decrease the cardinality of the support set.
We have
C q (Hom+ (T, G); R) = F 0,q ⊇ F 1,q ⊇ · · · ⊇ F |V (T )|−1,q ⊇ F |V (T )|,q = 0,
which is the Serre filtration associated to the support map.
Hence, the 0th tableau of the spectral sequence associated to the cochain complex
filtration F ∗ is given by
(4.3.2) E0p,q = C p+q (F p , F p+1 ) = C q (Hom (T [S], G); R).
S⊆V (T )
|S|=p+1
294 D. N. KOZLOV, MORPHISM COMPLEXES, AND STIEFEL-WHITNEY CLASSES
Furthermore, using the equation (4.2.5), we obtain the description of the first
tableau as well.
(4.3.3) E1p,q = H p+q (F p , F p+1 ) = H q (Hom (T [S], G); R).
S⊆V (T )
|S|=p+1
In the case of integer coefficients, R = Z, one needs more work to derive the
formula for dp,q
1 (σ) analogous to (4.3.4), since, additionally, the signs have to be
taken into consideration.
LECTURE 5
The Proof of the Lovász Conjecture
the target circle, whereas the sign of wind (ϕ) registers whether the orientation has
been changed or not. The usual way to define wind (ϕ) formally is to notice that
ϕ induces a group homomorphism ϕ∗ : H 1 (S 1 ; Z) → H 1 (S 1 ; Z). Any group homo-
morphism from Z to itself is uniquely determined by the image of 1. This image is
exactly the winding number.
As the proof of Theorem 3.3.5 suggests, we need to analyze the complexes
Hom (C2r+1 , K3 ) in some detail. One can see, by direct inspection, that the con-
nected components of Hom (C2r+1 , K3 ) can be indexed by the winding numbers α.
All one needs to see is that if two homomorphisms ϕ, ψ : C2r+1 → K3 have the same
winding number, then there is a sequence of edges in Hom (C2r+1 , K3 ) connecting ϕ
with ψ; and this is fairly straightforward.
We notice however, that these winding numbers cannot be arbitrary. Indeed, if
the number of times C2r+1 winds around K3 is α, then 2r + 1 = 3α + 2t, for some
nonnegative integer t ≤ r. Hence, α = (2r − 2t + 1)/3. It follows, that α must be
odd, and that it cannot exceed (2r + 1)/3. So α = ±1, ±3, . . . , ±(2s + 1), where
s = (r − 1)/3, in particular s ≥ 0.
Let ϕ : Hom (C2r+1 , K3 ) → {±1, ±3, . . . , ±(2s + 1)} map each point x ∈
Hom (C2r+1 , K3 ) to the point on the real line, indexing the connected component
of x. Clearly, ϕ is a Z2 -map. Since dim({±1, ±3, . . . , ±(2s + 1)}/Z2 ) = 0, we
have H 1 ({±1, ±3, . . . , ±(2s + 1)}/Z2 ; Z2 ) = 0, and the functoriality of the Stiefel-
Whitney classes implies 1 (Hom (C2r+1 , K3 )) = 0. The Conjecture 5.1.3 for this
case follows now from Theorem 3.3.5.
We have shown the Conjecture 5.1.3 for k = 0 using the Stiefel-Whitney classes,
but it is equally easy to prove the Lovász Conjecture for this case directly. Indeed,
following the lines of the proof of Theorem 3.3.5, we see that, a 3-coloring of G would
induce a Z2 -map from Hom (C2r+1 , G) to Hom (C2r+1 , K3 ). On the other hand, the
first one of these spaces is connected, by the conjecture assumption, whereas the
second one is not, and has no connected components preserved by the Z2 -action.
Clearly, this yields a contradiction.
Here the degenerate case t = 2 makes sense, if we let C2 be a graph with two
vertices, connected by an edge (or a double edge).
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1111
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6
4
2
Now, by Corollary 4.1.8, we have Hom+ (Ct , Kn ) Ind (Ct )∗n , hence we derive
an explicit description.
Corollary 5.2.2 ([BK04, Corollary 4.2]).
For any t ≥ 2, we have
nk−1
2n copies S , if t = 3k;
Hom+ (Ct , Kn )
S nk−1
, if t = 3k ± 1.
This is a very convenient situation for us, since we know that the spectral
sequence converges to something with a single nonzero entry.
H ∗ (Hom (C2r+1 , Kn ))
d1 d1 d1 d1
D2 2n − 4
d1 d1 d1 d1
D1 n−2
n−3 d2
d1 d1 d1 d1
D0 0
q
p 0 2r − 2 2r − 1 2r
Figure 5.2.2. The E1∗,∗ -tableau, for E1p,q ⇒ H p+q (Hom+ (C2r+1 , Kn ); Z).
3n − 6
2n − 4
n−2
i
2r
Figure 5.2.3. The possibly nonzero entries in E2∗,∗ -tableau, for E2p,q ⇒
H p+q (Hom+ (C2r+1 , Kn ); Z).
would be too large. Almost always this ensures that the entries of E2∗,∗ outside of
the shaded area on Figure 5.2.3 are equal to 0. There are two exceptional cases:
(n, t) = (5, 2), and n = 4. These cases can then be computed ”by hand”, using
rather specific observations, see [BK04, Subsections 4.6 and 4.7].
We remark here that the results presented in the Table 5.2.1 have been some-
what strengthened recently.
Theorem 5.2.3 ([CK04b, Corollary 4.6]).
For arbitrary integers r, n ≥ 3, the complex Hom (Cr , Kn ) is (n − 4)-connected.
Let us now return to Theorem 5.1.5. From the Table 5.2.1, we see that,
in most of the cases, 2 · ι∗Kn is a 0-map for a prosaic reason: the target group
H ∗ (Hom (C2r+1 , Kn ); Z) is isomorphic to Z2 . The only exception is the case n = 4,
LECTURE 5. THE PROOF OF THE LOVÁSZ CONJECTURE 301
r ≥ 4. The validity of the statement of Theorem 5.1.5 in this special case can
be verified by the direct analysis of the map d1 : E12r−1,2 → E12r,2 , see [BK04,
Subsection 4.8] for details.
We notice at this point that the support map supp : Hom+ (C2r+1 , Kn ) →
Δ[2r+1] is Z2 -equivariant and hence it induces the quotient map supp /Z2 :
Hom+ (C2r+1 , Kn )/Z2 → Δ[2r+1] /Z2 . In order to get simplicial structure on
Δ[2r+1] /Z2 , we subdivide Δ[2r+1] in a minimal way, so that every simplex preserved
by Z2 -action is fixed by this action pointwise.
One can think of this new subdivision as the one obtained by representing
simplex Δ[2r+1] as a topological join of one point and r intervals: {c} ∗ [a1 , b1 ] ∗
· · · ∗ [ar , br ], inserting an extra vertex ci into the middle of each of the [ai , bi ], and
then taking the join of {c} and the subdivided intervals. We denote the obtained
abstract simplicial complex by Δ̃[2r+1] .
The Z2 -quotient of this simplicial structure gives one on Δ[2r+1] /Z2 , and we
can consider the Serre filtration on Hom+ (C2r+1 , Kn )/Z2 associated with the map
supp /Z2 .
302 D. N. KOZLOV, MORPHISM COMPLEXES, AND STIEFEL-WHITNEY CLASSES
We can now state the analog of the formula (4.3.3) for the spectral sequence
of the quotient. The analogs of formulae (4.3.1), and (4.3.2), are straightforward,
and are omitted for the sake of space, see [BK04, Section 6] for further details.
E1p,q = H q−p (Hom (C2r+1 [ϑ(σ)], Kn )/Z2 ; Z2 )
σ
(5.3.1)
H q−p (Hom (C2r+1 [ϑ(τ )], Kn ); Z2 ),
τ
where the first sum is taken over all σ ⊆ C, such that |σ| = p + 1, and the second
sum is taken over all Z2 -orbits τ , such that τ ⊆ V (Δ̃[2r+1] ), |τ | = p + 1, and
τ \ C = ∅.
...
0 d4
0 Proposition 5.3.4
d3
n−2 0
n−3 d2 Z2 d2
0
d3
0
...
Figure 5.3.1. The E2∗,∗ -tableau, E2p,q ⇒ H p+q (Hom+ (C2r+1 , Kn )/Z2 ; Z2 ).
Corollary 6.1.6. Any connected bipartite graph T with a Z2 -action which flips
an edge is a Stiefel-Whitney test graph.
In particular, any even cycle with the Z2 -action which flips an edge is a Stiefel-
Whitney test graph.
Summary. The class of Stiefel-Whitney test graphs contains complete graphs, con-
nected bipartite graphs (in both cases one can take any involution which flips
an edge). Furthermore, it is closed under factorizations, as described in Proposi-
tion 6.1.5.
By Theorem 5.1.4, the odd cycles are Stiefel-Whitney n-test graphs, for odd
n ≥ 3. Conjecturally, see Conjecture 6.2.5, odd cycles are Stiefel-Whitney n-test
graphs, for all n ≥ 3.
308 D. N. KOZLOV, MORPHISM COMPLEXES, AND STIEFEL-WHITNEY CLASSES
1At the time of the writing of this survey, this conjecture has been proved and is now a theorem,
see [CK04b].
LECTURE 6. SUMMARY AND OUTLOOK 309
We remark here that the Conjecture 6.2.5, coupled with Theorem 3.3.5, implies
the Conjecture 5.1.3. Note also that, as previously remarked, for a fixed value of
n, if the equation (6.2.1) is true for C2r+1 , then it is true for any C2r̃+1 , if r ≥ r̃.
We finish with another conjecture by Lovász. In [BW04], Brightwell & Winkler
have shown the following result.
Theorem 6.2.6 (Brightwell & Winkler, [BW04]).
Let G be an arbitrary graph. If for any graph T , with maximal valency at most d,
the graph Hom 1 (T, G) is connected or empty, then χ(G) ≥ d2 + 2.
Lovász has suggested that this statement can be strengthened, and that fur-
thermore, a higher dimensional analog is true.
Conjecture 6.2.7 (Lovász). Let G be an arbitrary graph. If for any graph T , with
maximal valency at most d, the complex Hom (T, G) is k-connected or empty, then
χ(G) ≥ d + k + 2.
[BW04] G.R. Brightwell, P. Winkler, Graph homomorphisms and long range ac-
tion, Graphs, morphisms and statistical physics, DIMACS Ser. Discrete
Math. Theoret. Comput. Sci., 63, Amer. Math. Soc., Providence, RI,
2004, pp. 29–47.
[Cay78] A. Cayley, On the coloring of maps, Proc. London Math. Soc. vol. 9,
(1878), p.148.
[Co73] M. Cohen, A course in simple-homotopy theory, Graduate Texts in Math-
ematics, Vol. 10, Springer-Verlag, New York-Berlin, 1973.
[Cs04a] P. Csorba, Homotopy type of the box complexes, preprint, 11 pages, 2004.
arXiv:math.CO/0406118
[Cs04b] P. Csorba, private communication, 2004.
[CL04] P. Csorba, F. Lutz, private communication, 2004.
[CK04a] S.Lj. Čukić, D.N. Kozlov, The homotopy type of the complexes of graph
homomorphisms between cycles, Discrete Comp. Geometry, in press.
arXiv:math.CO/0408015
[CK04b] S.Lj. Čukić, D.N. Kozlov, Higher connectivity of graph coloring complexes,
Int. Math. Res. Not. 2005, no. 25, pp. 1543–1562.
arXiv:math.CO/0410335
[tD87] T. tom Dieck, Transformation groups, de Gruyter Studies in Mathematics,
8. Walter de Gruyter & Co., Berlin, 1987. x+312 pp.
[Dir52] G.A. Dirac, A property of 4-chromatic graphs and some remarks on critical
graphs, J. London Math. Soc. 27, (1952), pp. 85–92.
[Dol88] V.L. Dol’nikov, A combinatorial inequality, (Russian) Sibirsk. Mat. Zh. 29
(1988), no. 3, pp. 53–58, 219; translation in Siberian Math. J. 29 (1988),
no. 3, pp. 375–379.
[FFG86] A.T. Fomenko, D.B. Fuks, V.L. Gutenmacher, Homotopic topology, Trans-
lated from the Russian by K. Mályusz. Akadémiai Kiadó (Publishing
House of the Hungarian Academy of Sciences), Budapest, 1986.
[For98] R. Forman, Morse theory for cell complexes, Adv. Math. 134, (1998), no.
1, pp. 90–145.
[GJ76] M.R. Garey, D.S. Johnson, The complexity of near-optimal graph coloring,
J. Assoc. Comp. Mach. 23, (1976), pp. 43–49.
[GJ79] M.R. Garey, D.S. Johnson, Computers and Intractability, A guide to the
theory of NP-completeness, A Series of Books in the Mathematical Sci-
ences, W.H. Freeman and Co., San Francisco, 1979.
[GM96] S.I. Gelfand, Y.I. Manin, Methods of homological algebra, Springer-Verlag,
Berlin Heidelberg, 1996.
[GR01] C. Godsil, G. Royle, Algebraic Graph Theory, Graduate texts in mathe-
matics 207, Springer-Verlag, New York, 2001.
[Gr02] J. Greene, A new short proof of Kneser’s conjecture, Amer. Math. Monthly
109 (2002), no. 10, pp. 918–920.
[Gut80] F. Guthrie, Note on the colouring of maps, Proc. Roy. Soc. Edinburgh,
vol. 10, (1880), p. 729.
[Had43] H. Hadwiger, Über eine Klassifikation der Streckenkomplexe, Vierteljschr.
Naturforsch. Ges., Zürich, vol. 88, (1943), pp. 133–142.
[Har69] F. Harary, Graph Theory, Addison-Wesley Series in Mathematics, Read-
ing, MA, 1969.
BIBLIOGRAPHY 313
Robert MacPherson
IAS/Park City Mathematics Series
Volume 14, 2004
Robert MacPherson
Introduction
1 Institute
for Advanced Study, Princeton NJ 08540.
E-mail address: rdm@ias.edu.
c
2007 American Mathematical Society
319
320 R. MACPHERSON, EQUIVARIANT INVARIANTS AND LINEAR GEOMETRY
0.2 Guide to reading. The Lectures have been made independent of each other
as much as possible, so as to allow several different points of entry into the subject.
The following is the diagram of dependencies:
1.2 We can visualize the n-torus as an n-cube [0, 1]n with the opposite faces
identified. For example, if n = 1, we have S 1 = [0, 1]/ ∼ where ∼ identifies 0 and
1.
INTRODUCTION 321
Or, for example, the 2-torus is the square with the opposite edges identified,
1.3 Exercise. Show in general that an n-torus as an n-cube [0, 1]n with the
opposite faces identified. Hint: Show every Zn coset in Rn meets the unit cube
[0, 1]n ⊂ Rn , so Rn /Zn = [0, 1]n / ∼ where x ∼ y when x − y ∈ Zn . Check that ∼
identifies opposite faces.
1.4 Exercise*. Let T be the n-torus Rn /Zn and let T be the k-torus Rk /Zk .
Every group homomorphism h : Zn −→ Zk extends uniquely to a continuous group
homomorphism h̃ : Rn −→ Rk , and so it passes to a continuous group homomor-
phism h̄ : T −→ T . Show that the map Hom(Zn , Zk ) −→ Hom(T, T ) that sends h
to h̄ is an isomorphism. Here Hom(T, T ) is the set of continuous homomorphisms
from T to T .
322 R. MACPHERSON, EQUIVARIANT INVARIANTS AND LINEAR GEOMETRY
2.1 Definition. A linear graph is a finite set of points {vi } in a real vector space
V, called vertices, and a finite set of line segments {ek } in V, called edges such that
the two endpoints of each edge are both vertices.
2.2 For example, the following are linear graphs:
The first one, in R2 , has four vertices and six edges. Note that the edges do not have
to be disjoint: In this example, the two diagonals cross each other. The second one
has six vertices and twelve edges. It is just the vertices and edges of an octahedron
in R3 . Any convex polyhedron gives rise to a linear graph by taking the vertices
and the edges.
2.3 A topological graph is, of course, defined in a similar way, but without the
embedding into a vector space. (For our purposes, a topological graph has at most
INTRODUCTION 323
one edge between a pair of vertices, and has no edge going from a vertex to itself.)
So a linear graph is a graph together with a mapping into V in such a way that its
edges are mapped into straight lines.
2.6 Exercise. What is the dimension of the space of linear graphs equivalent to
the linear graphs pictured above?
2.8 Exercise. Consider the triangle graph with the direction data that assigns
to the three edges the following three directions D in R3 : (1, 0, 1), (−1, 1, 1), and
(0, −1, 1). Show that there is no linear graph with this direction data.
3.1 Our rings R will all be graded algebras over the real numbers R.
324 R. MACPHERSON, EQUIVARIANT INVARIANTS AND LINEAR GEOMETRY
Since all of our graded rings are zero in odd degree, it is conventional to introduce
the variable q = x2 .
Hilb(R) = (x2 )j dim(R2j ) = q j dim(R2j ).
j≥0 j≥0
2) Or, calculate the Hilbert series of the polynomial ring O(R) of polynomials in
one variable
1
Hilb(O(R)) = 1 + q + q 2 + · · · =
1−q
then justify the following manipulations:
× ·
Hilb(O(T)) = Hilb(O(R · · × R
) = Hilb(O(R) ⊗ · · · ⊗ O(R)) =
n factors n factors
n
1
= Hilb(O(R)) · · · Hilb(O(R)) =
1−q
n factors
3.7 Exercise. Let R be the ring of continuous functions on the real line R, whose
restriction to the positive reals R>0 and the negative reals R<0 are both polynomial
functions. Show that R is a graded ring isomorphic to the polynomial ring in two
variables x and y divided by the principal ideal generated by the polynomial xy,
i.e. R = R[x, y]/(xy), and that its Hilbert series is (1 + q)/(1 − q).
LECTURE 1
Equivariant Homology and Intersection Homology
(Geometry of Pseudomanifolds)
1.1. Introduction
1.1 In this lecture, we will give a geometric way of defining equivariant homology
and equivariant intersection homology. The standard definitions of these homology
theories, as found in the literature, are good for proving properties, but are perhaps
not so intuitive. In this lecture, we will consider G X: an action of a general Lie
group G on a space X, although in the other lectures we are interested mainly in
the case that G is a torus T .
1.2 The definitions we present are based on the notion of a pseudomanifold. A
k-dimensional manifold is a space that looks locally like k-dimensional Euclidean
space near every point. A k-dimensional pseudomanifold P is allowed to have
singularities, i.e. points where it doesn’t locally look like Euclidean space. However,
it must satisfy two properties:
(1) The part of P where it is a k-manifold is open and dense in P and it must
be oriented.
(2) The set of singularities has dimension at most k − 2 (i.e. codimension at
least 2).
There are several ways to make this intuitive notion of a pseudomanifold rigorous.
We will use simplicial complexes, because that is the one most in keeping with the
spirit of these notes. Readers who are comfortable with pseudomanifolds can skip
directly to §1.4
1.3 Equivariant homology theories are difficult to compute directly from the de-
finitions as given in this Lecture. However, the methods of Lectures 3 to 5 provide
effective computations in many interesting cases.
can draw an orientation by representing the basis vectors as arrows, and signaling
the ordering by placing the tail of each arrow at the head of the previous one.
An orientation O of k-simplex Δ determines an orientation of the k-dimensional
Euclidean space E containing Δ as follows: Suppose O = {p0 < p1 < · · · < pk }.
Then {p1 − p0 , p2 − p1 , . . . , pk − pk−1 } is the ordered basis.
Exercise. Show that two orientations of Δ are equivalent if and only if they
determine equivalent ordered bases of E.
2.6 If Δ is a k-simplex and Δ is a (k − 1)-simplex, an orientation O of Δ induces
an orientation O of Δ as follows: Pick an equivalent ordering such that the unique
vertex of Δ not in Δ is the last one of the ordering. Then O is the restriction of
that ordering to Δ . (This definition doesn’t work if Δ is a 1-simplex. In this case,
O is − if Δ is the first vertex of the ordering, and it is + if it is the second one.)
1.3. Pseudomanifolds
3.2 The following exercise shows why property 3 is called continuity of orientation.
Exercise. Suppose that Δ and Δ̃ are two k-simplices in a Euclidean k-space, and
that they intersect in a (k − 1)-simplex Δ . Show that the orientations O(Δ) and
O(Δ̃) induce opposite orientations on Δ if and only if the ordered basis for E
determined by Δ as in exercise 1.2.5 can be continuously deformed into the ordered
basis for E determined by O(Δ̃).
3.4 Exercise. Show that the continuity of orientation property for the boundary
B of a pseudomanifold with boundary follows from the other properties in the
definition.
The idea behind this definition is that if P1 and P2 are cobordant, they surround
the same holes in the same way. For example, if σ has appropriate differentiability
assumptions so that it makes sense, any closed differential i-form will have the same
integral on P1 and on P2 , by Stokes’ Theorem.
4.4 Proposition. Cobordism is an equivalence relation among i-cycles.
Exercise Prove this. For example, if S1 is a cobordism between P1 and P2 , and
S2 is a cobordism between P2 and P3 , then S1 and S2 can be glued together to
provide a cobordism between P1 and P3 .
332 R. MACPHERSON, EQUIVARIANT INVARIANTS AND LINEAR GEOMETRY
4.5 Definition – Proposition. The i-th homology, notated Hi (X), is the set
of cobordism classes of i-cycles. The operations + and − induce the structure of
an Abelian group on this set. The identity element is represented by the empty
pseudomanifold.
4.6 For example, if X is the annulus, H0 (X) is Z generated by a point, and
H1 (X) is Z generated by the cycle P1 or P2 as in the pictures above.
4.7 Exercise. Show for any X that H0 (X) is Zk where k is the number of path
connected components of X.
4.8 Convention. We write H∗ (X) for i Hi (X). It’s a summation convention:
wherever a star appears, it means a direct sum over the possible indices i that
might appear there.
π σ
P ←−−−− E −−−−→ X
6.2 When we want to refer to an i-cycle, we use the symbol P for the pseudo-
manifold, even though it is really a 4-tuple of data. The sum P1 + P2 of two
equivariant i-cycles P1 and P2 is P1 ∪ P2 ←− E1 ∪ E2 −→ X. The negative −P
of an equivariant i-cycle is the same i-cycle with the opposite orientation on P .
334 R. MACPHERSON, EQUIVARIANT INVARIANTS AND LINEAR GEOMETRY
The two nonzero equivariant homology groups are generated by the equivariant
cycles shown in the pictures above.
Similarly, the equivariant homology of the annulus is nonzero only in degree 0,
where it is generated by the 0-cycle shown in the picture.
In both of these examples, the equivariant homology is smaller than the ordi-
nary homology. For the cases we are going to study later, the reverse is true.
6.8 Exercise. Show that H0 (G X) = H0 (X), the free group generated by the
connected components of X.
6.9 Exercise. Show that Hi (1 X) = Hi (X), where 1 is the one element group.
Proposition. The composed map from the left to the right in this diagram is the
multiplication rule of a ring structure on equivariant cohomology.
7.6 Signs. The product in cohomology satisfies the following sign rule: If x ∈
H i (G X) and y ∈ H j (G X), then xy = (−1)ij yx. The reason for this it that if
P1 and P2 are pseudomanifolds of dimensions i and j, then P1 × P2 = (−1)ij P2 ×
P1 . A nice exercise is to figure out how to define precisely the product of two
pseudomanifolds, and to show this commutation rule.
7.7 Exercise. Use the map 1 X =⇒ G X to show that there is a canonical
map Hi (X) −→ Hi (G X), or dually H i (G X) −→ H i (X).
7.8 Exercise. Let A be the ring H ∗ (G pt), where pt is a point. Use the map
G X =⇒ G pt to show that there is a map A −→ H ∗ (G X), so H ∗ (G X)
is an A-module. We will be interested mainly in actions where this map is an
injection, unlike the illustrative examples considered in the last section.
7.9 Homology vs. cohomology. There is a psychological dilemma. An equi-
variant cohomology class is hard to imagine. It is an element in a dual space — it
eats a homology class and gives you a number. But the cohomology is an algebra,
which most people find to be an intuitive structure. An equivariant homology class
is easy to visualize, but the homology forms a co-algebra, which is hard to think
about.
Choosing the demons of dual spaces over the demons of co-algebras, our com-
putations will be in equivariant cohomology. Of course, the computation of equi-
variant cohomology is mathematically equivalent to the computation of equivariant
homology, so the information is the same in the end.
LECTURE 1. EQUIVARIANT HOMOLOGY AND INTERSECTION HOMOLOGY 337
Here S 2k+1 is the real unit real sphere in complex (k + 1)-space, given by the
equation |z0 |2 + |z1 |2 + · · · + |zk |2 = 1. The circle T 1 = S 1 ⊂ C acts on it freely
by scalar multiplication. The quotient space CPk is the complex projective k-space
(see §2.3.1), a pseudomanifold (indeed a manifold) of real dimension 2k.
8.2 It is useful to think about why this 2k-cycle is nonzero. 1. The 2k + 1 sphere
bounds the 2k + 2 ball |z0 |2 + |z1 |2 + · · · + |zk |2 ≤ 1. The S 1 action extends to the
ball. If k > 0, the quotient is a pseudomanifold with boundary CPk . Why isn’t this
a cobordism to zero? 2. The fibers of the map S 2k+1 −→ CPk are all circles S 1 .
There is a “trivial” example of a map to CPk whose fibers are all circles: if we had
a homeomorphism S 1 × CPk −→ CPk which is cobordant to zero, since we can take
S 1 × C(CPk ) −→ C(CPk ) where C(CPk ) is the cone on CPk . So, since our 2k-cycle
is not cobordant to zero, it must not be equivalent to the“trivial” example.
8.3 Exercise*. Prove that the 2k-cycle above is not 0 in H2k (T 1 pt). Hint:
Use characteristic classes.
The basis {1, t, t2 , . . .} of H ∗ (T 1 pt) is dual to the basis {CP0 , CP1 , CP2 , . . .} of
H∗ (T 1 pt).
T pt = T 1 pt ×T 1 pt × · · · × T 1 pt
n factors
338 R. MACPHERSON, EQUIVARIANT INVARIANTS AND LINEAR GEOMETRY
n factors
⎧ ⎫
⎪
⎨ ⎪
⎬
= polynomial functions on T1 × · · · × T1
⎪
⎩
⎪⎭
n factors
= {polynomial functions on T} = O(T)
9.1 Homology of the fixed point set N ∪ S. Suppose that the circle T 1 acts
on the 2-sphere X = S 2 by rotation as in §0.1.4. There are two fixed points, the
North pole N and the South pole S. The space N ∪ S is just two points, so its T 1
equivariant homology is just two copies of the equivariant homology of a point:
R[(CPk )N ] ⊕ R[(CPk )S ] = R2 if i = 2k
Hi (T 1 (N ∪ S)) =
0 if i is odd
Hi (T 1 (N ∪ S)) −→ Hi (T 1 X)
is an isomorphism for all i > 0. For i = 0, there is one relation [CP0N ] = [CP0S ]
given by the following cobordism whose boundary is CP0N − CP0S .
9.3 Exercise*. This result says that every equivariant cycle for T X is cobor-
dant to one that maps into just the two fixed points N and S. The corresponding
statement in ordinary homology (i.e. 1 X) is false. Can you see geometrically why
this is true?
9.4 Translating this calculation to equivariant cohomology. In summary,
the equivariant homology of X is a quotient of the equivariant homology of N ∪ S;
i.e. we have the exact sequence of graded vector spaces
q
0 −−−−→ R −−−−→ H∗ (T 1 (N ∪ S)) −−−−→ H∗ (T 1 X) −−−−→ 0
where the map q sends 1 to [CP0N ] − [CP0S ]
. Dualizing, the equivariant cohomology
of X is a sub of the equivariant cohomology of N ∪S; i.e. we have the exact sequence
of rings:
q∗
0 ←−−−− R ←−−−− H ∗ (T 1 (N ∪ S)) ←−−−− H ∗ (T 1 X) ←−−−− 0.
∗ ∗ ∗
Here H (T 1
(N ∪ S)) = H (T N ) ⊕ H (T
1
S) which is two copies of the
1
ring O(T1 ) of polynomials on T1 . The map q ∗ sends the difference of the identity
elements of the two copies of the polynomial ring 1N − 1S to 1 ∈ R.
In other words, H ∗ (T 1 (N ∪ S)) is the ring of pairs (fN , fS ) of polynomial
functions on T1 = R. The ring H ∗ (T 1 X) is pairs (fN , fS ) such that fN (0) =
fS (0).
9.5 The torus equivariant cohomology of a sphere. Now suppose that an
n torus T acts on the sphere X = S 2 by rotating it. By changing coordinates in
the torus, we can arrange things so that T = T 1 × T n−1 where the circle T 1 acts
on X as in the discussion above and the torus T n−1 acts trivially. Therefore, we
have the product of spaces with group action
T X = T1 X × T n−1 p.
Applying the Kunneth theorem, we get
H ∗ (T X) = H ∗ (T 1 X) ⊗ H ∗ (T n−1 pt)
= (fN , fS ) ∈ O(T 1 ) ⊕ O(T 1 ) such that fN |0 = fS |0 ⊗ O(Tn−1 )
= (fN , fS ) ∈ O(T 1 × Tn−1 ) such that fN |Tn−1 = fS |Tn−1
In Lecture 3, this simple calculation will lie at the root of all of our calculations
of equivariant cohomology (and ordinary cohomology) of many complicated spaces.
340 R. MACPHERSON, EQUIVARIANT INVARIANTS AND LINEAR GEOMETRY
The various types of homology groups of X that have been defined in this lecture
are given in the following table:
Type of
homology i odd i = 0 i=2 i even, i ≥ 4
Hi (X) 0 R R2 0
Hi (T X) 0 R R 3
R3
IHi (X) 0 R2 R2 0
IHi (T X) 0 R 2
R 4
R4
342 R. MACPHERSON, EQUIVARIANT INVARIANTS AND LINEAR GEOMETRY
Give explicit cycles generating these groups, and give plausibility arguments that
these calculations are correct. (The hardest ones are the 4 generators of IHi (T X)
for i ≥ 2 and even. Each generator of IH2 (T X) may be represented as a 3-sphere
with a free T action, mapped into X is such a way that the inverse image of a fixed
point in X is a single T orbit. This has codimension 2 in the 3-sphere, so it satisfies
the allowability condition 1.10.2.)
This example actually comes up. It is a generalized Schubert variety §4.7.3,
and it is a Springer variety §5.4.6. The calculation methods of Lectures 3, 4, and 5
all apply to this example.
LECTURE 2
Moment Graphs
(Geometry of Orbits)
1.3 Exercise. Suppose T acts on S 2 so that the orbits are the North pole
N = (0, 0, 1), the South pole S = (0, 0, −1) and the circles of constant latitude
z = c where c is a constant between −1 and 1. Show that T S 2 is equivalent to
a balloon.
1.4 Exercise*. Show that any action of a torus T on S 2 is either a balloon or
else it’s the trivial action, where every point of T leaves every point of S 2 fixed.
1.5 Definition. A balloon sculpture is a space with a torus action such that is a
finite union of balloons Bj such that any two balloons are either disjoint or intersect
a fixed point of the torus action.
A balloon sculpture Y
The graph Y /T
2.7 Exercise. Construct an example T X where the moment graph does not
exist. (Hint: Take X to be the balloon sculpture whose direction data coincides
with that of exercise 0.2.8.)
Stereographic projection takes rotation about the z axis to rotation in the complex
plane about 0, i.e. to multiplication by a complex number on the unit circle S 1 .
Now, suppose that the 2 torus acts on the projective line by the formula
z(x1 : x2 ) = (z1 x1 : z2 x2 ).
Proposition. With this action, P1 is a balloon B where the direction vector DB
is e1 − e2 in V. (Here ei is the standard basis as in §2.2.4.)
z2 x2 e2πit2 x2 x2 x1 x1
= 2πit1 = e2πi(t2 −t1 ) =e 2πi(e2 −e1 )(t1 ,t2 )
=e 2πi(e2 −e1 )t
z1 x1 e x1 x1 x2 x2
which means that t̄ gives a rotation of (e1 − e2 )(t) Alternatively, the proposition
can be seen by §2.2.5: If (e2 − e1 )(t1 , t2 ) = 0, then t1 = t2 so (e2πit1 x1 : e2πit2 x2 )
is the same point as (x1 ; x2 ) because both homogeneous coordinates are multiplied
by the same number.
3.3 Exercise. More generally, show that if an n-torus T acts on the projective
line by t̄(x1 : x2 ) = (e2πiφ1 (t) x1 : e2πiφ2 (t) x2 ) for φ1 , φ2 : T −→ R, and φ1 = φ2 ,
then P1 is a balloon with DB = φ1 − φ2 .
3.4 Almost all of the balloons in the 1-skeleta of the T X we will consider in
this Lecture are themselves a copy of P1 embedded in the space X. So the analysis
of this section will be used repeatedly in what follows.
4.1 The fixed points are the n points Fi where all the homogeneous coordinates
are zero except the i-th one.
4.2 The balloons. Let i and j be any pair of distinct indices 1 ≤ i, j ≤ n. Then
the balloon Bij is where all the homogeneous coordinates are zero except the i-th
one or the j-th one. It connects the fixed points Fi and Fj .
4.3 Remark: balloons and C∗ orbits. The action of T = S1n on Pn−1 extends
to an action of TC = (C∗ )n where C∗ is the nonzero complex numbers considered
as a group under multiplication. The action of TC is given by the same formula
z(x1 : · · · : xn ) = (z1 x1 : · · · : zn xn ) where zj ∈ C∗ . Each balloon consists of
three TC orbits: the two fixed points and one more, of complex dimension 1. So the
classification of balloons is the same as the classification of complex one dimensional
orbits of the TC action.
4.6 The proofs. The points Fi are fixed by the equivalence relation on homo-
geneous coordinates. The sets Bi,j are projective lines, so they are spheres. The
action of T on Bi,j is very similar to the action of §2.3.2, so the direction vectors
can be computed in a similar way. Alternatively, §2.3.3 can be used directly. The
only real challenge is to show that the 1-skeleton is the union of the balloons Bij .
This follows from the following exercise.
4.7 Exercise. Show that if x ∈ Pn has k nonzero homogeneous coordinates, then
the dimension of the orbit T x is k − 1.
5.9 Exercise. Show that the number of faces of dimension i in the cross-polytope
On is the coefficient of q i+1 in the polynomial (1 + 2q)n .
6.1 The space Gni is the Grassmannian variety whose points are the i-dimen-
sional subspaces of the n-dimensional complex vector space Cn . The n-torus acts
on it through its action on Cn : z(x1 , . . . , xn ) = (z1 x1 , . . . , zn xn ).
6.2 The fixed points. Suppose that S is a subset of {1, 2, . . . , n}. Let PS be
the coordinate plane corresponding to S, i.e. PS is the |S|-plane defined by the
condition that only the coordinates {xj | j ∈ S} can be nonzero. Here |S| is the
number of elements of S. The fixed points in Gni are the planes PS where |S| = i.
We denote PS by FS when thinking of it as a fixed point in Gni .
6.4 Here is a picture of the planes in the balloon connecting F{1,2} , and F{2,3} in
G32 . Since we can’t visualize C3 , we’re using a real picture, i.e. real planes in the
real vector space R3 instead of complex planes in the complex vector space C3 .
6.5 The hypersimplex Δni is the intersection of the n-cube [0, 1]n ⊂ Rn = V
with the plane v1 + v2 + · · · + vn = i. It is a convex polyhedron with vertices
νS = Σj∈S ej where S is an i element subset of {1, . . . , n}. The vertices νS and νS
are connected by an edge if S is obtained from S by deleting the number i and
adding number j, for i = j. Then the edge is parallel to ei − ej .
352 R. MACPHERSON, EQUIVARIANT INVARIANTS AND LINEAR GEOMETRY
6.6 The moment graph of the Grassmannian Gni is the 1-skeleton of the hyper-
simplex Δni . There is a rich theory surrounding hypersimplices and Grassmannians
[12], [11], [6].
6.7 Exercise. Show that the hypersimplices can be arranged in a polyhedral
version of Pascal’s triangle where the faces of each polyhedron are isomorphic to
one of the two polyhedra lying above it.
For dimension up to 4, this is illustrated in the following picture. The labels
of vertices show which coordinates are 1 (or equivalently, which coordinate axes
are in the corresponding plane representing a T fixed point of the Grassmannian).
The figures in last line, representing 4-dimensional hypersimplices, are projections
to R3 called Schlegel diagrams. Note that the polyhedra on the two upper edges of
the picture are ordinary simplices.
6.8 The Lagrangian Grassmannian and the cube. Consider C2n with co-
ordinates x1 , . . . , xn , y1 , . . . , yn and the alternating form Σi xi yi − xi yi . The La-
grangian Grassmannian Ln is the subvariety of the Grassmannian G2n n consisting
of n planes on which this alternating form vanishes identically. The torus T acts
on Ln by through its action on C2n by the formula
z(x1 , . . . , xn , y1 , . . . , yn ) = (z1 x1 , . . . , zn xn , z1−1 yn , . . . , zn−1 yn ).
The fixed points FS are the coordinate planes that lie in Ln . For any subset
S ⊂ {1, . . . , n}, FS is the plane whose nonzero coordinates are the xi for i ∈ S and
the yi for i ∈ / S. There are 2n of them.
Exercise. Show that the vertices of the moment graph of Ln are the vertices of
the n-cube [0, 1]n ⊂ V and the edges of the moment graph are the edges of the cube
together with the diagonals of the 2-dimensional faces.
7.1 The flag manifold. Consider Cn as R2n in the usual way, with the standard
real dot product ·R on it. A point in the flag manifold Fn is an ordered set of
n mutually orthogonal complex lines through the origin in Cn . Here mutually
orthogonal means that if x is in one of the complex lines and y is in another one,
then x ·R y = 0. (This is the same as their being orthogonal with respect to the
standard Hermitian inner product.)
The n-torus T acts on Fn through its standard action on Cn . This action
preserves the orthogonality condition.
7.2 Fixed points. A point is fixed if the n mutually orthogonal lines coincide
with the complex coordinate axes in Cn . There are n! of them, one for each ordering
of the coordinate axes.
7.3 The balloons. Pick two coordinate axes of Cn , say the xi axis and the xj
axis. A balloon is the set where all but two of the mutually orthogonal complex
lines are required to lie on a coordinate axis that is not the xi axis or the xj axis.
The remaining two complex lines are free to wander (staying orthogonal to each
other) in the 2-dimensional plane spanned by the xi axis and the xj axis.
7.4 The permutahedron. Fix n distinct real numbers a1 , . . . , an . The permu-
tahedron is the convex hull in Rn of the n! points (aσ(1) , . . . aσ(n) ) where σ runs
through the n! permutations of the numbers {1, . . . , n}. It is an (n− 1)-dimensional
354 R. MACPHERSON, EQUIVARIANT INVARIANTS AND LINEAR GEOMETRY
7.5 The moment graph of the flag manifold. The vertices of the moment
graph for Fn are the vertices of the permutahedron. Two vertices
are connected by
an edge if one is a reflection of the other in one of the n2 hyperplanes defined by
an equation vi = vj .
Moment graph
G(T F3 )
8.3 F (p). Given a point p ∈ P of a polyhedron, we write F (p) for the smallest
face of P containing p. If p is a vertex, then F (p) is p itself. F (p) = P , if and only
if p is an interior point of P .
P ×T
T(P ) =
∼
8.5 The T action. The torus T = T/L acts on the toric variety T(P ) as follows:
T acts on P × T by vector addition t(p, t ) = (p, t + t), and this action passes to
an action of T on the quotient space T(P ). On the quotient space, L acts trivially,
since if t ∈ L, then t(p, t ) ∼ (p, t ) So the quotient group T/L acts on the quotient
space T(P ).
8.6 The moment map. There is a map μ : T(P ) −→ P called the moment map
which is induced from the projection (P × T) −→ P . The reason the projection
passes to the quotient T(P ) is that the equivalence relation ∼ is compatible with
this map — it identifies points only if they lie in the same fiber. In fact, there is
an identification T(P )/T ≈ P , the moment map T(P ) −→ T(P )/T is the quotient
map for the group action T T(P ).
Proposition. The fiber μ−1 p ⊂ T(P ) over a point p ∈ P is a torus of the same
dimension as the face F (p).
So we can think of the toric variety T(P ) as a family of tori over the polyhedron
P whose fiber dimensions decrease as you get to smaller faces. To visualize it, here
are some pictures of fibers at various points of P .
356 R. MACPHERSON, EQUIVARIANT INVARIANTS AND LINEAR GEOMETRY
We must show that L/(L ∩ F (p)⊥ ) is a lattice in T/F (p). Since this quotient
space will itself be a torus: it will be the vector space T/F (p)⊥ modulo the lattice
L/(L ∩ F (p)⊥ )
8.8 Exercise. Show that the following conditions are equivalent, and they all
hold if the polytope P is rational:
(1) The vector space F ⊥ is a rational subspace of T for all faces F .
(2) The vector space F ⊥ is spanned by F ⊥ ∩ L for all faces F .
(3) The quotient space T/(L + F ⊥ ) is Hausdorff for all faces F .
(4) The subgroup L/(L ∩ F (p)⊥ ) is a lattice in the vector space T/F (p).
LECTURE 2. MOMENT GRAPHS 357
8.9 Proposition. The moment graph of the toric variety T (P ) is the 1-skeleton
P 1 of P .
Since the dimension of the orbit μ−1 (p) is the dimension of F (p), the 1-skeleton of
T T(P ) is the inverse image of the 1-skeleton of P . The inverse image of an edge
of P is a balloon.
It remains to see that the direction vector of this balloon is parallel to the edge.
This follows from §2.2.5.
8.10 Exercise. Show that the projective (n − 1)-space Pn−1 is a toric T(P )
where P is an (n − 1)-simplex.
8.11 Simple polytopes. A polytope is simple if the edges coming in to every
vertex, considered as vectors, are linearly independent. For example, a tetrahedron
and a cube are simple, whereas an octahedron is not. All 2-dimensional polyhedra
are simple.
Toric varieties of simple polytopes play a special role that will become apparent
later (§3.8.1).
1. These are all the 1 complex dimensional orbits of TC . If we are given a complex
algebraic action of TC on X, then our hypothesis that the 1-skeleton of T X is a
balloon sculpture is equivalent to the hypothesis that TC has finitely many orbits
of complex dimension 0 and 1.
9.3 The moment map. If X is nonsingular and projective, then it has a real
symplectic form ω called the Kähler form. By Weyl’s trick of averaging over T , we
can choose ω to be T invariant. We define a V-valued differential 1-form θ on X as
follows: For t ∈ T, let ξt be the corresponding vector field on X. If τ ∈ Tx X is a
tangent vector to X at x, then t → ω(τ, ξt (x)) is a linear map T −→ R, so it is an
element of V = T∗ . That element is θ(τ ). The moment map μ : X −→ V is defined
by the formula x
μ(x) = θ
x0
where x0 is a base point chosen in X. (If X is not connected, we define μ on each
connected component separately by this procedure.)
If X is singular, we proceed a little differently. We embed X in a complex
projective space in a way that is TC equivariant. Then we take the moment map on
the ambient complex projective space as constructed above, and restrict it to X.
If T X is a toric variety, then the moment map as defined here will coincide
with the moment map from its definition as a toric variety.
9.4 Proposition. If TC acts algebraically on X with finitely many orbits of
dimension 0 and 1, then the moment graph G(T X) is μ(X 1 ), the moment map
image of its 1-skeleton. The set of vertices of the moment graph is μ(X 0 ). The
image μ(X) of all of X will be the convex hull of the moment graph. There were
several choices in constructing the moment map (choice of a Kähler form, choice of
a base point). Different choices will result in different but equivalent linear graphs.
9.5 Exercise*. Suppose X is nonsingular and compact, and that TC acts al-
gebraically on X with finitely many fixed points F . Suppose further that at each
fixed F , the representation TC TF X on the tangent space has no representation of
multiplicity greater than 1. Show that TC acts with finitely many one dimensional
orbits, so that the 1-skeleton of T X is a balloon sculpture.
LECTURE 3
The Cohomology of a Linear Graph
359
360 R. MACPHERSON, EQUIVARIANT INVARIANTS AND LINEAR GEOMETRY
where Hi (G) = 0 if i is odd, and H2k (G) is the set of elements represented by sets
of polynomials {fν , . . .}, each of which is homogeneous of degree k (i.e. every term
of fν is of degree k). If α ∈ Hi (G) and β ∈ Hj (G), then the product αβ ∈ Hi+j (G).
1.4 Module structure. The ring H∗ (G) is a graded module over the graded
ring O(T) of polynomial functions on T. The module action of g ∈ O(T) sends
{fν , fν , . . .} ∈ H∗ (G) to {gfν , gfν , . . .}.
1.6 Exercise. Show that the graded structure, the module structure, and the re-
striction, as defined above, make sense – for example that they respect the condition
for each edge of G in the definition of H∗ (G).
1.7 Sections 3.2 to 3.7 will be devoted to the study of the cohomology ring of
a graph. The definition is simple enough, but it is not immediately clear from
the definition how you would compute it or how to think about it. The papers of
Guillemin, Holm, and Zara are recommended for further reading [18], [19], [20],
[21], [22].
2.2 The degree 2 part of the cohomology. The dimension of the vector space
H2 (G) is the dimension of the space of graphs in V that are equivalent to G (see
§0.2.4).
2.3 Proof. An element of vertices ν of G O(T) is a linear function on T for
2
= the graph G
2.4 Exercise. Determine the dimension of H2 (G) for all of the linear graphs
pictured in Lecture 2.
2.5 The degree 4 part of the co-
homology. The dimension of the vec-
tor space H4 (G) is the dimension of the
space C(G) of configurations of the fol-
lowing sort: For each vertex ν of the
graph G, we give an ellipsoid Eν in V
centered at ν. For each edge
νν , we
ask that when you take the projection
along the direction of
νν to an (n − 1)-
dimensional quotient space of V, the two
ellipsoids Eν and Eν should have the
same image. (Recall that an ellipsoid is
the zero set of a degree two polynomial
that is compact.) I am indebted to Vic- A configuration of ellipses in C(G)
tor Guillemin for this interpretation of
H4 (G).
2.6 Exercise. Prove this statement. More precisely, prove that the tangent
space to C(G) at any point is canonically H4 (G).
critical values are distinct, i.e. for any pair of vertices ν and ν of G, φ(ν) = φ(ν ).
It follows that φ is not constant on any edge of G.
Morse functions exist for any linear graph. In fact, if you choose a linear
function φ : V −→ R at random, you have to be infinitely unlucky to get one that
is not Morse.
4.2 The truncated graph. Now suppose c is a real number, which we call the
“cut-off value”. We define G ≤c to be the subgraph of G consisting of those vertices
ν such that the critical value φ(ν) ≤ c, together with all the edges
νν connecting
vertices ν and ν both of which have critical values ≤ c.
4.3 The Morse module. Suppose that c1 < c2 < · · · < ck are the critical values
of the Morse function φ, and c0 is a real number less than the smallest critical value
c1 . Then for all integers j ∈ {1, . . . , k}, we have
ij
G ≤cj−1 ⊂ G ≤cj
i∗
j
H∗ (G ≤cj−1 ) ←− H∗ (G ≤cj ) ←− Mj ←− 0
Here i∗j is the map on cohomology induced by the inclusion of graphs ij and Mj
the kernel of the map i∗j . The kernel Mj is a graded module over O(T) because it
is the kernel of a map of graded modules. It is a graded ideal in H∗ (G ≤cj ), but it
is more useful to think of it as a O(T)-module. The module Mj is called the Morse
module of the vertex νj whose critical value is cj .
364 R. MACPHERSON, EQUIVARIANT INVARIANTS AND LINEAR GEOMETRY
As a module over O(T), Mj is a free module generated by gνj , which lies in the
graded piece O(T)Index(νj ) .
4.6 Exercise. Finish the proof of this proposition.
4.7 Exercise*. Suppose that G is the 1-skeleton of a simple polyhedron. Show
that the ordering of the vertices given by a Morse function corresponds to a linear
shelling of the dual simplicial polytope.
But since the Morse module Mj is a free O(T) module on a generator of degree
Index(νj ), and the Hilbert series of O(T) is computed in §0.3.5, we have the follow-
ing:
5.3 Proposition. If φ is a perfect Morse function, the Hilbert series of the
cohomology of the graph is given by
k n k n
1 1
Hilb(H∗ (G)) = xIndex(νi ) = q Index(νi )/2
.
i=1
1 − x2 i=1
1−q
5.4 The Betti numbers of a graph. Suppose that G has a perfect Morse
function φ. Then we define the Betti numbers Bi of G to be the number of vertices
of G whose Morse index is i. Note that Bi is automatically zero if i is odd.
We define the Poincaré polynomial P to be
P (x) = Bi xi .
i
so we have n
1
Hilb(H∗ (G)) = P (x)
1 − x2
and
n
P (x) = Hilb(H∗ (G)) 1 − x2
where the last expression, which is a priori an infinite power series, is actually a
polynomial.
5.5 Exercise. Show that if the graph G has more than one different perfect Morse
function, the Betti numbers (and the Poincaré polynomials) are independent of the
choice of the Morse function.
!
5.6 Exercise. Show that the sum of the Betti numbers i Bi is the number of
vertices of the graph G.
!
5.7 Exercise. Show that the sum i (i/2)Bi is the number of edges of the graph
G.
5.8 Exercise*. Show that the homology groups of the topological graph G are
determined by the Betti numbers of H∗ (G).
7.1 Exercise. Suppose that G is k-valent and that it has a perfect Morse function
φ such that the Morse function −φ is also perfect. Show that
Bj (G) = B2k−j (G).
As usual in mathematics, it is better to have a canonical isomorphism or a duality
of vector spaces than an equality of their dimensions. We want something of the
kind for Poincaré duality. First, we need some preliminaries on graded rings.
7.2 The canonical filtration of a graded R-module. Consider a graded
module M over a graded ring R. Let M ≤k be the sum of the graded pieces M 0 ⊕
M 1 ⊕· · ·⊕M k . This is not an R-module, but it generates one; call it Fk M = R·M ≤k .
Then M has an increasing filtration of R-submodules F0 M ⊆ F1 M ⊆ · · · .
7.3 Exercise. If G has the free module property, then Bi is the dimension of the
i graded piece of Fi H∗ (G)/Fi−1 H∗ (G).
7.4 Internal Hom. Suppose that M and N are two graded R modules. Then
the space HomR (M, N ) has the structure of a graded R module. The i-th graded
piece is the elements of HomR (M, N ) that map each M j into N j+i .
7.5 Proposition. Functorial Poincaré duality. Now, suppose that G is con-
nected, k-valent, and that it is universally perfect (i.e. all Morse functions are per-
fect). Then H∗ (G)/F2k−1 H∗ (G) is a free O(T) module on one generator in degree
2k. Call it D. The pairing
H∗ (G)
H∗ (G) ⊗O(T) H∗ (G) −→ H∗ (G) −→ F2k−1 H∗ (G) =D
x × y → xy
is perfect in the sense that the induced map
H∗ (G) −→ Hom(H∗ (G), D)
is an isomorphism of O(T) modules.
7.6 Exercise. Show that functorial Poincaré duality implies numerical Poincaré
duality.
8.5 We can pause to marvel at the statements. The data in moment graph of
T X depends only on a very small part of X – its 1-skeleton. Yet by these
theorems, all of the homology and equivariant homology of X is encoded in this
data.
The proofs of these three propositions are beyond our ambitions here. The
reader is referred to [13] and the references given there. However, we have given
enough information in our explicit construction of generators and relations for the
equivariant cohomology of the 2-sphere, we have to construct the map
∼
H ∗ (T X) ←− H∗ (G(T X))
in Theorem 3.8.2.
8.6 Exercise. Construct a map H ∗ (T X) → H∗ (G(T X)).
8.7 Exercise. Show that the free generators α1 , α2 , . . . for H∗ (G(X)) as a module
over O(T) pass in the quotient to generators of H ∗ (X) as a vector space, i.e. as a
module over R.
8.8 Morse theory and Poincaré duality for our examples. In Lecture 2,
we gave many examples of spaces with a torus action: projective spaces, quadric
hypersurfaces, Grassmann manifolds, Lagrangian Grassmannians, flag manifolds,
and toric varieties for simple polyhedra. These examples all satisfy the hypotheses
of the theorems above. Furthermore, they are all universally perfect (every Morse
function is perfect), so they satisfy Poincaré duality. (This may be seen using
topological methods.) Many other examples in this favorable class will be mentioned
in §4.1.2.
8.9* Morse theory and moment maps. Suppose that X is a nonsingular
algebraic variety, and the action T X and the moment map μ : X −→ V are as
in §2.9. If ϕ : V −→ R is a Morse function for the moment graph of T X in the
sense of this Lecture, then ϕ ◦ μ : X −→ R is a Morse function in the usual sense
of differential topology. In this case, the Morse function will be perfect. In this
case, Morse theory we have described is a reflection of the usual topological Morse
theory.
LECTURE 3. THE COHOMOLOGY OF A LINEAR GRAPH 369
8.10* The Schubert basis. Suppose X is a generalized flag manifold, i.e. a pro-
jective space, a quadric hypersurface, a Grassmann manifold, a Lagrangian Grass-
mannian, a flag manifold, or more generally a space of §4.7.2. Then the Morse
function ϕ ◦ μ is perfect on ordinary cohomology H ∗ (X). The basis of cohomology
it provides is called the Schubert basis, and the study of the properties of this basis
in the ring H ∗ (X) is called Schubert calculus, an interesting combinatorial study in-
volving such things as the Littlewood-Richardson rule, Schubert polynomials, etc.
By Exercise 3.8.7, the H ∗ (X) and its Schubert basis is encoded in the moment
graph, so in principle questions in Schubert calculus reduce to questions about the
moment graph.
8.11* A general Lie group. Here’s a brief account. Suppose G X is an
action of a general connected Lie group. Then H ∗ (G X) = H ∗ (K X) where K
is a maximal compact subgroup of G. Then, by a theorem of Borel, H ∗ (K X) =
H ∗ (T X)W where T is a maximal torus of K and W is the Weyl group of K and
the superscript means taking the invariants. Now, suppose the T action satisfies our
hypotheses, so it has moment graph in G(T X) ⊂ V. The Weyl group W acts on V
preserving the moment graph, so we can calculate H ∗ (G X) = H∗ (G(T X))W .
LECTURE 4
Computing Intersection Homology
Or, when V = Rn , H could be the n2 planes xi = xj in Rn where two coordinates
are equal. A finite reflection group W is the group of maps of V to itself generated
by reflections in hyperplanes in H.
371
372 R. MACPHERSON, EQUIVARIANT INVARIANTS AND LINEAR GEOMETRY
1.3 Exercise. All of the linear graphs pictured in sections 2.3 to 2.7 arise in this
way. Construct the family of hyperplanes H for each of them.
1.4 Crystallographic reflection groups. If there is some lattice L ⊂ V such
that reflection in each of the hyperplanes in H takes this lattice into itself, then H
is called crystallographic. This is true for most, but not all of the possible choices
for H. If H is crystallographic, then G(H, v) arises as a moment graph, as described
in §4.7.2.
1.5 The linear graphs G(H, v) are all universally perfect. In other words, all
Morse functions on these graphs are perfect. (This may be seen using a topological
argument if H is crystallographic. In general, it follows from [18].) In fact, every
graph we have considered so far is universally perfect, with the exception of a few
counterexamples and 1-skeleta of non-simple polytopes. We will now construct a
large class of examples with non-perfect Morse functions.
2.1 Consider a linear graph arising from a finite reflection group G(H, v) ⊂ V
and a Morse function φ : V −→ R (a linear function that takes distinct values on
different vertices of G). Recall (§3.4.4) that if ν is a vertex of G, we define L− (ν) to
be the edges going down from ν and L+ (ν) to be the edges going up from ν, where
“up” and “down” are measured by φ.
2.2 Definition. We call a subgraph G of G upward saturated with respect to φ
if whenever ν is in G then every edge in L+ (ν) is in G .
2.3 The Morse function φ is not usually perfect on upward saturated subgraphs.
In fact, for two of the examples above, the inverted V §3.5.10 and the Egyptian
pyramid §3.5.9 the function φ has already been shown not to be perfect. However,
2.4 Exercise. Show that −φ is perfect for an upward saturated subgraph. (Use
the fact that G(H, v) is universally perfect.)
2.5 Exercise*. The Morse function φ turns the set of vertices of G(H, v) into a
poset where ν ≤ ν if there is a sequence of edges from ν to ν such that φ increases
along each edge. The partial order of this poset is called the Bruhat order. Show
that an upward saturated subgraph can be characterized as a complete subgraph
on a set of vertices that is an ideal in this poset.
3.5 Proposition. The cohomology H∗ (G) of the graph G is the global sections
Γ(A, E(G)) of the sheaf A.
This is just a slightly disguised presentation of the definition of H∗ (G).
4.5 If you want to understand what makes a Morse function perfect, it is worth-
while to pause to appreciate this proposition. An element e in Γ(A, L− (ν)) is just a
collection of polynomials on the hyperplanes
⊥ ⊂ T for
∈ L− (ν). The condition
that e be in the image of Γ(A, L− (ν)) ←− Γ(A, E <c ) is a potentially complicated
compatibility condition on these polynomials, coming from the structure of the
graph. According to the proposition, it is perfect if and only if a set of polynomials
⊥ satisfying this compatibility condition is necessarily the restriction of a single
polynomial on T.
4.6 For example, take the Egyptian pyramid. As in §3.3.4, an element of the
image of
Γ(A, L− (ν)) ←− Γ(A, E <c )
is a continuous piecewise polynomial function on configuration consisting of the
four upper planes on the right below. We are asking whether such a thing is
the restriction of a polynomial in 3-space. We can see that it is not, just by a
dimension count. For example, there is a 4-dimensional space of linear polynomials
on the configuration (it can be anything on each of the 4 lines). But there is only a
3-dimensional space of linear polynomials in 3-space. So the height function is not
perfect. (Compare §3.5.9.)
• If ν is a vertex and M has already been constructed on all of the vertices and
edges of G <ν , then Mν is the free cover of
" #
Image Γ(M , L− (ν)) ←− Γ(M , E <ν (G))
and the maps mν are determined by
F " #
Mν −→ Image Γ(M , L− (ν)) ←− Γ(M , E <ν (G )) ⊂ M
∈L− (ν)
5.9 Exercise. Carry out the inductive construction for the inverted V graph and
the Egyptian pyramid. In both cases, M will coincide with A until the top vertex.
At the top vertex ν, for the inverted V graph, Mν will be as in the first example of
Exercise 4.5.8 above. For the Egyptian pyramid, it will have a generator in degree
2 reflecting the phenomenon for linear functions explained in §4.4.6.
6.2 Theorem. All Morse functions on G(H, v) are perfect for the sheaf M .
6.3 Theorem. Γ(M , G(H, v)) is a free module over O(T).
Let IBi be the number of free generators in degree i.
6.4 Theorem. Γ(M , G(H, v)) satisfies Poincaré duality: IBi = IB−i . Moreover,
the canonical pairing
Γ(M , G(H, v)) ⊗O(T) Γ(M , G(H, v)) −→ O(T)
is perfect in the sense that the induced map
Γ(M , G(H, v)) −→ Hom(Γ(M , G(H, v)), O(T))
is an isomorphism of O(T) modules.
One advantage is that this paradigm treats certain spaces whose 1-skeleton
is not a balloon sculpture, as was required up until now. But the main point is
that the arrangements of linear spaces that arise in this way seem interesting in
themselves.
π s1 s2 · · ·
T
379
380 R. MACPHERSON, EQUIVARIANT INVARIANTS AND LINEAR GEOMETRY
1.3 A diagram of spaces. Now suppose we have a space with a torus action
T X with finitely many fixed points F1 , F2 , . . . Then we have the following diagram
of spaces with a T action and T equivariant maps:
X
p i1 i2 · · ·
pt
here pt is a point (with a trivial T action), p : X −→ pt is the only thing it could
be, and ij : pt −→ X sends pt to the fixed point Fj .
1.4 The fixed point arrangement. We apply second equivariant homology
functor H2 (·) to this diagram.
Definition. The fixed point arrangement of X is the arrangement of sections
E = H2 (T X), T = H2 (T pt), π = p∗ : H2 (T X) −→ H2 (T pt) and
sj = (ij )∗ : H2 (T pt) −→ H2 (T X).
Conversely, if we know the linear functions fjk , that determines the configuration
up to automorphisms of E commuting with π, which is enough for our purposes.
In fact, since fjk + fkl = fkl , if we’re really efficient about it, we only need to know
m − 1 of them, where m is the number of fixed points.
2.3 Determining fkj . Suppose that Fj and Fk lie in a balloon B: a 2-sphere
that is taken into itself by T . Since there are only finitely many fixed points, T the
orbits of T on B will all be circles, except for Fj and Fk . Chose such a circle, S.
There will be a linear map g : T −→ R and an identification of S with R/Z such
that the action of t̄ corresponds to addition of g(t). The trouble is that there are
two such functions g which are negatives of each other, corresponding to opposite
identifications of S with R/Z. If we had an orientation of S, we could specify that
the orientation should correspond to the natural orientation of R.
Choose an orientation O for B. That does two things for us. First, it makes
B into a cycle, so it gives us a class [B] in homology. Second, it restricts to an
orientation of the disk bounded by S containing Fj . This induces an orientation
on its boundary, S, solving the problem above. With these conventions, we get
fjk = g[B]
2.4 Exercise. Show that this definition is independent of the orientation O of
B chosen. Show directly from this definition that fjk = −fkj .
3.1 The function ring of the configuration. Let A be the union of the linear
subspaces in the fixed point configuration. In other words, a ∈ A if and only if
a = sj (t) for some section sj in our arrangement and some t ∈ T. The set A
is a real algebraic variety - it has a function ring O(A), which is the polynomial
functions on E, two being considered equivalent if they take the same values on
every point of A. In other words,
O(E)
O(A) =
I(A)
where I(A) is the ideal of polynomials vanishing on the set A.
382 R. MACPHERSON, EQUIVARIANT INVARIANTS AND LINEAR GEOMETRY
4.3 Exercise. Prove that the torus T preserves the Springer variety Xγ . Use
the fact that the matrix t commutes with the matrix γ.
4.5 The Springer action. The symmetric group Sk on k letters acts on the
configuration A of linear subspaces (by permuting the diagonal entries in the last
matrix). It follows that Sk acts on the equivariant cohomology of Xγ ([17]). The
Sk action on A preserves the fibers of the map π to T. Therefore, it passes to an
action of Sk on the ordinary cohomology of Xγ . This is the usual Springer action.
A similar equivariant cohomology construction for Springer actions on the co-
homology of Springer varieties for loop groups is found in [14]. Historically, the
effort to solve the problem addressed in [14] was what originally led to the whole
body of material in this lecture series.
384 R. MACPHERSON, EQUIVARIANT INVARIANTS AND LINEAR GEOMETRY
Note that this configuration has an obvious action of S3 , the symmetric group
which permutes the 3 sections. Thus we get an action of S3 on the equivariant
cohomology of Xγ and also the ordinary cohomology. This is not induced by an
action of S3 on Xγ itself: Xγ is less symmetric than its fixed point configuration.
LECTURE 5. COHOMOLOGY AS FUNCTIONS ON A VARIETY 385
4.7 Exercise. Verify the statements in the last section about the Springer variety
for the partition 3 = 2 + 1.
Δ η1 η2 · · ·
T∗
If we dualize this diagram, we get the arrangement of sections of §5.1.1.
5.1 Exercise. In this situation, construct a map
O(A) −→ H∗ (G(T X)).
∗
Assuming that H (G(T X)) is generated by H (G(T 2
X)), show that this map
is an isomorphism.
BIBLIOGRAPHY
Richard P. Stanley
IAS/Park City Mathematics Series
Volume 14, 2004
An Introduction to Hyperplane
Arrangements
Richard P. Stanley
LECTURE 1
Basic Definitions, the Intersection Poset and the
Characteristic Polynomial
c
2007 American Mathematical Society
391
392 R. STANLEY, HYPERPLANE ARRANGEMENTS
even if we’re only interested in this case it is useful to consider other fields as well.
To make sure that the definition of a hyperplane arrangement is clear, we define a
linear hyperplane to be an (n − 1)-dimensional subspace H of V , i.e.,
H = {v ∈ V : α · v = 0},
where α is a fixed nonzero vector in V and α · v is the usual dot product:
(α1 , . . . , αn ) · (v1 , . . . , vn ) = αi vi .
An affine hyperplane is a translate J of a linear hyperplane, i.e.,
J = {v ∈ V : α · v = a},
where α is a fixed nonzero vector in V and a ∈ K.
If the equations of the hyperplanes of A are given by L1 (x) = a1 , . . . , Lm (x) =
am , where x = (x1 , . . . , xn ) and each Li (x) is a homogeneous linear form, then we
call the polynomial
QA (x) = (L1 (x) − a1 ) · · · (Lm (x) − am )
the defining polynomial of A. It is often convenient to specify an arrangement
by its defining polynomial. For instance, the arrangement A consisting of the n
coordinate hyperplanes has QA (x) = x1 x2 · · · xn .
Let A be an arrangement in the vector space V . The dimension dim(A) of
A is defined to be dim(V ) (= n), while the rank rank(A) of A is the dimension
of the space spanned by the normals to the hyperplanes in A. We say that A is
essential if rank(A) = dim(A). Suppose that rank(A) = r, and take V = K n . Let
Y be a complementary space in K n to the subspace X spanned by the normals to
hyperplanes in A. Define
W = {v ∈ V : v · y = 0 ∀y ∈ Y }.
If char(K) = 0 then we can simply take W = X. By elementary linear algebra we
have
(1) codimW (H ∩ W ) = 1
for all H ∈ A. In other words, H ∩ W is a hyperplane of W , so the set AW :=
{H ∩W : H ∈ A} is an essential arrangement in W . Moreover, the arrangements A
and AW are “essentially the same,” meaning in particular that they have the same
intersection poset (as defined in Definition 1.1). Let us call AW the essentialization
of A, denoted ess(A). When K = R and we take W = X, then the arrangement A
is obtained from AW by “stretching” the hyperplane H ∩ W ∈ AW orthogonally to
W . Thus if W ⊥ denotes the orthogonal complement to W in V , then H ∈ AW if
and only if H ⊕ W ⊥ ∈ A. Note that in characteristic p this type of reasoning fails
since the orthogonal complement of a subspace W can intersect W in a subspace
of dimension greater than 0.
Example 1.1. Let A consist of the lines x = a1 , . . . , x = ak in K 2 (with coordinates
x and y). Then we can take W to be the x-axis, and ess(A) consists of the points
x = a1 , . . . , x = ak in K.
Now let K = R. A region of an arrangement A is a connected component of
the complement X of the hyperplanes:
X = Rn − H.
H∈A
LECTURE 1. BASIC DEFINITIONS 393
It is a simple exercise to show that every region R ∈ R(A) is open and convex
(continuing to assume K = R), and hence homeomorphic to the interior of an n-
dimensional ball Bn (Exercise 1). Note that if W is the subspace of V spanned by
the normals to the hyperplanes in A, then R ∈ R(A) if and only if R ∩W ∈ R(AW ).
We say that a region R ∈ R(A) is relatively bounded if R ∩ W is bounded. If A
is essential, then relatively bounded is the same as bounded. We write b(A) for
the number of relatively bounded regions of A. For instance, in Example 1.1 take
K = R and a1 < a2 < · · · < ak . Then the relatively bounded regions are the
regions ai < x < ai+1 , 1 ≤ i ≤ k − 1. In ess(A) they become the (bounded) open
intervals (ai , ai+1 ). There are also two regions of A that are not relatively bounded,
viz., x < a1 and x > ak .
A (closed) half-space is a set {x ∈ Rn : x · α ≥ c} for some α ∈ Rn , c ∈ R. If
H is a hyperplane in Rn , then the complement Rn − H has two (open) components
whose closures are half-spaces. It follows that the closure R̄ of a region R of A is
a finite intersection of half-spaces, i.e., a (convex) polyhedron (of dimension n). A
bounded polyhedron is called a (convex) polytope. Thus if R (or R̄) is bounded,
then R̄ is a polytope (of dimension n).
An arrangement A is in general position if
{H1 , . . . , Hp } ⊆ A, p ≤ n ⇒ dim(H1 ∩ · · · ∩ Hp ) = n − p
{H1 , . . . , Hp } ⊆ A, p > n ⇒ H1 ∩ · · · ∩ Hp = ∅.
For instance, if n = 2 then a set of lines is in general position if no two are parallel
and no three meet at a point.
Let us consider some interesting examples of arrangements that will anticipate
some later material.
Example 1.2. Let Am consist of m lines in general position in R2 . We can compute
r(Am ) using the sweep hyperplane method. Add a L line to Ak (with AK ∪ {L} in
general position). When we travel along L from one end (at infinity) to the other,
every time we intersect a line in Ak we create a new region, and we create one new
394 R. STANLEY, HYPERPLANE ARRANGEMENTS
region at the end. Before we add any lines we have one region (all of R2 ). Hence
r(Am ) = #intersections + #lines + 1
m
= + m + 1.
2
Example 1.3. The braid arrangement Bn in K n consists of the hyperplanes
Bn : xi − xj = 0, 1 ≤ i < j ≤ n.
n
Thus Bn has 2 hyperplanes. To count the number of regions when K = R, note
that specifying which side of the hyperplane xi − xj = 0 a point (a1 , . . . , an ) lies
on is equivalent to specifying whether ai < aj or ai > aj . Hence the number of
regions is the number of ways that we can specify whether ai < aj or ai > aj for
1 ≤ i < j ≤ n. Such a specification is given by imposing a linear order on the
ai ’s. In other words, for each permutation w ∈ Sn (the symmetric group of all
permutations of 1, 2, . . . , n), there corresponds a region Rw of Bn given by
Rw = {(a1 , . . . , an ) ∈ Rn : aw(1) > aw(2) > · · · > aw(n) }.
Hence r(Bn ) = n!. Rarely is it so easy to compute the number of regions!
Note that the braid arrangement Bn is not essential; indeed, rank(Bn ) = n − 1.
When char(K) does not divide n the space W ⊆ K n of equation (1) can be taken
to be
W = {(a1 , . . . , an ) ∈ K n : a1 + · · · + an = 0}.
The braid arrangement has a number of “deformations” of considerable interest.
We will just define some of them now and discuss them further later. All these
arrangements lie in K n , and in all of them we take 1 ≤ i < j ≤ n. The reader who
likes a challenge can try to compute their number of regions when K = R. (Some
are much easier than others.)
• generic braid arrangement : xi − xj = aij , where the aij ’s are “generic”
(e.g., linearly independent over the prime field, so K has to be “sufficiently
large”). The precise definition of “generic” will be given later. (The prime
field of K is its smallest subfield, isomorphic to either Q or Z/pZ for some
prime p.)
• semigeneric braid arrangement : xi −xj = ai , where the ai ’s are “generic.”
• Shi arrangement : xi − xj = 0, 1 (so n(n − 1) hyperplanes in all).
• Linial arrangement : xi − xj = 1.
• Catalan arrangement : xi − xj = −1, 0, 1.
• semiorder arrangement : xi − xj = −1, 1.
• threshold arrangement : xi + xj = 0 (not really a deformation of the braid
arrangement, but closely related).
An arrangement A is central if H∈A H
= ∅. Equivalently, A is a translate
of a linear arrangement (an arrangement of linear hyperplanes, i.e., hyperplanes
passing through theorigin). Many other writers call an arrangement
central, rather
than linear, if 0 ∈ H∈A H. If A is central with X = H∈A H, then rank(A) =
codim(X). If A is central, then note also that b(A) = 0 [why?].
There are two useful arrangements closely related to a given arrangement A.
If A is a linear arrangement in K n , then projectivize A by choosing some H ∈ A
LECTURE 1. BASIC DEFINITIONS 395
n−1
to be the hyperplane at infinity in projective space PK . Thus if we regard
n−1
PK = {(x1 , . . . , xn ) : xi ∈ K, not all xi = 0}/ ∼,
where u ∼ v if u = αv for some 0
= α ∈ K, then
H = ({(x1 , . . . , xn−1 , 0) : xi ∈ K, not all xi = 0}/ ∼) ∼ n−2
= PK .
The remaining hyperplanes in A then correspond to “finite” (i.e., not at infinity)
n−1
projective hyperplanes in PK . This gives an arrangement proj(A) of hyperplanes
in PK . When K = R, the two regions R and −R of A become identified in
n−1
proj(A). Hence r(proj(A)) = 12 r(A). When n = 3, we can draw PR2 as a disk with
antipodal boundary points identified. The circumference of the disk represents the
hyperplane at infinity. This provides a good way to visualize three-dimensional real
linear arrangements. For instance, if A consists of the three coordinate hyperplanes
x1 = 0, x2 = 0, and x3 = 0, then a projective drawing is given by
1
2
3
1
2
3
1
396 R. STANLEY, HYPERPLANE ARRANGEMENTS
12 24
14
34
23
13
Regarding this diagram as a planar graph, the dual graph is the 3-cube (i.e., the
vertices and edges of a three-dimensional cube) [why?].
For a more complicated example of projectivization, Figure 1 shows proj(B4 )
(where we regard B4 as a three-dimensional arrangement contained in the hyper-
plane x1 + x2 + x3 + x4 = 0 of R4 ), with the hyperplane xi = xj labelled ij, and
with x1 = x4 as the hyperplane at infinity.
We now define an operation which is “inverse” to projectivization. Let A be
an (affine) arrangement in K n , given by the equations
L1 (x) = a1 , . . . , Lm (x) = am .
Introduce a new coordinate y, and define a central arrangement cA (the cone over
A) in K n × K = K n+1 by the equations
L1 (x) = a1 y, . . . , Lm (x) = am y, y = 0.
For instance, let A be the arrangement in R1 given by x = −1, x = 2, and x = 3.
The following figure should explain why cA is called a cone.
−1 2
3
LECTURE 1. BASIC DEFINITIONS 397
Proof. Since L(A) has a unique minimal element 0̂ = V , it suffices to show that
(a) if xy in L(A) then dim(x)−dim(y) = 1, and (b) all maximal elements of L(A)
have dimension n − rank(A). By linear algebra, if H is a hyperplane and x an affine
subspace, then H ∩ x = x or dim(x) − dim(H ∩ x) = 1, so (a) follows. Now suppose
that x has the largest codimension of any element of L(A), say codim(x) = d. Thus
x is an intersection of d linearly independent hyperplanes (i.e., their normals are
linearly independent) H1 , . . . , Hd in A. Let y ∈ L(A) with e = codim(y) < d. Thus
y is an intersection of e hyperplanes, so some Hi (1 ≤ i ≤ d) is linearly independent
from them. Then y ∩ Hi
= ∅ and codim(y ∩ Hi ) > codim(y). Hence y is not a
maximal element of L(A), proving (b).
If P has a 0̂, then we write μ(x) = μ(0̂, x). Figure 3 shows the intersection poset
L of the arrangement A in K 3 (for any field K) defined by QA (x) = xyz(x + y),
together with the value μ(x) for all x ∈ L.
A important application of the Möbius function is the Möbius inversion for-
mula. The best way to understand this result (though it does have a simple direct
proof) requires the machinery of incidence algebras. Let I(P ) = I(P, K) denote
LECTURE 1. BASIC DEFINITIONS 399
−2
2 1 1 1
−1 −1 −1 −1
1
Figure 3. An intersection poset and Möbius function values
the vector space of all functions f : Int(P ) → K. Write f (x, y) for f ([x, y]). For
f, g ∈ I(P ), define the product f g ∈ I(P ) by
f g(x, y) = f (x, z)g(z, y).
x≤z≤y
It is easy to see that this product makes I(P ) an associative Q-algebra, with mul-
tiplicative identity δ given by
1, x = y
δ(x, y) =
0, x < y.
Define the zeta function ζ ∈ I(P ) of P by ζ(x, y) = 1 for all x ≤ y in P . Note that
the Möbius function μ is an element of I(P ). The definition of μ (Definition 1.2) is
equivalent to the relation μζ = δ in I(P ). In any finite-dimensional algebra over a
field, one-sided inverses are two-sided inverses, so μ = ζ −1 in I(P ).
Theorem 1.1. Let P be a finite poset with Möbius function μ, and let f, g : P → K.
Then the following two conditions are equivalent:
f (x) = g(y), for all x ∈ P
y≥x
g(x) = μ(x, y)f (y), for all x ∈ P.
y≥x
Proof. The set K P of all functions P → K forms a vector space on which I(P )
acts (on the left) as an algebra of linear transformations by
(ξf )(x) = ξ(x, y)f (y),
y≥x
where f ∈ K P and ξ ∈ I(P ). The Möbius inversion formula is then nothing but
the statement
ζf = g ⇔ f = μg.
We now come to the main concept of this section.
400 R. STANLEY, HYPERPLANE ARRANGEMENTS
Exercises
We will (subjectively) indicate the difficulty level of each problem as follows:
[1] easy: most students should be able to solve it
[2] moderately difficult: many students should be able to solve it
[3] difficult: a few students should be able to solve it
[4] horrendous: no students should be able to solve it (without already
knowing how)
[5] unsolved.
Further gradations are indicated by + and −. Thus a [3–] problem is about the
most difficult that makes a reasonable homework exercise, and a [5–] problem is an
unsolved problem that has received little attention and may not be too difficult.
Note. Unless explicitly stated otherwise, all graphs, posets, lattices, etc., are
assumed to be finite.
(1) [2] Show that every region R of an arrangement A in Rn is an open convex set.
Deduce that R is homeomorphic to the interior of an n-dimensional ball.
LECTURE 1. BASIC DEFINITIONS 401
(2) [1+] Let A be an arrangement and ess(A) its essentialization. Show that
tdim(ess(A)) χA (t) = tdim(A) χess(A) (t).
(3) [2+] Let A be the arrangement in Rn with equations
x1 = x2 , x2 = x3 , . . . , xn−1 = xn , xn = x1 .
Compute the characteristic polynomial χA (t), and compute the number r(A) of
regions of A.
(4) [2+] Let A be an arrangement in Rn with m hyperplanes. Find the maximum
possible number f (n, m) of regions of A.
(5) [2] Let A be an arrangement in the n-dimensional vector space V whose normals
span a subspace W , and let B be another arrangement in V whose normals span
a subspace Y . Suppose that W ∩ Y = {0}. Show that
χA∪B (t) = t−n χA (t)χB (t).
(6) [2] Let A be an arrangment in a vector space V . Suppose that χA (t) is divisible
by tk but not tk+1 . Show that rank(A) = n − k.
(7) Let A be an essential arrangement in Rn . Let Γ be the union of the bounded
faces of A.
(a) [3] Show that Γ is contractible.
(b) [2] Show that Γ need not be homeomorphic to a closed ball.
(c) [2+] Show that Γ need not be starshaped. (A subset S of Rn is starshaped
if there is a point x ∈ S such that for all y ∈ S, the line segment from x to
y lies in S.)
(d) [3] Show that Γ is pure, i.e., all maximal faces of Γ have the same dimension.
(This was an open problem solved by Xun Dong at the PCMI Summer
Session in Geometric Combinatorics, July 11–31, 2004.)
(e) [5] Suppose that A is in general position. Is Γ homeomorphic to an n-
dimensional closed ball?
LECTURE 2
Properties of the Intersection Poset and Graphical
Arrangements
K
x
Ax
K
A K
A
H0
A
A’
A"
The main goal of this section is to give a formula in terms of χA (t) for r(A)
and b(A) when K = R (Theorem 2.5). We first establish recurrences for these two
quantities.
Lemma 2.1. Let (A, A , A ) be a triple of real arrangements with distinguished
hyperplane H0 . Then
r(A) = r(A ) + r(A )
b(A ) + b(A ), if rank(A) = rank(A )
b(A) =
0, if rank(A) = rank(A ) + 1.
Note. If rank(A) = rank(A ), then also rank(A) = 1 + rank(A ). The figure
below illustrates the situation when rank(A) = rank(A ) + 1.
H0
Proof. Note that r(A) equals r(A ) plus the number of regions of A cut into two
regions by H0 . Let R be such a region of A . Then R ∩ H0 ∈ R(A ). Conversely,
if R ∈ R(A ) then points near R on either side of H0 belong to the same region
R ∈ R(A ), since any H ∈ R(A ) separating them would intersect R . Thus R is
cut in two by H0 . We have established a bijection between regions of A cut into
two by H0 and regions of A , establishing the first recurrence.
The second recurrence is proved analogously; the details are omitted.
We now come to the fundamental recursive property of the characteristic poly-
nomial.
Lemma 2.2. (Deletion-Restriction) Let (A, A , A ) be a triple of real arrange-
ments. Then
χA (t) = χA (t) − χA (t).
LECTURE 2. PROPERTIES OF THE INTERSECTION POSET 405
For the proof of this lemma, we will need some tools. (A more elementary proof
could be given, but the tools will be useful later.)
Let P be a poset. An upper bound of x, y ∈ P is an element z ∈ P satisfying
z ≥ x and z ≥ y. A least upper bound or join of x and y, denoted x ∨ y, is an upper
bound z such that z ≤ z for all upper bounds z . Clearly if x ∨ y exists, then it
is unique. Similarly define a lower bound of x and y, and a greatest lower bound
or meet, denoted x ∧ y. A lattice is a poset L for which any two elements have a
meet and join. A meet-semilattice is a poset P for which any two elements have
a meet. Dually, a join-semilattice is a poset P for which any two elements have a
join. Figure 1 shows two non-lattices, with a pair of elements circled which don’t
have a join.
Lemma 2.3. A finite meet-semilattice L with a unique maximal element 1̂ is a
lattice. Dually, a finite join-semilattice L with a unique minimal element 0̂ is a
lattice.
Proof. Let L be a finite meet-semilattice. If x, y ∈ L then the set of upper bounds
of x, y is nonempty since 1̂ is an upper bound. Hence
x∨y = z.
z≥x
z≥y
Since L(A) = L (A) or L(A) = L (A) − {1̂}, it follows that L(A) is always a meet-
semilattice, and is a lattice if A is central. If A isn’t central, then x∈L(A) x does
not exist, so L(A) is not a lattice.
We now come to a basic formula for the Möbius function of a lattice.
Theorem 2.2. (the Cross-Cut Theorem) Let L be a finite lattice. Let X be a subset
of L such that 0̂
∈ X, and such that if y ∈ L, y
= 0̂, then some x ∈ X satisfies
x ≤ y. Let Nk be the number of k-element subsets of X with join 1̂. Then
μL (0̂, 1̂) = N0 − N1 + N2 − · · · .
We will prove Theorem 2.2 by an algebraic method. Such a sophisticated proof
is unnecessary, but the machinery we develop will be used later (Theorem 4.13).
Let L be a finite lattice and K a field. The Möbius algebra of L, denoted A(L), is
the semigroup algebra of L over K with respect to the operation ∨. (Sometimes
the operation is taken to be ∧ instead of ∨, but for our purposes, ∨ is more con-
venient.) In other words, A(L) = KL (the vector space with basis L) as a vector
space. If x, y ∈ L then we define xy = x ∨ y. Multiplication is extended to all
of A(L) by bilinearity (or distributivity). Algebraists will recognize that A(L) is
a finite-dimensional commutative algebra with a basis of idempotents, and hence
is isomorphic to K #L (as an algebra). We will show this by exhibiting an explicit
∼
=
isomorphism A(L) → K #L. For x ∈ L, define
(8) σx = μ(x, y)y ∈ A(L),
y≥x
where μ denotes the Möbius function of L. Thus by the Möbius inversion formula,
(9) x= σy , for all x ∈ L.
y≥x
= (x ∨ y) .
Hence the linear transformation ϕ : A(L) → A (L) defined by ϕ(x) = x is an
algebra isomorphism. Since ϕ(σx ) = σx , it follows that σx σy = δxy σx .
LECTURE 2. PROPERTIES OF THE INTERSECTION POSET 407
c d
a
b
408 R. STANLEY, HYPERPLANE ARRANGEMENTS
The following table shows all central subsets B of A and the values of #B and
rank(B).
B #B rank(B)
∅ 0 0
a 1 1
b 1 1
c 1 1
d 1 1
ac 2 2
ad 2 2
bc 2 2
bd 2 2
cd 2 2
acd 3 2
It follows that χA (t) = t2 − 4t + (5 − 1) = t2 − 4t + 4.
Proof of Theorem 2.4. Let z ∈ L(A). Let
Λz = {x ∈ L(A) : x ≤ z},
the principal order ideal generated by z. Recall the definition
Az = {H ∈ A : H ≤ z (i.e., z ⊆ H)}.
By the Crosscut Theorem (Theorem 2.2), we have
μ(z) = (−1)k Nk (z),
k
where Nk (z) is the number of k-subsets of Az with join z. In other words,
μ(z) = (−1)#B .
Ì
B⊆Az
z= H∈B H
Note that z = H∈B H implies that rank(B) = n − dim z. Now multiply both sides
by tdim(z) and sum over z to obtain equation (10).
We have now assembled all the machinery necessary to prove the Deletion-
Restriction Lemma (Lemma 2.2) for χA (t).
Proof of Lemma 2.2. Let H0 ∈ A be the hyperplane defining the triple
(A, A , A ). Split the sum on the right-hand side of (10) into two sums, depending
on whether H0
∈ B or H0 ∈ B. In the former case we get
(−1)#B tn−rank(B) = χA (t).
H0
∈B⊆A
B central
In the latter case, set B1 = (B−{H0 })H0 , a central arrangement in H0 ∼= K n−1 and
a subarrangement of A = A . Since #B1 = #B− 1 and rank(B1 ) = rank(B)− 1,
H0
we get
(−1)#B tn−rank(B) = (−1)#B1 +1 t(n−1)−rank(B1 )
H0 ∈B⊆A B1 ∈A
B central
= −χA (t),
and the proof follows.
LECTURE 2. PROPERTIES OF THE INTERSECTION POSET 409
First proof. Equation (11) holds for A = ∅, since r(∅) = 1 and χ∅ (t) = tn .
By Lemmas 2.1 and 2.2, both r(A) and (−1)n χA (−1) satisfy the same recurrence,
so the proof follows.
Now consider equation (12). Again it holds for A = ∅ since b(∅) = 1. (Recall
that b(A) is the number of relatively bounded regions. When A = ∅, the entire
ambient space Rn is relatively bounded.) Now
(13) ψ(Δ) = f0 − f1 + f2 − · · · .
We take (13) as the definition of ψ(Δ). For “nice” spaces and decompositions, it is
independent of the decomposition. In particular, ψ(Rn ) = (−1)n . Write R̄ for the
closure of a region R ∈ R(A).
Write F(A) for the set of faces of A, and let relint denote relative interior. Then
Rn = relint(F ),
F ∈F(A)
where denotes disjoint union. If fk (A) denotes the number of k-faces of A, it
follows that
(−1)n = ψ(Rn ) = f0 (A) − f1 (A) + f2 (A) − · · · .
Every k-face is a region of exactly one Ay for y ∈ L(A). Hence
fk (A) = r(Ay ).
y∈L(A)
dim(y)=k
Putting x = Rn gives
(−1)n r(A) = (−1)dim(y) μ(y) = χA (−1),
y∈L(A)
a a
b
c d c d
Figure 2. Two arrangements with the same intersection poset
joining x and y lies in Γ. This would imply that Γ is contractible, and hence (since
Γ is compact when A is essential) ψ(Γ) = 1. A counterexample to Zaslavsky’s
conjecture appears as an exercise in [7, Exer. 4.29], but nevertheless Björner and
Ziegler showed that Γ is indeed contractible. (See [7, Thm. 4.5.7(b)] and Lecture 1,
Exercise 7.) The argument just given for r(A) now carries over mutatis mutandis
to b(A). There is also a direct argument that ψ(Γ) = 1, circumventing the need to
show that Γ is contractible. We will omit proving here that ψ(Γ) = 1.
Corollary 2.1. Let A be a real arrangement. Then r(A) and b(A) depend only on
L(A).
Figure 2 shows two arrangements in R2 with different “face structure” but
the same L(A). The first arrangement has for instance one triangular and one
quadrilateral face, while the second has two triangular faces. Both arrangements,
however, have ten regions and two bounded regions.
We now give two basic examples of arrangements and the computation of their
characteristic polynomials.
Proposition 2.4. (general position) Let A be an n-dimensional arrangement of m
hyperplanes in general position. Then
m n−2 n m
χA (t) = t − mt
n n−1
+ t − · · · + (−1) .
2 n
In particular, if A is a real arrangement, then
m m
r(A) = 1 + m + + ···+
2 n
m n m
b(A) = (−1) 1 − m +
n
− · · · + (−1)
2 n
m−1
= .
n
Proof. Every B ⊆ A with #B ≤ n defines an element xB = H∈B H of L(A).
Hence L(A) is a truncated boolean algebra:
L(A) ∼ = {S ⊆ [m] : #S ≤ n},
412 R. STANLEY, HYPERPLANE ARRANGEMENTS
ordered by inclusion. Figure 3 shows the case n = 2 and m = 4, i.e., four lines in
general position in R2 . If x ∈ L(A) and rk(x) = k, then [0̂, x] ∼
= Bk , a boolean
algebra of rank k. By equation (4) there follows μ(x) = (1)k . Hence
χA (t) = (−1)#S tn−#S
S⊆[m]
#S≤n
m
= tn − mtn−1 + · · · + (−1)n . 2
n
Note. Arrangements whose hyperplanes are in general position were formerly
called free arrangements. Now, however, free arrangements have another meaning
discussed in the note following Example 4.11.
Our second example concerns generic translations of the hyperplanes of a lin-
ear arrangement. Let L1 , . . . , Lm be linear forms, not necessarily distinct, in the
variables v = (v1 , . . . , vn ) over the field K. Let A be defined by
L1 (v) = a1 , . . . , Lm (v) = am ,
where a1 , . . . , am are generic elements of K. This means if Hi = ker(Li (v) − ai ),
then
Hi1 ∩ · · · ∩ Hik
= ∅ ⇔ Li1 , . . . , Lik are linearly independent.
For instance, if K = R and L1 , . . . , Lm are defined over Q, then a1 , . . . , am are
generic whenever they are linearly independent over Q.
nongeneric generic
1 6 3 12 4 12
where B ranges over all linearly independent subsets of A. (We say that a set of hy-
perplanes are linearly independent if their normals are linearly independent.) Thus
χA (t), or more precisely (−t)n χA (−1/t), is the generating function for linearly
independent subsets of L1 , . . . , Lm according to their number of elements. For in-
stance, if A is given by Figure 2 (either arrangement) then the linearly independent
subsets of hyperplanes are ∅, a, b, c, d, ac, ad, bc, bd, cd, so χA (t) = t2 − 4t + 5.
Consider the more interesting example xi − xj = aij , 1 ≤ i < j ≤ n, where the
aij are generic. We could call this arrangement the generic braid arrangement Gn .
Identify the hyperplane xi − xj = aij with the edge ij on the vertex set [n]. Thus
a subset B ⊆ Gn corresponds to a simple graph GB on [n]. (“Simple” means that
there is at most one edge between any two vertices, and no edge from a vertex to
itself.) It is easy to see that B is linearly independent if and only if the graph GB
has no cycles, i.e., is a forest. Hence we obtain the interesting formula
(14) χGn (t) = (−1)e(F ) tn−e(F ) ,
F
where F ranges over all forests on [n] and e(F ) denotes the number of edges of
F . For instance, the isomorphism types of forests (with the number of distinct
labelings written below the forest) on four vertices are given by Figure 4. Hence
χG4 (t) = t4 − 6t3 + 15t2 − 16t.
Equation (11) can be rewritten as
r(A) = (−1)rk(x) μ(x).
x∈L(A)
(Theorem 3.10 will show that (−1)rk(x) μ(x) > 0, so we could also write |μ(x)| for
this quantity.) It is easy to extend this result to count faces of A of all dimensions,
not just the top dimension n. Let fk (A) denote the number of k-faces of the real
arrangement A.
Theorem 2.6. We have
(15) fk (A) = (−1)dim(x)−dim(y) μ(x, y)
x≤y in L(A)
dim(x)=k
(16) = |μ(x, y)|.
x≤y in L(A)
dim(x)=k
where we are dealing with the poset L(A). Summing over all x ∈ L(A) of dimension
k yields (15), and (16) then follows from Theorem (3.10) below.
6 8
9
5 4 4 3
7
2
1 2
1 3
G H
Figure 5. Two graphs
LECTURE 2. PROPERTIES OF THE INTERSECTION POSET 415
κ(4). On the other hand, if κ(1)
= κ(3), then there are q choices for κ(1), then
q − 1 choices for κ(3), and then q − 2 choices each for κ(2) and κ(4). Hence
χH (q) = q(q − 1)2 + q(q − 1)(q − 2)2
= q(q − 1)(q 2 − 3q + 3).
For further information on graphs whose chromatic polynomial can be evaluated
one vertex at a time, see Corollary 4.10 and the note following it.
It is easy to see directly that χG (q) is a polynomial function of q. Let ei (G)
denote the number of surjective proper colorings κ : [n] → [i] of G. We can choose
an arbitrary proper
coloring κ : [n] → [q] by first choosing the size i = #κ([n]) of
its image in qi ways, and then choose κ in ei ways. Hence
n
q
(18) χG (q) = ei .
i=0
i
Since qi = q(q−1) · · · (q−i+1)/i!, a polynomial in q (of degree i), we see that χG (q)
is a polynomial. We therefore write χG (t), where t is an indeterminate. Moreover,
any surjection (= bijection) κ : [n] → [n] is proper. Hence en = n!. It follows from
equation (18) that χG (t) is monic of degree n. Using more sophisticated methods
we will later derive further properties of the coefficients of χG (t).
Theorem 2.7. For any graph G, we have χAG (t) = χG (t).
First proof. The first proof is based on deletion-restriction (which in the
context of graphs is called deletion-contraction). Let e = ij ∈ E(G). Let G − e
(also denoted G\e) denote the graph G with edge e deleted, and let G/e denote G
with the edge e contracted to a point and all multiple edges replaced by a single
edge (i.e., whenever there is more than one edge between two vertices, replace these
edges by a single edge). (In some contexts we want to keep track of multiple edges,
but they are irrelevant in regard to proper colorings.)
2 4 2 4 4
e 5 5 23 5
1 3 1 3 1
G G−e G/e
Let H0 ∈ A = AG be the hyperplane xi = xj . It is clear that A−{H0 } = AG−e .
We claim that
(19) AH0 = AG/e ,
so by Deletion-Restriction (Lemma 2.2) we have
χAG (t) = χAG−e (t) = χAG/e (t).
∼
=
To prove (19), define an affine isomorphism ϕ : H0 → Rn−1 by
(x1 , x2 , . . . , xn ) → (x1 , . . . , xi , . . . , xˆj , . . . , xn ),
where xˆj denotes that the jth coordinate is omitted. (Hence the coordinates in
Rn−1 are 1, 2, . . . , ĵ, . . . , n.) Write Hab for the hyperplane xa = xb of A. If neither
416 R. STANLEY, HYPERPLANE ARRANGEMENTS
abcd
a c ab ac bd cd
bc
It is easily seen that |σ| = dim Xσ , so comparing equation (21) with Definition 1.3
shows that χG (t) = χAG (t).
Corollary 2.2. The characteristic polynomial of the braid arrangement Bn is given
by
χBn (t) = t(t − 1) · · · (t − n + 1).
Proof. Since Bn = AKn (the graphical arrangement of the complete graph Kn ),
we have from Theorem 2.7 that χBn (t) = χKn (t). The proof follows from equation
(17).
There is a further invariant of a graph G that is closely connected with the
graphical arrangement AG .
Definition 2.7. An orientation o of a graph G is an assignment of a direction
i → j or j → i to each edge ij of G. A directed cycle of o is a sequence of vertices
i0 , i1 , . . . , ik of G such that i0 → i1 → i2 → · · · → ik → i0 in o. An orientation o is
acyclic if it contains no directed cycles.
A graph G with no loops (edges from a vertex to itself) thus has 2#E(G) orien-
tations. Let R ∈ R(AG ), and let (x1 , . . . , xn ) ∈ R. In choosing R, we have specified
for all ij ∈ E(G) whether xi < xj or xi > xj . Indicate by an arrow i → j that
xi < xj , and by j → i that xi > xj . In this way the region R defines an orientation
oR of G. Clearly if R
= R , then oR
= oR . Which orientations can arise in this
way?
Proposition 2.5. Let o be an orientation of G. Then o = oR for some R ∈ R(AG )
if and only if o is acyclic.
Proof. If oR had a cycle i1 → i2 → · · · → ik → i1 , then a point (x1 , . . . , xn ) ∈ R
would satisfy xi1 < xi2 < · · · < xik < xi1 , which is absurd. Hence oR is acyclic.
Conversely, let o be an acyclic orientation of G. First note that o must have a
sink, i.e., a vertex with no arrows pointing out. To see this, walk along the edges
of o by starting at any vertex and following arrows. Since o is acyclic, we can never
return to a vertex so the process will end in a sink. Let jn be a sink vertex of o.
When we remove jn from o the remaining orientation is still acyclic, so it contains
a sink jn−1 . Continuing in this manner, we obtain an ordering j1 , j2 , . . . , jn of [n]
such that ji is a sink of the restriction of o to j1 , . . . , ji . Hence if x1 , . . . , xn ∈ R
satisfy xj1 < xj2 < · · · < xjn then the region R ∈ R(A) containing (x1 , . . . , xn )
satisfies o = oR .
Note. The transitive, reflexive closure ō of an acyclic orientation o is a par-
tial order. The construction of the ordering j1 , j2 , . . . , jn above is equivalent to
constructing a linear extension of o.
Let AO(G) denote the set of acyclic orientations of G. We have constructed a
bijection between AO(G) and R(AG ). Hence from Theorem 2.5 we conclude:
Corollary 2.3. For any graph G with n vertices, we have
#AO(G) = (−1)n χG (−1).
Corollary 2.3 was first proved by Stanley in 1973 by a “direct” argument based
on deletion-contraction (see Exercise 7). The proof we have just given based on
arrangements is due to Greene and Zaslavsky in 1983.
Note. Given a graph G on n vertices, let A# G be the arrangement defined by
xi − xj = aij , ij ∈ E(G),
LECTURE 2. PROPERTIES OF THE INTERSECTION POSET 419
where the aij ’s are generic. Just as we obtained equation (14) (the case G = Kn )
we have
χA# (t) = (−1)e(F ) tn−e(F ) ,
G
F
where F ranges over all spanning forests of G.
Exercises
(1) [3–] Show that for any arrangement A, we have χcA (t) = (t − 1)χA (t), where cA
denotes the cone over A. (Use Whitney’s theorem.)
(2) [2–] Let G be a graph on the vertex set [n]. Show that the bond lattice LG is a
sub-join-semilattice of the partition lattice Πn but is not in general a sublattice
of Πn .
(3) [2–] Let G be a forest (graph with no cycles) on the vertex set [n]. Show that
LG ∼ = BE(G) , the boolean algebra of all subsets of E(G).
(4) [2] Let G be a graph with n vertices and AG the corresponding graphical arrange-
ment. Suppose that G has a k-element clique, i.e., k vertices such that any two
are adjacent. Show that k!|r(A).
(5) [2+] Let G be a graph on the vertex set [n] = {1, 2, . . . , n}, and let AG be the
corresponding graphical arrangement (over any field K, but you may assume
K = R if you wish). Let Cn be the coordinate hyperplane arrangement, con-
sisting of the hyperplanes xi = 0, 1 ≤ i ≤ n. Express χAG ∪Cn (t) in terms of
χAG (t).
(6) [4] Let G be a planar graph, i.e., G can be drawn in the plane without crossing
edges. Show that χAG (4)
= 0.
(7) [2+] Let G be a graph with n vertices. Show directly from the the deletion-
contraction recurrence (20) that
(−1)n χG (−1) = #AO(G).
(8) [2+] Let χG (t) = tn − cn−1 tn−1 + · · · + (−1)n−1 c1 t be the chromatic polynomial
of the graph G. Let i be a vertex of G. Show that c1 is equal to the number of
acyclic orientations of G whose unique source is i. (A source is a vertex with no
arrows pointing in. In particular, an isolated vertex is a source.)
(9) [5] Let A be an arrangement with characteristic polynomial χA (t) = tn −
cn−1 tn−1 + cn−2 tn−2 − · · · + (−1)n c0 . Show that the sequence c0 , c1 , . . . , cn = 1
is unimodal, i.e., for some j we have
c0 ≤ c1 ≤ · · · ≤ cj ≥ cj+1 ≥ · · · ≥ cn .
(10) [2+] Let f (n) be the total number of faces of the braid arrangement Bn . Find
a simple formula for the generating function
xn x2 x3 x4 x5 x6
f (n) = 1 + x + 3 + 13 + 75 + 541 + 4683 + · · · .
n! 2! 3! 4! 5! 6!
n≥0
More generally, let fk (n) denote the number of k-dimensional faces of Bn . For
instance, f1 (n) = 1 (for n ≥ 1) and fn (n) = n!. Find a simple formula for the
generating function
xn x2 x3
fk (n)y k = 1 + yx + (y + 2y 2 ) + (y + 6y 2 + 6y 3 ) + · · · .
n! 2! 3!
n≥0 k≥0
LECTURE 3
Matroids and Geometric Lattices
3.1. Matroids
A matroid is an abstraction of a set of vectors in a vector space (for us, the normals
to the hyperplanes in an arrangement). Many basic facts about arrangements
(especially linear arrangements) and their intersection posets are best understood
from the more general viewpoint of matroid theory. There are many equivalent
ways to define matroids. We will define them in terms of independent sets, which
are an abstraction of linearly independent sets. For any set S we write
2S = {T : T ⊆ S}.
Definition 3.8. A (finite) matroid is a pair M = (S, I), where S is a finite set and
I is a collection of subsets of S, satisfying the following axioms:
(1) I is a nonempty (abstract) simplicial complex, i.e., I
= ∅, and if J ∈ I and
I ⊂ J, then I ∈ I.
(2) For all T ⊆ S, the maximal elements of I ∩ 2T have the same cardinality.
In the language of simplicial complexes, every induced subcomplex of I is
pure.
The elements of I are called independent sets. All matroids considered here will
be assumed to be finite. By standard abuse of notation, if M = (S, I) then we write
x ∈ M to mean x ∈ S. The archetypal example of a matroid is a finite subset S of
a vector space, where independence means linear independence. A closely related
matroid consists of a finite subset S of an affine space, where independence now
means affine independence.
It should be clear what is meant for two matroids M = (S, I) and M = (S , I )
to be isomorphic, viz., there exists a bijection f : S → S such that {x1 , . . . , xj } ∈ I
if and only if {f (x1 ), . . . , f (xj )} ∈ I . Let M be a matroid and S a set of points in
Rn , regarded as a matroid with independence meaning affine independence. If M
and S are isomorphic matroids, then S is called an affine diagram of M . (Not all
matroids have affine diagrams.)
Example 3.7. (a) Regard the configuration in Figure 1 as a set of five points in the
two-dimensional affine space R2 . These five points thus define the affine diagram
of a matroid M . The lines indicate that the points 1,2,3 and 3,4,5 lie on straight
421
422 R. STANLEY, HYPERPLANE ARRANGEMENTS
1 5
2 4
3
Figure 1. A five-point matroid in the affine space R2
lines. Hence the sets {1, 2, 3} and {3, 4, 5} are affinely dependent in R2 and therefore
dependent (i.e., not independent) in M . The independent sets of M consist of all
subsets of [5] with at most two elements, together with all three-element subsets of
[5] except 123 and 345 (where 123 is short for {1, 2, 3}, etc.).
(b) Write I = S1 , . . . , Sk for the simplicial complex I generated by S1 , . . . , Sk ,
i.e.,
S1 , . . . , Sk = {T : T ⊆ Si for some i}
= 2 S1 ∪ · · · ∪ 2 Sk .
Then I = 13, 14, 23, 24 is the set of independent sets of a matroid M on [4]. This
matroid is realized by a multiset of vectors in a vector space or affine space, e.g., by
the points 1,1,2,2 in the affine space R. The affine diagam of this matroid is given
by
1,2 3,4
(c) Let I = 12, 23, 34, 45, 15. Then I is not the set of independent sets of a
matroid. For instance, the maximal elements of I ∩ 2{1,2,4} are 12 and 4, which do
not have the same cardinality.
(d) The affine diagram below shows a seven point matroid.
1 2
3
LECTURE 3. MATROIDS AND GEOMETRIC LATTICES 423
If we further require the points labelled 1,2,3 to lie on a line (i.e., remove 123
from I), we still have a matroid M , but not one that can be realized by real vectors.
In fact, M is isomorphic to the set of nonzero vectors in the vector space F32 , where
F2 denotes the two-element field.
010
110 011
111
1 2 3 4 5
L(M ) then rk(y) = 1 + rk(x). We now define the characteristic polynomial χM (t),
in analogy to the definition (3) of χA (t), by
(22) χM (t) = μ(0̂, x)tm−rk(x) ,
x∈L(M)
where μ denotes the Möbius function of L(M ) and m = rk(M ). Figure 2 shows the
lattice of flats of the matroid M of Figure 1. From this figure we see easily that
χM (t) = t3 − 5t2 + 8t − 4.
Let M be a matroid and x ∈ M . If the set {x} is dependent (i.e., if rk({x}) = 0)
then we call x a loop. Thus ¯∅ is just the set of loops of M . Suppose that x, y ∈ M ,
neither x nor y are loops, and rk({x, y}) = 1. We then call x and y parallel points.
A matroid is simple if it has no loops or pairs of parallel points. It is clear that the
following three conditions are equivalent:
• M is simple.
• ¯
∅ = ∅ and x̄ = x for all x ∈ M .
• rk({x, y}) = 2 for all points x
= y of M (assuming M has at least two
points).
For any matroid M and x, y ∈ M , define x ∼ y if x̄ = ȳ. It is easy to see that ∼ is
an equivalence relation. Let
(23) = {x̄ : x ∈ M, x
∈ ¯∅},
M
with an obvious definition of independence, i.e.,
) ⇔ {x1 , . . . , xk } ∈ I(M ).
{x̄1 , . . . , x̄k } ∈ I(M
Then M is simple, and L(M ) ∼ = L(M). Thus insofar as intersection lattices L(M )
are concerned, we may assume that M is simple. (Readers familiar with point set
topology will recognize the similarity between the conditions for a matroid to be
simple and for a topological space to be T0 .)
Example 3.8. Let S be any finite set and V a vector space. If f : S → V , then
define a matroid Mf on S by the condition that given I ⊆ S,
I ∈ I(M ) ⇔ {f (x) : x ∈ I} is linearly independent.
LECTURE 3. MATROIDS AND GEOMETRIC LATTICES 425
Then a loop is any element x satisfying f (x) = 0, and x ∼ y if and only if f (x) is
a nonzero scalar multiple of f (y).
Note. If M = (S, I) is simple, then L(M ) determines M . For we can identify
S with the set of atoms of L(M ), and we have
{x1 , . . . , xk } ∈ I ⇔ rk(x1 ∨ · · · ∨ xk ) = k in L(M ).
See the proof of Theorem 3.8 for further details.
We now come to the primary connection between hyperplane arrangements and
matroid theory. If H is a hyperplane, write nH for some (nonzero) normal vector
to H.
Proposition 3.6. Let A be a central arrangement in the vector space V . Define
a matroid M = MA on A by letting B ∈ I(M ) if B is linearly independent (i.e.,
{nH : H ∈ B} is linearly independent). Then M is simple and L(M ) ∼
= L(A).
Proof. M has no loops, since every H ∈ A has a nonzero normal. Two distinct
nonparallel hyperplanes have linearly independent normals, so the points of M are
closed. Hence M is simple.
Let B, B ⊆ A, and set
X= H = XB , X = H = XB .
H∈B H∈B
Definition 3.9. A finite lattice L satisfying condition (1) or (2) above is called
(upper) semimodular. A finite lattice L is atomic if every x ∈ L is a join of atoms
(where we regard 0̂ as an empty join of atoms). Equivalently, if x ∈ L is join-
irreducible (i.e., covers a unique element), then x is an atom. Finally, a finite
lattice is geometric if it is both semimodular and atomic.
To illustrate these definitions, Figure 3(a) shows an atomic lattice that is not
semimodular, (b) shows a semimodular lattice that is not atomic, and (c) shows a
graded lattice that is neither semimodular nor atomic.
We are now ready to characterize the lattice of flats of a matroid.
Theorem 3.8. Let L be a finite lattice. The following two conditions are equivalent.
(1) L is a geometric lattice.
(2) L ∼
= L(M ) for some (simple) matroid M .
Proof. Assume
L is geometric, and let A be the set of atoms of L. If T ⊆ A
that
then write T = x∈T x, the join of all elements of T . Let
I = {I ⊆ A : rk(∨I) = #I}.
Note any S ⊆ A and x ∈ A that rk(( S)∨x) ≤
that by semimodularity, we have for
rk( S) + 1. (Hence in particular, rk( S) ≤ #S.) It follows that I is a simplicial
complex. Let S ⊆ A, and let T, T be maximal elements of 2S ∩ I. We need to show
that #T = #T .
Assume #T < #T , say. If y ∈ S then y ≤ T , else T = T ∪ y satisfies
rk( T ) = #T , contradicting
the maximality of T . Since #T < #T and T ⊆ S,
it follows that T < T [why?]. Since L is atomic, there exists y ∈ S such that
y ∈ S but y
≤ T . But then rk( (T ∪ y)) = 1 + #T , contradicting the maximality
of T . Hence M = (A, I) is a matroid, and L ∼ = L(M ).
Conversely, given a matroid M , which we may assume is simple, we need to
show that L(M ) is a geometric lattice. Clearly L(M ) is atomic, since every flat is
the join of its elements. Let S, T ⊆ M . We will show that
(24) rk(S) + rk(T ) ≥ rk(S ∩ T ) + rk(S ∪ T ).
LECTURE 3. MATROIDS AND GEOMETRIC LATTICES 427
Note that Theorem 3.9 gives a “shortening” of the recurrence (2) defining μ.
Normally we take a to be an atom, since that produces fewer terms in (25) than
choosing any b > a. As an example, let L = Bn , the boolean algebra of all subsets
of [n], and let a = {n}. There are two elements x ∈ Bn such that x ∨ a = 1̂ = [n],
viz., x1 = [n − 1] and x2 = [n]. Hence μ(x1 ) + μ(x2 ) = 0. Since [0̂, x1 ] = Bn−1 and
[0̂, x2 ] = Bn , we easily obtain μBn (1̂) = (−1)n , agreeing with (4).
If x ≤ y in a graded lattice L, write rk(x, y) = rk(y) − rk(x), the length of
every saturated chain from x to y. The next result may be stated as “the Möbius
function of a geometric lattice strictly alternates in sign.”
Theorem 3.10. Let L be a finite geometric lattice with Möbius function μ, and let
x ≤ y in L. Then
(−1)rk(x,y) μ(x, y) > 0.
Proof. Since every interval of a geometric lattice is a geometric lattice (Exercise 3),
it suffices to prove the theorem for [x, y] = [0̂, 1̂]. The proof is by induction on the
rank of L. It is clear if rk(L) = 1, in which case μ(0̂, 1̂) = −1. Assume the result
for geometric lattices of rank < n, and let rk(L) = n. Let a be an atom of L in
Theorem 3.9. For any y ∈ L we have by semimodularity that
rk(y ∧ a) + rk(y ∨ a) ≤ rk(y) + rk(a) = rk(y) + 1.
428 R. STANLEY, HYPERPLANE ARRANGEMENTS
The sum on the right is nonempty since L is atomic, and by induction every x
indexing the sum satisfies (−1)n−1 μ(0̂, x) > 0. Hence (−1)n μ(0̂, 1̂) > 0.
Combining Proposition 3.8 and Theorem 3.10 yields the following result.
Corollary 3.4. Let A be any arrangement and x ≤ y in L(A). Then
(−1)rk(x,y) μ(x, y) > 0,
where μ denotes the Möbius function of L(A).
Similarly, combining Theorem 3.10 with the definition (22) of χM (t) gives the
next corollary.
Corollary 3.5. Let M be a matroid of rank n. Then the characteristic polynomial
χM (t) strictly alternates in sign, i.e., if
χM (t) = an tn + an−1 tn−1 + · · · + a0 ,
then (−1)n−i ai > 0 for 0 ≤ i ≤ n.
Let A be an n-dimensional arrangement of rank r. If MA is the matroid
corresponding to A, as defined in Proposition 3.6, then
(26) χA (t) = tn−r χM (t).
It follows from Corollary 3.5 and equation (26) that we can write
χA (t) = bn tn + bn−1 tn−1 + · · · + bn−r tn−r ,
where (−1)n−i bi > 0 for n − r ≤ i ≤ n.
Exercises
(1) (a) [1+] Let χG (t) be the characteristic polynomial of the graphical arrange-
ment AG . Suppose that χG (i) = 0, where i ∈ Z, i > 1. Show that
χG (i − 1) = 0.
(b) [2] Is the same conclusion true for any central arrangement A?
(2) [2] Show that if F and F are flats of a matroid M , then so is F ∩ F .
(3) [2] Prove the assertion in the Note following the proof of Theorem 3.8 that an
interval [x, y] of a geometric lattice L is also a geometric lattice.
(4) [2–] Let A be an arrangement (not necessarily central), and let cA denote the
cone over A. Show that there exists an atom a of L(cA) such that L(A) ∼ =
L(cA) − Va , where Va = {x ∈ L : x ≥ a}.
(5) [2–] Let L be a geometric lattice of rank n, and define the truncation T (L) to
be the subposet of L consisting of all elements of rank
= n − 1. Show that T (L)
is a geometric lattice.
(6) Let Wi be the number of elements of rank i in a geometric lattice (or just in the
intersection poset of a central hyperplane arrangement, if you prefer) of rank n.
(a) [3] Show that for k ≤ n/2,
W1 + W2 + · · · + Wk ≤ Wn−k + Wn−k+1 + · · · + Wn−1 .
(b) [2–] Deduce from (a) and Exercise 5 that W1 ≤ Wk for all 1 ≤ k ≤ n − 1.
LECTURE 3. MATROIDS AND GEOMETRIC LATTICES 429
(c) [5] Show that Wi ≤ Wn−i for i < n/2 and that the sequence W0 , W1 , . . . , Wn
is unimodal. (Compare Lecture 2, Exercise 9.)
(7) [3–] Let x ≤ y in a geometric lattice L. Show that μ(x, y) = ±1 if and only if
the interval [x, y] is isomorphic to a boolean algebra. (Use Weisner’s theorem.)
Note. This problem becomes much easier using Theorem 4.12 (the Broken
Circuit Theorem); see Exercise 4.13.
LECTURE 4
Broken Circuits, Modular Elements, and Supersolvability
This lecture is concerned primarily with matroids and geometric lattices. Since
the intersection lattice of a central arrangement is a geometric lattice, all our results
can be applied to arrangements.
1 5 3 5
2 4 2 4
3 1
431
432 R. STANLEY, HYPERPLANE ARRANGEMENTS
have BCO (M ) = 135, 145, 235, 245, while BCO (M ) = 135, 235, 245, 345. These
simplicial complexes have geometric realizations as follows:
2
1 3
1
5 5
4 2 4 3
Note that the two simplicial complexes BCO (M ) and BCO (M ) are not iso-
morphic (as abstract simplicial complexes); in fact, their geometric realizations are
not even homeomorphic. On the other hand, if fi (Δ) denotes the number of i-
dimensional faces (or faces of cardinality i − 1) of the abstract simplicial complex
Δ, then for Δ given by either BCO (M ) or BCO (M ) we have
f−1 (Δ) = 1, f0 (Δ) = 5, f1 (Δ) = 8, f2 (Δ) = 4.
Note, moreover, that
χM (t) = t3 − 5t2 + 8t − 4.
In order to generalize this observation to arbitrary matroids, we need to introduce
a fair amount of machinery, much of it of interest for its own sake. First we give
a fundamental formula, known as Philip Hall’s theorem, for the Möbius function
value μ(0̂, 1̂).
Lemma 4.4. Let P be a finite poset with 0̂ and 1̂, and with Möbius function μ.
Let ci denote the number of chains 0̂ = y0 < y1 < · · · < yi = 1̂ in P . Then
μ(0̂, 1̂) = −c1 + c2 − c3 + · · · .
Proof. We work in the incidence algebra I(P ). We have
μ(0̂, 1̂) = ζ −1 (0̂, 1̂)
= (δ + (ζ − δ))−1 (0̂, 1̂)
= δ(0̂, 1̂) − (ζ − δ)(0̂, 1̂) + (ζ − δ)2 (0̂, 1̂) − · · · .
This expansion is easily justified since (ζ −δ)k (0̂, 1̂) = 0 if the longest chain of P has
length less than k. By definition of the product in I(P ) we have (ζ − δ)i (0̂, 1̂) = ci ,
and the proof follows.
Note. Let P be a finite poset with 0̂ and 1̂, and let P = P − {0̂, 1̂}. Define
Δ(P ) to be the set of chains of P , so Δ(P ) is an abstract simplicial complex. The
reduced Euler characteristic of a simplicial complex Δ is defined by
χ̃(P ) = −f−1 + f0 − f1 + · · · ,
where fi is the number of i-dimensional faces F ∈ Δ (or #F = i + 1). Comparing
with Lemma 4.4 shows that
μ(0̂, 1̂) = χ̃(Δ(P )).
Readers familiar with topology will know that χ̃(Δ) has important topological sig-
nificance related to the homology of Δ. It is thus natural to ask whether results
LECTURE 4. BROKEN CIRCUITS AND MODULAR ELEMENTS 433
3 2 1
1 1 2 1
1 1
2 1 3 1 2
3
1 2 3 2 2
1 3 1 2
1 2 3
Proof. Since λ restricted to [x, y] (i.e., to E([x, y])) is an E-labeling, we can assume
[x, y] = [0̂, 1̂] = P . Let S = {a1 , a2 , . . . , aj−1 } ⊆ [n − 1], with a1 < a2 < · · · < aj−1 .
Define αP (S) to be the number of chains 0̂ < y1 < · · · < yj−1 < 1̂ in P such that
rk(yi ) = ai for 1 ≤ i ≤ j − 1. The function αP is called the flag f -vector of P .
Claim. αP (S) is the number of maximal chains 0̂ = x0 x1 · · · xn = 1̂ such
that
(27) λ(xi−1 , xi ) > λ(xi , xi+1 ) ⇒ i ∈ S, 1 ≤ i ≤ n.
To prove the claim, let 0̂ = y0 < y1 < · · · < yj−1 < yj = 1̂ with rk(yi ) = ai for
1 ≤ i ≤ j − 1. By the definition of E-labeling, there exists a unique refinement
0̂ = y0 = x0 x1 · · · xa1 = y1 xa1 +1 · · · xa2 = y2 · · · xn = yj = 1̂
satisfying
λ(x0 , x1 ) ≤ λ(x1 , x2 ) ≤ · · · ≤ λ(xa1 −1 , xa1 )
λ(xa1 , xa1 +1 ) ≤ λ(xa1 +1 , xa1 +2 ) ≤ · · · ≤ λ(xa2 −1 , xa2 )
···
Thus if λ(xi−1 , xi ) > λ(xi , xi+1 ), then i ∈ S, so (27) is satisfied. Conversely, given
a maximal chain 0̂ = x0 x1 · · · xn = 1̂ satisfying the above conditions on λ,
let yi = xai . Therefore we have a bijection between the chains counted by αP (S)
and the maximal chains satisfying (27), so the claim follows.
Now for S ⊆ [n − 1] define
(28) βP (S) = (−1)#(S−T ) αP (T ).
T ⊆S
for all S ⊆ [n−1]. It follows from the claim and equation (29) that βP (T ) is equal to
the number of maximal chains 0̂ = x0 x1 · · · xn = 1̂ such that λ(xi ) > λ(xi+1 )
if and only if i ∈ T . In particular, βP ([n − 1]) is equal to the number of strictly
decreasing maximal chains 0̂ = x0 x1 · · · xn = 1̂ of P , i.e.,
λ(x0 , x1 ) > λ(x1 , x2 ) > · · · > λ(xn−1 , xn ).
Now by (28) we have
βP ([n − 1]) = (−1)n−1−#T αP (T )
T ⊆[n−1]
= (−1)n−k
k≥1 0̂=y0 <y1 <···<yk =1̂
= (−1)n (−1)k ck ,
k≥1
where ci is the number of chains 0̂ = y0 < y1 < · · · < yi = 1̂ in P . The proof now
follows from Philip Hall’s theorem (Lemma 4.4).
We come to the main result of this section, a combinatorial interpretation of
the coefficients of the characteristic polynomial χM (t) for any matroid M .
LECTURE 4. BROKEN CIRCUITS AND MODULAR ELEMENTS 435
Theorem 4.12. Let M be a matroid of rank n with a linear ordering x1 < x2 <
· · · < xm of its points (so the broken circuit complex BC(M ) is defined), and let
0 ≤ i ≤ n. Then
(−1)i [tn−i ]χM (t) = fi−1 (BC(M )).
Proof. We may assume M is simple since the “simplification” M has the same
lattice of flats and same broken circuit complex as M (Exercise 1). The atoms xi of
L(M ) can then be identified with the points of M . Define a labeling λ̃ : E(L(M )) →
P as follows. Let x y in L(M ). Then set
(30) λ̃(x, y) = max{i : x ∨ xi = y}.
Note that λ̃(x, y) is defined since L(M ) is atomic.
As an example, Figure 3 shows the lattice of flats of the matroid M of Figure 1
with the edge labeling (30).
5 5 4 5 4 2
4 1 5 1 42 5 2
3 55 4
3 2
1 2 3 4 5
1 2 3 4 5
B2(3)
B3(2)
Example 4.10. Before proceeding to the proof of Theorem 4.13, let us consider
an example. The illustration below is the affine diagram of a matroid M of rank
3, together with its lattice of flats. The two lines (flats of rank 2) labelled x and y
are modular by Example 4.9(c).
y x y
x
440 R. STANLEY, HYPERPLANE ARRANGEMENTS
Our proof of Theorem 4.13 will depend on the following lemma of Greene [19].
We give a somewhat simpler proof than Greene.
Lemma 4.5. Let L be a finite lattice with Möbius function μ, and let z ∈ L. The
following identity is valid in the Möbius algebra A(L) of L:
⎛ ⎞⎛ ⎞
(33) σ0̂ := μ(x)x = ⎝ μ(v)v ⎠ ⎝ μ(y)y ⎠ .
x∈L v≤z y∧z=0̂
Proof. Let σs for s ∈ L be given by (8). The right-hand side of equation (33) is
then given by
μ(v)μ(y)(v ∨ y) = μ(v)μ(y) σs
v≤z v≤z s≥v∨y
y∧z=0̂ y∧z=0̂
= σs μ(v)μ(y)
s v≤s,v≤z
y≤s,y∧z=0̂
⎛ ⎞
⎛ ⎞
⎜ ⎟
⎜ ⎜
⎟⎜
⎟ ⎟
= σs ⎜ μ(v)⎟ ⎜ μ(y)⎟
⎜ ⎟⎝ ⎠
s ⎝v≤s∧z ⎠ y≤s
y∧z=0̂
δ0̂,s∧z
⎛ ⎞
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
= ⎜
σs ⎜ μ(y)⎟
⎟
⎜ ⎟
s∧z=0̂ ⎜ y≤s ⎟
⎝y∧z=0̂ (redundant) ⎠
δ0̂,s
= σ0̂ .
Proof of Theorem 4.13. We are assuming that z is a modular element of
the geometric lattice L.
Claim 1. Let v ≤ z and y ∧ z = 0̂ (so v ∧ y = 0̂). Then z ∧ (v ∨ y) = v (as
illustrated below).
LECTURE 4. BROKEN CIRCUITS AND MODULAR ELEMENTS 441
zv y
z vvy
v y
vy → tn−rk(v)−rk(y) = tn−rk(v∨y) .
Now v ∨ y is just vy in the Möbius algebra A(L). Hence if we further substi-
tute μ(x)x → μ(x)tn−rk(x) in the left-hand side of (33), then the product will be
preserved. We thus obtain
⎛ ⎞
⎜ ⎟⎛ ⎞
⎜ ⎟
⎜ ⎟
μ(x)tn−rk(x) =⎜ μ(v)trk(z)−rk(v) ⎟ ⎝ μ(y)tn−rk(y)−rk(z) ⎠ ,
⎜ ⎟
x∈L
⎝v≤z ⎠ y∧z=0̂
χL (t) χz (t)
as desired.
442 R. STANLEY, HYPERPLANE ARRANGEMENTS
q i − 1 q i−1 − 1
= −
q−1 q−1
= q i−1 .
Hence
χBn (q) (t) = (t − 1)(t − q)(t − q 2 ) · · · (t − q n−1 ).
In particular, setting t = 0 gives
n
μBn (q) (1̂) = (−1)n q ( 2 ) .
$ %
Note. The expression kj is called a q-binomial coefficient. It is a
polynomial in q with many interesting properties. For the most basic
properties, see e.g. [31, pp. 27–30].
(c) Let L = Πn , the lattice of partitions of the set [n] (a geometric lattice of
rank n − 1). By Proposition 4.9, a maximal chain of Πn is modular if and
only if it has the form 0̂ = π0 π1 · · · πn−1 = 1̂, where πi for i > 0 has
exactly one nonsingleton block Bi (necessarily with i + 1 elements), with
B1 ⊂ B2 · · · ⊂ Bn−1 = [n]. In particular, Πn is supersolvable and has
exactly n!/2 modular chains for n > 1. The atoms covered by πi are the
444 R. STANLEY, HYPERPLANE ARRANGEMENTS
lattice LG of the graph G. A graph H with at least one edge is doubly connected if
it is connected and remains connected upon the removal of any vertex (and all in-
cident edges). A maximal doubly connected subgraph of a graph G is called a block
of G. For instance, if G is a forest then its blocks are its edges. Two different blocks
of G intersect in at most one vertex. Figure 5 shows a graph with eight blocks, five
of which consist of a single edge. The following proposition is straightforward to
prove (Exercise 16).
Proposition 4.11. Let G be a graph with blocks G1 , . . . , Gk . Then
LG ∼= LG × · · · × LG .
1 k
It is also easy to see that if L1 and L2 are geometric lattices, then L1 and
L2 are supersolvable if and only if L1 × L2 is supersolvable (Exercise 18). Hence
in characterizing supersolvable graphs G (i.e., graphs whose bond lattice LG is
supersolvable) we may assume that G is doubly connected. Note that for any
connected (and hence a fortiori doubly connected) graph G, any coatom π of LG
has exactly two blocks.
Proposition 4.12. Let G be a doubly connected graph, and let π = {A, B} be a
coatom of the bond lattice LG , where #A ≤ #B. Then π is a modular element of
LG if and only if #A = 1, say A = {v}, and the neighborhood N (v) (the set of
vertices adjacent to v) forms a clique (i.e., any two distinct vertices of N (v) are
adjacent).
Proof. The proof parallels that of Proposition 4.9, which is a special case. Suppose
that #A > 1. Since G is doubly connected, there exist u, v ∈ A and u , v ∈ B such
that u
= v, u
= v , uu ∈ E(G), and vv ∈ E(G). Set σ = {(A∪u )−v, (B∪v)−u }.
If G has n vertices then rk(π) = rk(σ) = n−2, rk(π∨σ) = n−1, and rk(π∧σ) = n−4.
Hence π is not modular.
Assume then that A = {v}. Suppose that av, bv ∈ E(G) but ab
∈ E(G). We
need to show that π is not modular. Let σ = {A − {a, b}, {a, b, v}}. Then
σ ∨ π = 1̂, σ ∧ π = {A − {a, b}, a, b, v}
rk(σ) = rk(π) = n − 2, rk(σ ∨ π) = n − 1, rk(σ ∧ π) = n − 4.
Hence π is not modular.
446 R. STANLEY, HYPERPLANE ARRANGEMENTS
Conversely, let π = {A, v}. Assume that if av, bv ∈ E(G) then ab ∈ E(G).
It is then straightforward to show (Exercise 8) that π is modular, completing the
proof.
As an immediate consequence of Propositions 4.10(b) and 4.12 we obtain a
characterization of supersolvable graphs.
Corollary 4.10. A graph G is supersolvable if and only if there exists an ordering
v1 , v2 , . . . , vn of its vertices such that if i < k, j < k, vi vk ∈ E(G) and vj vk ∈ E(G),
then vi vj ∈ E(G). Equivalently, in the restriction of G to the vertices v1 , v2 , . . . , vi ,
the neighborhood of vi is a clique.
Note. Supersolvable graphs G had appeared earlier in the literature under the
names chordal, rigid circuit, or triangulated graphs. One of their many characteri-
zations is that any circuit of length at least four contains a chord. Equivalently, no
induced subgraph of G is a k-cycle for k ≥ 4.
Exercises
),
(1) [2–] Let M be a matroid on a linearly ordered set. Show that BC(M ) = BC(M
where M is defined by equation (23).
(2) [2+] Let M be a matroid of rank at least one. Show that the coefficients of the
polynomial χM (t)/(t − 1) alternate in sign.
(3) (a) [2+] Let L be finite lattice for which every element has a unique comple-
ment. Show that L is isomorphic to a boolean algebra Bn .
(b) [3] A lattice L is distributive if
x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z)
x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z)
for all x, y, z ∈ L. Let L be an infinite lattice with 0̂ and 1̂. If every element
of L has a unique complement, then is L a distributive lattice?
(4) [3–] Let x be an element of a geometric lattice L. Show that the following four
conditions are equivalent.
(i) x is a modular element of L.
(ii) If x ∧ y = 0̂, then
rk(x) + rk(y) = rk(x ∨ y).
(iii) If x and y are complements, then rk(x) + rk(y) = n.
(iv) All complements of x are incomparable.
(5) [2+] Let x, y be modular elements of a geometric lattice L. Show that x ∧ y is
also modular.
(6) [2] Let L be a geometric lattice. Prove or disprove: if x is modular in L and y
is modular in the interval [x, 1̂], then y is modular in L.
(7) [2–] Let L and L be finite lattices. Show that if both L and L are geometric
(respectively, atomic, semimodular, modular) lattices, then so is L × L .
(8) [2] Let G be a (loopless) connected graph and v ∈ V (G). Let A = V (G) − v and
π = {A, v} ∈ LG . Suppose that whenever av, bv ∈ E(G) we have ab ∈ E(G).
Show that π is a modular element of LG .
(9) [2+] Generalize the previous exercise as follows. Let G be a doubly-connected
graph with lattice of contractions LG . Let π ∈ LG . Show that the following two
conditions are equivalent.
LECTURE 4. BROKEN CIRCUITS AND MODULAR ELEMENTS 447
χL (t) = μ(y)χLy (t)tn−rk(x∨y) ,
y∈L
x∧y=0̂
Now for any r × s matrix A, we have rank(A) ≥ t if and only if some t × t submatrix
B satisfies det(B)
= 0. It follows that L(A) ∼
L(Ap ) if and only if at least one
=
member S of a certain finite collection S of subsets of integer matrices B satisfies
the following condition:
(∀B ∈ S) det(B)
= 0 but det(B) ≡ 0 (mod p).
This can only happen for finitely many p, viz., for certain B we must have p| det(B),
so L(A) ∼
= L(Ap ) for p sufficiently large.
The main result of this section is the following. Like many fundamental results
in combinatorics, the proof is easy but the applicability very broad.
Theorem 5.15. Let A be an arrangement in Qn , and suppose that L(A) ∼
= L(Aq )
for some prime power q. Then
⎛ ⎞
χA (q) = # ⎝Fnq − H⎠
H∈Aq
= qn − # H.
H∈Aq
Proof. Let x ∈ L(Aq ) so #x = q dim(x) . Here dim(x) can be computed either over
Q or Fq . Define two functions f, g : L(Aq ) → Z by
f (x) = #x
& '
g(x) = # x− y .
y>x
In particular,
⎛ ⎞
g(0̂) = g(Fnq ) = # ⎝Fnq − H⎠ .
H∈Aq
Clearly
f (x) = g(y).
y≥x
Put x = 0̂ to get
g(0̂) = μ(y)q dim(y) = χA (q).
y
For the remainder of this lecture, we will be concerned with applications of
Theorem 5.15 and further interesting examples of arrangements.
LECTURE 5. FINITE FIELDS 451
A(B 2 ) A(D2 )
Figure 1. The arrangements A(B2 ) and A(D2 )
10 0
5 9 1 2
8 2 3
7 3
4 6
6 4
5
1
Let αi be the position (element of Fp ) at which i was placed. For our example
we have
(α1 , α2 , α3 , α4 , α5 , α6 ) = (6, 1, 2, 7, 9, 3).
It is easily verified that we have defined a bijection from the (p−n)n−1 weak ordered
partitions π = (B1 , . . . , Bp−n ) of [n] into p−n blocks such that 1 ∈ B1 , together with
the choice of α1 ∈ Fp , to the set Fnp −∪H∈(Sn )p H. There are (p−n)n−1 choices for π
and p choices for α1 , so it follows from Theorem 5.15 that χSn (t) = t(t − n)n−1 .
We obtain the following corollary immediately from Theorem 2.5.
Corollary 5.11. We have r(Sn ) = (n + 1)n−1 and b(Sn ) = (n − 1)n−1 .
Note. Since r(Sn ) and b(Sn ) have such simple formulas, it is natural to ask
for a direct bijective proof of Corollary 5.11. A number of such proofs are known;
a sketch that r(Sn ) = (n + 1)n−1 is given in Exercise 3.
Note. It can be shown that the cone cSn is not supersolvable for n ≥ 3 (Ex-
ercise 4) but is free in the sense of Theorem 4.14.
454 R. STANLEY, HYPERPLANE ARRANGEMENTS
Then equation (39) holds. A straightforward proof can be given by considering the
expansion
xn xn
exp f (n) = exp f (n)
n! n!
n≥1 n≥1
⎛ ⎞
xkn
= ⎝ f (n)k k ⎠ .
n! k!
n≥1 k≥0
Let A = (A1 , A2 , . . . ), and let B ⊆ An for some n. Define π(B) ∈ Πn to have blocks
that are the vertex sets of the connected components of the graph G on [n] with
edges
Define
χ̃An (t) = (−1)#B tn−rk(B) .
B⊆A
B central
π(B)=[n]
Then
χAn (t) = (−1)#B tn−rk(B)
π={B1 ,...,Bk }∈Πn B⊆A
B central
π(B)=π
= χ̃A#B1 (t)χ̃A#B2 (t) · · · χ̃A#Bk (t).
π={B1 ,...,Bk }∈Πn
But π(B) = [n] if and only if rk(B) = n − 1, so χ̃An (t) = cn t for some cn ∈ Z. We
therefore get
xn xn
χAn (t) = exp t cn
n! n!
n≥0 n≥1
⎛ ⎞t
xn
= ⎝ bn ⎠ ,
n!
n≥0
n
n
where exp x
n≥1 cn n! = . Put t = −1 to get
x
n≥0 bn n!
⎛ ⎞−1
xn xn
(−1)n r(An ) =⎝ bn ⎠ ,
n! n!
n≥0 n≥0
For a generalization of Theorem 5.17, see Exercise 10.
to the number of antichains A of strict intervals of [n], i.e., sets A of intervals [i, j],
where 1 ≤ i < j ≤ n, such that no interval in A is contained in another. (“Strict”
means that i = j is not allowed.) It is known (equivalent to [32, Exer. 6.19(bbb)])
2n
1
that the number of such antichains is the Catalan number Cn = n+1 n . For
the sake of completeness we give a bijection between these antichains and a stan-
dard combinatorial structure counted by Catalan numbers, viz., lattice paths from
(0, 0) to (n, n) with steps (1, 0) and (0, 1), never rising above the line y = x ([32,
Exer. 6.19(h)]). Given an antichain A of intervals of [n], there is a unique lattice
path of the claimed type whose “outer corners” (a step (1, 0) followed by (0, 1))
consist of the points (j, i − 1) where [i, j] ∈ A, together with the points (i, i − 1)
where no interval in A contains i. Figure 4 illustrates this bijection for n = 8 and
A = {[1, 4], [3, 5], [7, 8]}.
We have therefore proved the following result. For a refinement, see Exercise 11.
Proposition 5.14. The number of regions of the Catalan arrangement Cn is given
by r(Cn ) = n!Cn . Each region of Bn contains Cn regions of Cn .
In fact, there is a simple formula for the characteristic polynomial χCn (t).
Theorem 5.18. We have
χCn (t) = t(t − n − 1)(t − n − 2)(t − n − 3) · · · (t − 2n + 1).
458 R. STANLEY, HYPERPLANE ARRANGEMENTS
86
65
52
40
Figure 4. A bijection corresponding to A = {[1, 4], [3, 5], [7, 8]}
One method for expanding this series is to use the Lagrange inversion formula
[32, Thm. 5.4.2]. Let F (x) = a1 x + a2 x2 + · · · be a formal power series over K,
where char(K) = 0 and a1
= 0. Then there exists a unique formal power series
F −1 = a−1
1 x + · · · satisfying
a c f
b d
e
f
c d e
a b
Figure 5. An example of an interval order
1 6 3 3 6
Figure 6. The number of labelings of semiorders with three elements
The following proposition collects some basic results on interval orders. We sim-
ply state them without proof. Only part (a) is needed in what follows (Lemma 5.6).
We use the notation i to denote an i-element chain and P + Q to denote the disjoint
union of the posets P and Q.
Proposition 5.15.
(a) A finite poset is an interval order if and only if it has no induced subposet
isomorphic to 2 + 2.
(b) A finite poset is a semiorder if and only if it has no induced subposet
isomorphic to 2 + 2 or 3 + 1.
(c) A finite poset P is a semiorder if and only if its elements can be ordered as
I1 , . . . , In so that the incidence matrix of P (i.e., the matrix M = (mij ),
where mij = 1 if Ii < Ij and mij = 0 otherwise) has the form shown
below. Moreover, all such semiorders are nonisomorphic.
LECTURE 5. FINITE FIELDS 461
1 n
1
1
0
n
In (c) above, the southwest boundary of the positions of the 1’s in M form a
lattice path which by suitable indexing goes from (0, 0) to (n, n) with steps (0, 1)
and (1, 0), never rising above y = x. Since the number of such lattice paths is
the Catalan number Cn , it follows that the number of nonisomorphic n-element
semiorders is Cn . Later (Proposition 5.17) we will give a proof based on properties
of a certain arrangement. Figure 7 illustrates Proposition 5.15(c) when n = 3. It
shows the matrices M , the corresponding set of unit intervals, and the associated
semiorder.
Let 1 , . . . , n > 0 and set η = (1 , . . . , n ). Let Pη denote the set of all interval
orders P on [n] such that there exist a set I1 , . . . , In of intervals corresponding to
P
P (with Ii corresponding to i ∈ P ) such that (Ii ) = i . In other words, i < j if
and only if Ii lies entirely to the left of Ij . For instance, it follows from Figure 6
that #P(1,1,1) = 19.
We now come to the connection with arrangements. Given η = (1 , . . . , n ) as
above, define the arrangement Iη in Rn by letting its hyperplanes be given by
xi − xj = i , i
= j.
(Note the condition i
= j, not i < j.) Thus Iη has rank n − 1 and n(n − 1)
hyperplanes (since i > 0). Figure 8 shows the arrangement I(1,1,1) in the space
ker(x1 + x2 + x3 ).
462 R. STANLEY, HYPERPLANE ARRANGEMENTS
x3 = x2 x3 = x2 + 1
3
2 3
2
1 1
3 2
x 2 = x1 +1
3
1
1 2 3 1 2
x2 = x1
An : xi − xj = a1 , . . . , am , i = j,
Proof. Let c(n, k) denote the number of permutations w of n objects with k cycles
(in the disjoint cycle decomposition of w). The integer c(n, k) is known as a signless
Stirling number of the first kind and for fixed k has the exponential generating
function
xn 1 k
(44) c(n, k) = log(1 − x)−1 .
n! k!
n≥0
We have
F (x) = G(1 − e−x ) ⇔ G(x) = F (log(1 − x)−1 )
1 k
= r(Ak ) log(1 − x)−1
k!
k≥1
xn
= r(Ak ) c(n, k) .
n!
k≥1 n≥0
(1,5,2) (6,8) 1 5 2 6 8
( P,w ) Q = ρ ( P,w )
Given an unlabelled n-element semiorder Q, such as
we now show that there are exactly n! pairs (P, w) for which ρ(P, w) ∼
= Q. Call a
pair of elements x, y ∈ Q autonomous if for all z ∈ Q we have
x < z ⇔ y < z, x > z ⇔ y > z.
Equivalently, the map τ : Q → Q transposing x, y and fixing all other z ∈ Q is an
automorphism of Q. Clearly the relation of being autonomous is an equivalence
relation. Partition Q into its autonomous equivalence classes. Regard the elements
of Q as being distinguished, and choose a bijection (labeling) ϕ : Q → [n] (in n!
ways). Fix a linear ordering (independent of ϕ) of the elements in each equivalence
LECTURE 5. FINITE FIELDS 465
class. (The linear ordering of the elements in each equivalence class in the diagram
below is left-to-right.)
5 4 1
3 7 6 2 8
In each class, place a left parenthesis before each left-to-right maximum, and
place a right parenthesis before each left parenthesis and at the end. (This is the
bijection Sn → Sn , ŵ → w, in [31, p. 17].) Merge the elements c1 , c2 , . . . , cj
(appearing in that order) between each pair of parentheses into a single element
labelled with the cycle (c1 , c2 , . . . , cj ).
Q P
We have thus obtained a poset P whose elements are labelled by the cycles of
a permutation w ∈ Sn , such that ρ(P, w) = Q. For each unlabelled Q, there are
exactly n! pairs (P, w) (where the poset P is labelled by the cycles of w ∈ Sn )
for which ρ(P, w) ∼= Q. Since by Proposition 5.17 there are Cn nonisomorphic
n-element semiorders, we get
n
n!Cn = c(n, k)r(Ak ).
k=1
Note. Theorem 5.19 can also be proved using Burnside’s lemma (also called
the Cauchy-Frobenius lemma) from group theory.
To test one’s understanding of the proof of Theorem 5.19, consider why it
doesn’t work for all posets. In other words, let f (n) denote the number of posets on
n
the number of nonisomorphic n-element posets. Set F (x) = f (n) xn!
[n] and g(n)
and G(x) = g(n)xn . Why doesn’t the above argument show that G(x) = F (1 −
e−x )? Let Q = 2 + 2 (the unique obstruction to being an interval order, by
Proposition 5.15(a)). The autonomous classes have one element each. Consider the
two labelings ϕ : Q → [4] and the corresponding ρ−1 :
466 R. STANLEY, HYPERPLANE ARRANGEMENTS
2 4 2 4
ρ −1
1 3 1 3
4 2 4 2
ρ −1
3 1 3 1
We obtain the same labelled posets in both cases, so the proof of Theorem 5.19
fails. The key property of interval orders that the proof of Theorem 5.19 uses
implicitly is the following.
Lemma 5.6. If σ : P → P is an automorphism of the interval order P and
σ(x) = σ(y), then x and y are autonomous.
Proof. Assume not. Then there exists an element s ∈ P satisfying s > x, s
> y (or
dually). Since σ(x) = y, there must exist t ∈ P satisfying t > y, t
> x. But then
{x, s, y, t} form an induced 2 + 2, so by Proposition 5.15(a) P is not an interval
order.
Specializing m = 1 and a1 = 1 in Theorem 5.19 yields the following corollary,
due first (in an equivalent form) to Chandon, Lemaire and Pouget [12].
Corollary 5.12. Let f (n) denote the number of semiorders on [n] (or n-element
labelled semiorders). Then
xn
f (n) = C(1 − e−x ),
n!
n≥0
where √
1− 1 − 4x
n
C(x) = Cn x = .
2x
n≥0
Exercises
(1) [2] Verify equation (37), viz.,
χA(Dn ) (t) = (t − 1)(t − 3) · · · (t − (2n − 3)) · (t − n + 1).
(2) [2] Draw a picture of the projectivization of the Coxeter arrangement A(B3 ),
similar to Figure 1 of Lecture 1.
(3) (a) [2] An embroidered permutation of [n] consists of a permutation w of [n]
together with a collection E of ordered pairs (i, j) such that:
• 1 ≤ i < j ≤ n for all (i, j) ∈ E.
• If (i, j) and (h, k) are distinct elements of E, then it is false that
i ≤ h ≤ k ≤ j.
• If (i, j) ∈ E then w(i) < w(j).
For instance, the three embroidered permutations (w, E) of [2] are given
by (12, ∅), (12, {(1, 2)}), and (21, ∅). Give a bijective proof that the num-
ber r(Sn ) of regions of the Shi arrangement Sn is equal to the number of
embroidered permutations of [n].
(b) [2+] A parking function of length n is a sequence (a1 , . . . , an ) ∈ Pn whose
increasing rearrangement b1 ≤ b2 ≤ · · · ≤ bn satisfies bi ≤ i. For instance,
the parking functions of length three are 11, 12, 21. Give a bijective proof
that the number of parking functions of length n is equal to the number of
embroidered permutations of [n].
(c) [3–] Give a combinatorial proof that the number of parking functions of
length n is equal to (n + 1)n−1 .
(4) [2+] Show that if Sn denotes the Shi arrangement, then the cone cSn is not
supersolvable for n ≥ 3.
(5) [2] Show that if f : P → R and h : N → R are related by equation (40) (with
h(0) = 1), then equation (39) holds.
(6) (a) [2] Compute the characteristic polynomial of the arrangement Bn in Rn
with defining polynomial
Q(x) = (x1 − xn − 1) (xi − xj ).
1≤i<j≤n
Show that
xn G(x)(t+1)/2
(48) χAn (t) = .
n≥0
n! F (x)(t−1)/2
c d
c e
b
b d
a
a
W3
Let Pn be the poset of intervals [i, j], 1 ≤ i < j ≤ n, ordered by reverse
inclusion.
[1,2] [2,3] [3,4]
[1,2] [2,3]
[1,3] [2,4]
[1,3]
[1,4]
P3 P4
Show that Wn ∼ = J(Pn ), the lattice of order ideals of Pn . (An order ideal of a
poset P is a subset I ⊆ P such that if x ∈ I and y ≤ x, then y ∈ I. Define J(P )
to be the set of order ideals of P , ordered by inclusion. See [31, Thm. 3.4.1].)
(12) [2] Use the finite field method to prove that
xi ± xj = 0, 1, 1≤i<j≤n
2xi = 0, 1, 1 ≤ i ≤ n,
called the Shi arrangement of type B. Find the characteristic polynomial and
number of regions of SB
n . Is there a “nice” bijective proof of the formula for the
number of regions?
LECTURE 5. FINITE FIELDS 471
(15) [5–] Let 1 ≤ k ≤ n. Find the number of regions (or more generally the charac-
teristic polynomial) of the arrangement (in Rn )
1, 1 ≤ i ≤ k
xi − xj =
2, k + 1 ≤ i ≤ n,
for all i
= j. Thus we are counting interval orders on [n] where the elements
1, 2, . . . , k correspond to intervals of length one, while k + 1, . . . , n correspond
to intervals of length two. Is it possible to count such interval orders up to
isomorphism (i.e., the unlabelled case)? What if the length 2 is replaced instead
by a generic length a?
(16) [2+] A double semiorder on [n] consists of two binary relations < and on [n]
that arise from a set x1 , . . . , xn of real numbers as follows:
i<j if xi < xj − 1
ij if xi < xj − 2.
If we associate the interval Ii = [xi − 2, xi ] with the point xi , then we are
specifying whether Ii lies to the left of the midpoint of Ij , entirely to the left of
Ij , or neither. It should be clear what is meant for two double semiorders to be
isomorphic.
(a) [2] Draw interval diagrams of the 12 nonisomorphic double semiorders on
{1, 2, 3}.
(b) [2] Let ρ2 (n) denote the number of double semiorders on [n]. Find an
(2) (2)
arrangement In satisfying r(In ) = ρ2 (n).
(c) [2+] Show that 3nthe
number of nonisomorphic double semiorders on [n] is
1
given by 2n+1 n
.
1
3n n
(d) [2–] Let F (x) = n≥0 2n+1 n x . Show that
xn
ρ2 (n) = F (1 − e−x ).
n!
n≥0
(e) [2] Generalize to “k-semiorders,” where ordinary semiorders (or unit interval
orders) correspond to k = 1 and double semiorders to k = 2.
(17) [1+] Show that intervals of lengths 1, 1.0001, 1.001, 1.01, 1.1 cannot form an in-
terval order isomorphic to 4 + 1, but that such an interval order can be formed
if the lengths are 1, 10, 100, 1000, 10000.
(18) [5–] What more can be said about interval orders with generic interval lengths?
For instance, consider the two cases: (a) interval lengths very near each other
(e.g., 1, 1.001, 1.01, 1.1), and (b) interval lengths superincreasing (e.g., 1, 10, 100,
1000). Are there finitely many obstructions to being such an interval order? Can
the number of unlabelled interval orders of each type be determined? (Perhaps
the numbers are the same, but this seems unlikely.)
(19) (a) [3] Let Ln denote the Linial arrangement, say in Rn . Show that
n
t n
χLn (t) = n (t − k)n−1 .
2 k
k=1
1 3 2 4 2 4 1 3
1 4 2 3 3 4 1 2
2 3 1 4
1 2
4
2 3
1 3
Figure 10. The seven alternating trees on the vertex set [4]
(20) (a) [3–] An alternating tree on the vertex set [n] is a tree on [n] such that
every vertex is either less than all its neighbors or greater than all its neigh-
bors. Figure 10 shows the seven alternating trees on [4]. Deduce from
Exercise 19(a) that r(Ln ) is equal to the number of alternating trees on
[n + 1].
(b) [5] Find a bijective proof of (a), i.e., give an explicit bijection between the
regions of Ln and the alternating trees on [n + 1].
(21) [3–] Let
χLn (t) = an tn − an−1 tn−1 + · · · + (−1)n−1 a1 t.
Deduce from Exercise 19(a) that ai is the number of alternating trees on the
vertex set 0, 1, . . . , n such that vertex 0 has degree (number of adjacent vertices)
i.
(22) (a) [2+] Let P (t) ∈ C[t] have the property that every (complex) zero of P (t)
has real part a. Let z ∈ C satisfy |z| = 1. Show that every zero of the
polynomial P (t − 1) + zP (t) has real part a + 12 .
(b) [2+] Deduce from (a) and Exercise 19(a) that every zero of the polynomial
χLn (t)/t has real part n/2. This result is known as the “Riemann hypothesis
for the Linial arrangement.”
(23) (a) [2–] Compute limn→∞ b(Sn )/r(Sn ), where Sn denotes the Shi arrangement.
(b) [3] Do the same for the Linial arrangement Ln .
(24) [2+] Let Ln denote the Linial arrangement in Rn . Fix an integer r
= 0, ±1, and
let Mn (r) be the arrangement in Rn defined by xi = rxj , 1 ≤ i < j ≤ n, together
with the coordinate hyperplanes xi = 0. Find a relationship between χLn (t) and
χMn (r) (t) without explicitly computing these characteristic polynomials.
(25) (a) [3–] A threshold graph on [n] may be defined recursively as follows: (i) the
empty graph ∅ is a threshold graph, (ii) if G is a threshold graph, then so is
the disjoint union of G and a single vertex, and (iii) if G is a threshold graph,
then so is the graph obtained by adding a new vertex v and connecting it
to every vertex of G. Let Tn denote the threshold arrangement. Show that
r(Tn ) is the number of threshold graphs on [n].
LECTURE 5. FINITE FIELDS 473
the generating function for rooted labelled trees on n vertices. Show that
1/4
xn 1 1 + R(x)
r(An ) = eT (x)− 2 R(x)
n! 1 − R(x)
n≥0
x2 x3 x4 x5 x6
= 1+x+2 + 8 + 54 + 533 + 6934 + · · · .
2! 3! 4! 5! 6!
(29) [2+] Fix n ≥ 1. Let f (k, n, r) be the number of k × n (0, 1)-matrices A over
the rationals such that all rows of A are distinct, every row has at least one 1,
and rank(A) = r. Let gn (q) be the number of n-tuples (a1 , . . . , an ) ∈ Fnq such
that no nonempty subset of the entries sums to 0 (in Fq ). Show that for p 0,
where q = pd , we have
(−1)k
gn (q) = f (k, n, r)q n−r .
k!
k,r
474 R. STANLEY, HYPERPLANE ARRANGEMENTS
(The case k = 0 is included, corresponding to the empty matrix, which has rank
0.)
LECTURE 6
Separating Hyperplanes
We simply write DA (t) if no confusion will result. Also define the weak order (with
respect to R0 ) of A to be the partial order WA on R(A) given by
R ≤ R if sep(R0 , R) ⊆ sep(R0 , R ).
It is easy to see that WA is a partial ordering of R(A). The poset WA is graded
by distance from R0 , i.e., R0 is the 0̂ element of R(A), and all saturated chains
between R0 and R have length d(R0 , R).
Figure 1 shows three arrangements in R2 , with R0 labelled 0 and then each
R
= R0 labelled d(R0 , R). Under each arrangement is shown the corresponding
weak order WA . The first arrangement is the braid arrangement B3 (essentialized).
Here the choice of base region does not affect the distance enumerator 1 + 2t + 2t2 +
t3 = (1 + t)(1 + t + t2 ) nor the weak order. On the other hand, the second two
475
476 R. STANLEY, HYPERPLANE ARRANGEMENTS
1 0 1 1 2
2 0
1 2 0 1
3 1
2 2 3 1 2
arrangements of Figure 1 are identical, but the choice of R0 leads to different weak
orders and different distance enumerators, viz., 1 + 2t + 2t2 + t3 and 1 + 3t + 2t2 .
Consider now the braid arrangement Bn . We know from Example 1.3 that the
regions of Bn are in one-to-one correspondence with the permutations of [n], viz.,
R(Bn ) ↔ Sn
xw(1) > xw(2) > · · · > xw(n) ↔ w.
Given w = a1 a2 · · · an ∈ Sn , define an inversion of w to be a pair (i, j) such that
i < j and ai > aj . Let (w) denote the number of inversions of w. The inversion
sequence IS(w) of w is the vector (c1 , · · · , cn ), where
cj = #{i : i < j, w−1 (j) < w−1 (i)}.
Note that the condition w−1 (j) < w−1 (i) is equivalent to i appearing to the right
of j in w. For instance, IS(461352) = (0, 0, 1, 3, 1, 4). The inversion sequence is a
modified form of the inversion table or of the code of w, as defined in the literature,
e.g., [31, p. 21][32, solution to Exer. 6.19(x)]. For our purposes the inversion
sequence is the most convenient. It is clear from the definition of IS(w) that if
IS(w) = (c1 , . . . , cn ) then (w) = c1 + · · · + cn . Moreover, is easy to see (Exercise 2)
that a sequence (c1 , . . . , cn ) ∈ Nn is the inversion sequence of a permutation w ∈ Sn
if and only if ci ≤ i − 1 for 1 ≤ i ≤ n. It follows that
t(w) = tc1 +···+cn
w∈Sn (c1 ,...,cn )
0≤ci ≤i−1
& ' & n−1 '
0
= t c1
··· t cn
c1 =0 cn =0
and Rv are separated by a single hyperplane H, and R0 and Ru lie on the same
side of H. Suppose that H is given by xi = xj with i < j. Then i and j appear
consecutively in u written as a word a1 · · · an (since H is a bounding hyperplane of
the region Ru ) and i appears to the left of j (since R0 and Ru lie on the same side
of H). Thus v is obtained from u by transposing the adjacent pair ij of letters. It
follows that (v) = (u) + 1. If u(k) = i and we let sk = (k, k + 1), the adjacent
transposition interchanging k and k + 1, then v = usk .
The following result is an immediate consequence of equation (49) and mathe-
matical induction.
Proposition 6.18. Let R0 = Rid as above. If w ∈ Sn then d(R0 , Rw ) = (w).
Moreover,
DBn (t) = (1 + t)(1 + t + t2 ) · · · (1 + t + · · · + tn−1 ).
There is a somewhat different approach to Proposition 6.18 which will be gen-
eralized to the Shi arrangement. We label each region R of Bn recursively by a
vector λ(R) = (c1 , . . . , cn ) ∈ Nn as follows.
• λ(R0 ) = (0, 0, . . . , 0)
• Let ei denote the ith unit coordinate vector in Rn . If the regions R and R
of Bn are separated by the single hyperplane H with the equation xi = xj ,
i < j, and if R and R0 lie on the same side of H, then λ(R ) = λ(R) + ej .
Figure 2 shows the labels λ(R) for B3 .
x1 = x3 x2 = x3
001
002 000
x1 = x 2
012 010
011
counterclockwise until encountering an empty space, in which case the car parks
there. Note the following:
• All the cars can always park, since they drive in a circle and will always
find an empty space.
• After all cars have parked there will be one empty space.
• The sequence (a1 , . . . , an ) is a parking function if and only if the empty
space after all the cars have parked is n + 1.
• If the preference sequence (a1 , . . . , an ) produces the empty space i at the
end, then the sequence (a1 + k, . . . , an + k) (taking entries modulo n + 1 so
they always lie in the set [n + 1]) produces the empty space i + k (modulo
n + 1).
It follows that exactly one of the sequences (a1 + k, . . . , an + k) (modulo n + 1),
where 1 ≤ k ≤ n+1, is a parking function. There are (n+1)n sequences (a1 , . . . , an )
in all, so exactly (n + 1)n /(n + 1) = (n + 1)n−1 are parking functions.
Many readers will have recognized that the number (n + 1)n−1 is closely related
to the enumeration of trees. Indeed, there is an intimate connection between trees
and parking functions. We therefore now present some background material on
trees. A tree on [n] is a connected graph without cycles on the vertex set [n]. A
rooted tree is a pair (T, i), where T is a tree and i is a vertex of T , called the root.
We draw trees in the standard computer science manner with the root at the top
and all edges emanating downwards. A forest on [n] is a graph F on the vertex set
[n] for which every (connected) component is a tree. Equivalently, F has no cycles.
A rooted forest (also called a planted forest ) is a forest for which every component
has a root, i.e., for each tree T of the forest select a vertex iT of T to be the root of
T . A standard result in enumerative combinatorics (e.g., [32, Prop. 5.3.2]) states
that the number of rooted forests on [n] is (n + 1)n−1 .
An inversion of a rooted forest F on [n] is a pair (i, j) of vertices such that i < j
and j appears on the (unique) path from i to the root of the tree in which i occurs.
Write inv(F ) for the number of inversions of F . For instance, the rooted forest F
of Figure 3 has the inversions (6, 7), (1, 7), (5, 7), (1, 5), and (2, 4), so inv(F ) = 5.
7 8 3 4
5
6 10 2 11
9 1 12
Figure 3. A rooted forest on [12]
1 2 3 1 3 1 2 2 3 2 1 3 2 3 1 1
2 3 1 3 1 2 2 3
0 0 0 1 0 1 1 0
2 3
1 1 2 2 3 3
1 3 1 2 2 3 1 3 1 2
3 2 3 1 2 1
1 2 0 1 1 2 2 3
We collect below the three main results on In (t). They are theorems in “pure”
enumeration and have no direct connection with arrangements. The first result,
due to Mallows and Riordan [23], gives a remarkable connection with connected
graphs.
Theorem 6.21. We have
In (1 + t) = te(G)−n ,
G
where G ranges over all connected (simple) graphs on the vertex set
[0, n] = {0, 1, . . . , n} and e(G) denotes the number of edges of G.
For instance,
I3 (1 + t) = 16 + 15t + 6t2 + t3 .
Thus, for instance, there are 15 connected graphs on [0, 3] with four edges. Three of
these are 4-cycles and twelve consist of a triangle with an incident edge. The enu-
meration of connected graphs is well-understood [32, Exam. 5.2.1]. In particular,
if
Cn (t) = te(G) ,
G
where G ranges over all connected (simple) graphs on [n], then
xn n x
n
(50) Cn (t) = log (1 + t)( 2 ) .
n! n!
n≥0 n≥1
Thus Theorem 6.21 “determines” In (t). There is an alternative way to state this
result that doesn’t involve the logarithm function.
Corollary 6.13. We have
n
t(
n+1
2 )x
xn n≥0
n!
(51) In (t)(t − 1)n = n n
n! x
n≥0 t( 2 )
n!
n≥0
LECTURE 6. SEPARATING HYPERPLANES 481
The third result, due to Kreweras [22], connects inversion enumerators with
parking functions. Let PFn denote the set of parking function of length n.
Theorem 6.22. Let n ≥ 1. Then
n
(52) t( 2 ) In (1/t) = ta1 +···+an −n .
(a1 ,...,an )∈PFn
We now give proofs of Theorem 6.21, Corollary 6.13, and Theorem 6.22.
Proof of Theorem 6.21 (sketch). The following elegant proof is due to Gessel
and Wang [18]. Let G be a connected graph on [0, n]. Start at vertex 0 and let
T be the “depth-first spanning tree,” i.e., move to the largest unvisited neighbor
or else (if there is no unvisited neighbor) backtrack. The edges traversed when all
vertices are visited are the edges of the spanning tree T . Remove the vertex 0 and
root the trees that remain at the neighbors of 0. Denote this rooted forest by FG .
0 4 6
1 0
3 5 5 1 5 1
2 3 4 3 4
G 2 6 2 6
T F
Given a spanning forest F on [n], what connected graphs G on [0, n] satisfy
F = FG ? The answer, whose straightforward verification we leave to the reader, is
the following. Add the vertex 0 to F and connect it the roots of F , obtaining T .
Clearly G consists of T with some added edges ij. The edge ij can be added to T
if and only if the path from 0 to j contains i (or vice versa), and if i is the next
vertex after i on the path from i to j, then (j, i ) is an inversion of F . Thus each
inversion of F corresponds to a possible edge that can be added to T , and these
edges can be added or not added independently. It follows that
te(G) = te(T ) (1 + t)inv(F )
G : F =FG
= tn (1 + t)inv(F ) .
Summing on all rooted forests F on [n] gives
te(G) = tn (1 + t)inv(F )
G F
= tn In (1 + t),
where G ranges over all connected graphs on [0, n].
Proof of Corollary 6.13. By equation (50) and Theorem 6.21 we have
xn n x
n
tn−1 In−1 (1 + t) = log (1 + t)( 2 ) .
n! n!
n≥0 n≥0
Claim #1:
n
n
(53) Jn+1 (t) = (1 + t + t2 + · · · + ti )Ji (t)Jn−i (t).
i=0
i
Proof of claim. We give a proof due to G. Kreweras [22]. Let F be a rooted forest
on S ⊆ [n], #S = i, and let G be a rooted forest on S̄ = [n] − S. Let u1 < · · · < ui
be the vertices of F , and set ui+1 = n + 1. Choose 1 ≤ j ≤ i + 1. For all m ≥ j
replace um by um+1 . (If j = i + 1, then do nothing.) This gives a labelled forest F
on (S ∪ {n + 1}) − {uj }. Let T be the labelled tree obtained from F by adjoining
the root uj and connecting it to the roots of F . Keep G the same. We obtain a
rooted forest H on [n + 1] satisfying
inv(H) = j − 1 + inv(F ) + inv(G).
LECTURE 6. SEPARATING HYPERPLANES 483
7 11 10 7 10 4
j=4
4
8 3 8 12
5 1 6 9 1 6 9
2 5 11 3
F G 2 H
This process gives a bijection (S, F, G, j) → H, where S ⊆ [n], F is a rooted
forest on S, G is a rooted forest on S̄, 1 ≤ j ≤ 1 + #S, and H is a rooted forest on
[n + 1]. Hence
n
Ii (t)In−i (t)(1 + t + · · · + ti ) = In+1 (t),
i=0 S⊆[n]
#S=i
x2 = x 3
x2 = x3 + 1
201
101 200
102 210
x1 = x 2 + 1
001 100
002 110
000
x1 = x 2
010
012 120
011 020
021
x1 = x3 + 1
x1 = x3
521769348
LECTURE 6. SEPARATING HYPERPLANES 485
48
348
6348
16348
169348
1769348
21769348
521769348
Figure 6. Constructing a valid pair (w, I) from the parking function α = (2, 3, 0, 0, 7, 2, 3, 0, 3)
(5) Insert 1. Using the lemma we obtain 16348. Since α1 = 2, there is an arc
over 1 and two other elements to the right to 1. This gives the fifth row
of Figure 6.
(6) Insert 9. Placing 9 before 1 or 6 yields α9 ≥ 4, contradicting α9 = 3.
Placing 9 after 3,4, or 8 is excluded by the lemma. Hence we get the sixth
row of Figure 6.
(7) Insert 7. Placing 7 at the beginning yields four terms j < 7 appearing to
the right of 7, giving α7 ≥ 4, a contradiction. Placing 7 after 6,9,3,4,8 will
violate the lemma, so we get the partial permutation 1769348. In order
that α7 = 3, we must have 7 and 8 appearing under the same arc. Hence
the arc from 6 to 8 must be extended to 7, yielding row seven of Figure 6.
(8) Insert 2 and 5. By now we hope it is clear that there is always a unique
way to proceed.
The uniqueness of the above procedure shows that the map from the regions
R of Sn (or the valid pairs (w, I) that index the regions) to parking functions α
is injective. Since the number of valid pairs and number of parking functions are
both (n + 1)n−1 , the map is bijective, completing the (sketched) proof. In fact,
it’s not hard to show surjectivity directly, i.e., that the above procedure produces
a valid pair (w, I) for any parking function, circumventing the need to know that
r(Sn ) = #PFn in advance.
Corollary 6.14. The distance enumerator of Sn is given by
(57) DSn (t) = ta1 +···+an −n .
(a1 ,...,an )∈PFn
2
1 3
H
6 4
5
no!
Since rank(A) > rank(A0 ), we have #π −1 (R ) > 1 (for H ∈ A does not bisect
R if and only if rank(A0 ∪ H) = rank(A0 )). Thus there are two distinct regions
R1 , R2 ∈ π −1 (R ) that are endpoints of the “chain of regions.”
Let ed have the meaning of equation (34), i.e.,
ed = #{H ∈ A : H
∈ A0 } = #A1 .
Then π (R ) is a chain of regions of length ed , so #π −1 (R ) = 1 + ed . We now
−1
come to the key definition of this section. The definition is recursive by rank, the
base case being rank at most 2.
Definition 6.16. Let A be a real supersolvable central arrangement of rank d, and
let A0 be a supersolvable subarrangement of rank d − 1 (which always exists by the
definition of supersolvability). A region R0 ∈ R(A) is called canonical if either (1)
d ≤ 2, or else (2) d ≥ 3, π(R0 ) ∈ R(A0 ) is canonical, and R0 is an endpoint of the
chain F(R0 ).
Since every chain has two endpoints and a central arrangement of rank 1 has
two (canonical) regions, it follows that there are at least 2d canonical regions.
The main result on distance enumerators of supersolvable arrangements is the
following, due to Björner, Edelman, and Ziegler [8, Thm. 6.11].
Theorem 6.24. Let A be a supersolvable central arrangement of rank d in Rn . Let
R0 ∈ R(A) be canonical, and suppose that
χA (t) = (t − e1 )(t − e2 ) · · · (t − ed )tn−d .
(There always exist such positive integers ei by Corollary 4.9.) Then
d
DA,R0 (t) = (1 + t + t2 + · · · + tei ).
i=1
Since R0 is canonical, for all R ∈ R(A0 ) we have that π −1 (R ) is a chain of length
ed . Hence if R ∈ R(A) and h(R) denotes the rank of R in the chain F(R), then
dA (R0 , R) = dA0 (π(R)) + h(R).
Therefore
DA (t) = DA0 (t)(1 + t + · · · + ted ),
and the proof follows by induction.
Note. The following two results were also proved in [8]. We simply state them
here without proof.
• If A is a real supersolvable central arrangement and R0 is canonical, then
WA is a lattice (Exercise 7).
• If A is any real central arrangement and WA is a lattice, then R0 is
simplicial (bounded by exactly rk(A) hyperplanes, the minimum possible).
In other words, the closure R̄0 is a simplex. As a partial converse, if every
region R is simplicial, then WA is a lattice (Exercise 8).
1 2
1 2 3
3
4
5 7
6
Then
1 2 3 4 5 6 7
1 1 a1 a1 a2 a1 a3 a3 a2 a3 a1 a2 a3
2 a1 1 a2 a3 a1 a3 a1 a2 a3 a2 a3
3 a1 a2 a2 1 a2 a3 a1 a2 a3 a1 a3 a3
V =
4 a1 a3 a3 a2 a3 1 a1 a1 a2 a2
5 a3 a1 a3 a1 a2 a3 a1 1 a2 a1 a2
6 a2 a3 aa a2 a3 a1 a3 a1 a2 a2 1 a1
7 a1 a2 a3 a1 a3 a3 a2 a1 a2 a1 1
The determinant of this matrix happens to be given by
3 3 3
det(V ) = 1 − a21 1 − a22 1 − a23 .
LECTURE 6. SEPARATING HYPERPLANES 491
Proof. Omitted.
Exercises
(1) Let A be a central arrangement in Rn with distance enumerator DA (t) (with
respect to some base region R0 ). Define a graph GA on the vertex set R(A)
by putting an edge between R and R if #sep(R, R ) = 1 (i.e., R and R are
separated by a unique hyperplane).
(a) [2–] Show that GA is a bipartite graph.
(b) [2] Show that if #A is odd, then DA (−1) = 0.
(c) [2] Show that if #A is even and r(A) ≡ 2 (mod 4), then DA (−1) ≡ 2 (mod 4)
(so DA (−1)
= 0).
(d) [2] Give an example of (c), i.e., find A so that #A is even and r(A) ≡
2 (mod 4).
(e) [2] Show that (c) cannot hold if A is supersolvable. (It is not assumed that
the base region R0 is canonical. Try to avoid the use of Section 6.0.4.)
(f) [2+] Show that if #A is even and r(A) ≡ 0 (mod 4), then it is possible for
DA (−1) = 0 and for DA (−1)
= 0. Can examples be found for rank(A) ≤ 3?
(2) [2–] Show that a sequence (c1 , . . . , cn ) ∈ Nn is the inversion sequence of a per-
mutation w ∈ Sn if and only if ci ≤ i − 1 for 1 ≤ i ≤ n.
492 R. STANLEY, HYPERPLANE ARRANGEMENTS
(3) [2] Show that all cars can park under the scenario following Definition 6.15 if
and only if the sequence (a1 , . . . , an ) of preferred parking spaces is a parking
function.
(4) [5] Find a bijective proof of Theorem 6.22, i.e., find a bijection ϕ between the
and the set PFn of all parking functions of length
set of all rooted forestson [n]
n satisfying inv(F ) = n+1 2 − a1 − · · · − an when ϕ(F ) = (a1 , . . . , an ). Note.
In principle a bijection ϕ can be obtained by carefully analyzing the proof of
Theorem 6.22. However, this bijection will be of a messy recursive nature. A
“nonrecursive” bijection would be greatly preferred.
(5) [3] There is a natural two-variable refinement of the distance enumerator (57)
of Sn . Given R ∈ R(Sn ), define d0 (R0 , R) to be the number of hyperplanes
xi = xj separating R0 from R, and d1 (R0 , R) to be the number of hyperplanes
xi = xj + 1 separating R0 from R. (Here R0 is given by (55) as usual.) Set
Dn (q, t) = q d0 (R0 ,R) td1 (R0 ,R) .
R∈R(Sn )
What can be said about the polynomial Dn (q, t)? Can its coefficients be inter-
preted in a simple way in terms of tree or forest inversions? Are there formulas
or recurrences for Dn (q, t) generalizing Theorem 6.21, Corollary 6.13, or equa-
tion (53)? The table below give the coefficients of q i tj in Dn (q, t) for 2 ≤ n ≤ 4.
t\
q
0 1 2 3 4 5 6
0 1 1 2 3 3 3 1
t\
q
0 1 2 3
1 3 3 6 7 6 3
t\
q
0 1 0 1 1 2 1
2 5 5 8 9 5
0 1 1 1 2 2 2
3 6 7 9 6
1 1 2 2 2
4 5 6 5
3 1
5 3 3
6 1
(6) [5–] Let Gn denote the generic braid arrangement
xi − xj = aij , 1 ≤ i < j ≤ n,
in R . Can anything interesting be said about the distance enumerator DGn (t)
n
(which depends on the choice of base region R0 and possibly on the aij ’s)? Gen-
eralize if possible to generic graphical arrangments, especially for supersolvable
(or chordal) graphs.
(7) [3–] Let A be a real supersolvable arrangement and R0 a canonical region of A.
Show that the weak order WA (with respect to R0 ) is a lattice.
(8) (a) [2+] let A be a real central arrangement of rank d. Suppose that the weak
order WA (with respect to some region R0 ∈ R(A)) is a lattice. Show that
R0 is simplicial, i.e., bounded by exactly d hyperplanes.
(b) [3–] Let A be a real central arrangement. Show that if every region R ∈
R(A) is simplicial, then WA is a lattice.
(9) (a) [2] Set each aH = q in the Varchenko matrix V of an arrangement R in Rn ,
obtaining a matrix V (q). Let r = r(A). The entries of V (q) belong to the
principal ideal domain Q[q], so V (q) has a Smith normal form AV (q)B =
diag(p1 , . . . , pr ), where A, B are r × r matrices whose entries belong to Q[q]
and whose determinants are nonzero elements of Q, and where p1 , . . . , pr ∈
LECTURE 6. SEPARATING HYPERPLANES 493
Q[q] such that pi | pi+1 for 1 ≤ i ≤ r − 1. The Smith normal form is unique
up to multiplication of the pi ’s by nonzero elements of Q. For instance, if
A = B3 , then
AV (q)B = diag(1, q 2 − 1, q 2 − 1, q 2 − 1, (q 2 − 1)2 , (q 2 − 1)2 (q 4 + q 2 + 1)).
Show that each pi is a polynomial in q 2 .
(b) [3+] Let ai be the number of j’s for which (q 2 − 1)i | pj but (q 2 − 1)i+1 pj .
Show that
χA (t) = (−1)n−i ai q n−i .
i≥0
(c) [5] What more can be said about the polynomials pi ? By Theorem 6.25
they are products of cyclotomic polynomials, so one could begin by asking
for the largest powers of q 2 + 1 or q 4 + q 2 + 1 dividing each pi .
BIBLIOGRAPHY
Michelle L. Wachs
IAS/Park City Mathematics Series
Volume 14, 2004
Michelle L. Wachs
Introduction
The theory of poset topology evolved from the seminal 1964 paper of Gian-Carlo
Rota on the Möbius function of a partially ordered set. This theory provides a deep
and fundamental link between combinatorics and other branches of mathematics.
Early impetus for this theory came from diverse fields such as
• commutative algebra (Stanley’s 1975 proof of the upper bound conjecture)
• group theory (the work of Brown (1974) and Quillen (1978) on p-subgroup
posets)
• combinatorics (Björner’s 1980 paper on poset shellability)
• representation theory (Stanley’s 1982 paper on group actions on the ho-
mology of posets)
• topology (the Orlik-Solomon theory of hyperplane arrangements (1980))
• complexity theory (the 1984 paper of Kahn, Saks, and Sturtevant on the
evasiveness conjecture).
Later developments have kept the theory vital. I mention just a few examples:
Goresky-MacPherson formula for subspace arrangements, Björner-Lovász-Yao com-
plexity theory results, Björner-Wachs extension of shellability to nonpure com-
plexes, Forman’s discrete version of Morse theory, and Vassiliev’s work on knot
invariants and graph connectivity.
So, what is poset topology? By the topology of a partially ordered set (poset)
we mean the topology of a certain simplicial complex associated with the poset,
called the order complex of the poset. In these lectures I will present some of
the techniques that have been developed over the years to study the topology of a
poset, and discuss some of the applications of poset topology to the fields mentioned
above as well as to other fields. In particular, I will discuss tools for computing
homotopy type and (co)homology of posets, with an emphasis on group equivariant
(co)homology. Although posets and simplicial complexes can be viewed as essen-
tially the same topological object, we will narrow our focus, for the most part,
1 Departmentof Mathematics, University of Miami, Coral Gables, Fl 33124.
E-mail address: wachs@math.miami.edu.
This work was partially supported by NSF grant DMS 0302310.
c
2007 Michelle L. Wachs
499
500 WACHS, POSET TOPOLOGY
to tools that were developed specifically for posets; for example, lexicographical
shellability, recursive atom orderings, Whitney homology techniques, (co)homology
bases/generating set techniques, and fiber theorems.
Research in poset topology is very much driven by the study of concrete ex-
amples that arise in various contexts both inside and outside of combinatorics.
These examples often turn out to have a rich and interesting topological struc-
ture, whose analysis leads to the development of new techniques in poset topology.
These lecture notes are organized according to techniques rather than applications.
A recurring theme is the use of original examples in demonstrating a technique,
where by original example I mean the example that led to the development of the
technique in the first place. More recent examples will be discussed as well.
With regard to the choice of topics, I was primarily motivated by my own
research interests and the desire to provide the students at the PCMI graduate
school with concrete skills in this subject. Due to space and time constraints and
my decision to focus on techniques specific to posets, there are a number of very
important tools for general simplicial complexes that I have only been able to
mention in passing (or not at all). I point out, in particular, discrete Morse theory
(which is a major part of the lecture series of Robin Forman, its originator) and
basic techniques from algebraic topology such as long exact sequences and spectral
sequences. For further techniques and applications, still of current interest, we
strongly recommend the influential 1995 book chapter of Anders Björner [29].
The exercises vary in difficulty and are there to reinforce and supplement the
material treated in these notes. There are many open problems (simply referred to
as problems) and conjectures sprinkled throughout the text.
I would like to thank the organizers (Ezra Miller, Vic Reiner and Bernd Sturm-
fels) of the 2004 PCMI Graduate Summer School for inviting me to deliver these
lectures. I am very grateful to Vic Reiner for his encouragement and support. I
would also like to thank Tricia Hersh for the help and support she provided as my
overqualified teaching assistant. Finally, I would like to express my gratitude to
the graduate students at the summer school for their interest and inspiration.
LECTURE 1
Basic Definitions, Results, and Examples
3 2
1 3
2 5 6
4
5
6
1 4
P Δ (P)
^
1
123 123
2
12 13 23 34 12 13 23 34
1 3
1 2 3 4 1 2 3 4
4 ^
0
Δ P(Δ ) L(Δ )
drop the and let Δ denote an abstract simplicial complex as well as its geometric
realization.
To every poset P , one can associate an abstract simplicial complex Δ(P ) called
the order complex of P . The vertices of Δ(P ) are the elements of P and the faces
of Δ(P ) are the chains (i.e., totally ordered subsets) of P . (The order complex
of the empty poset is the empty simplicial complex {∅}.) For example, the Hasse
diagram of a poset P and the geometric realization of its order complex are given
in Figure 1.1.1.
To every simplicial complex Δ, one can associate a poset P (Δ) called the face
poset of Δ, which is defined to be the poset of nonempty faces ordered by inclusion.
The face lattice L(Δ) is P (Δ) with a smallest element 0̂ and a largest element 1̂
attached. An example is given in Figure 1.1.2.
If we start with a simplicial complex Δ, take its face poset P (Δ), and then take
the order complex Δ(P (Δ)), we get a simplicial complex known as the barycentric
LECTURE 1. BASIC DEFINITIONS, RESULTS, AND EXAMPLES 503
2 2
123
12 23
123
12 13 23 34
1 3 1 3
13
34
1 2 3 4
4 4
Δ P(Δ ) Δ (P(Δ ))
2
12 13 23
12 23
1 2 3 1 3
13
B3 Δ (B3)
subdivision of Δ; see Figure 1.1.3. The geometric realizations are always homeo-
morphic,
Δ∼= Δ(P (Δ)).
When we attribute a topological property to a poset, we mean that the geomet-
ric realization of the order complex of the poset has that property. For instance, if
we say that the poset P is homeomorphic to the n-sphere Sn we mean that Δ(P )
is homeomorphic to Sn .
Example 1.1.1. The Boolean algebra. Let Bn denote the lattice of subsets of
[n] := {1, 2, . . . , n} ordered by containment, and let B̄n := Bn − {∅, [n]}. Then
B̄n ∼
= Sn−2
because Δ(B̄n ) is the barycentric subdivision of the boundary of the (n−1)-simplex.
See Figure 1.1.4.
We now review some basic poset terminology. An m-chain of a poset P is a
totally ordered subset c = {x1 < x2 < · · · < xm+1 } of P . We say the length l(c) of
c is m. We consider the empty chain to be a (−1)-chain. The length l(P ) of P is
defined to be
l(P ) := max{l(c) : c is a chain of P }.
Thus, l(P ) = dim Δ(P ) and l(P (Δ)) = dim Δ.
A chain of P is said to be maximal if it is inclusionwise maximal. Thus, the set
M(P ) of maximal chains of P is the set of facets of Δ(P ). A poset P is said to be
504 WACHS, POSET TOPOLOGY
pure (also known as ranked or graded) if all maximal chains have the same length.
Thus, P is pure if and only if Δ(P ) is pure. Also a simplicial complex Δ is pure
if and only if its face poset P (Δ) is pure. The posets and simplicial complexes of
Figures 1.1.1 and 1.1.2 are all nonpure, while the poset and simplicial complex of
Figure 1.1.4 are both pure.
For x ≤ y in P , let (x, y) denote the open interval {z ∈ P : x < z < y} and let
[x, y] denote the closed interval {z ∈ P : x ≤ z ≤ y}. Half open intervals (x, y] and
[x, y) are defined similarly.
If P has a unique minimum element, it is usual to denote it by 0̂ and refer
to it as the bottom element. Similarly, the unique maximum element, if it exists,
is denoted 1̂ and is referred to as the top element. Note that if P has a bottom
element 0̂ or top element 1̂ then Δ(P ) is contractible since it is a cone. We usually
remove the top and bottom elements and study the more interesting topology of
the remaining poset. Define the proper part of a poset P , for which |P | > 1, to be
P̄ := P − {0̂, 1̂}.
In the case that |P | = 1, it will be convenient to define Δ(P̄ ) to be the degenerate
empty complex ∅. We will also say Δ((x, y)) = ∅ and l((x, y)) = −2 if x = y.
For posets with a bottom element 0̂, the elements that cover 0̂ are called atoms.
For posets with a top element 1̂, the elements that are covered by 1̂ are called
coatoms.
A poset P is said to be bounded if it has a top element 1̂ and a bottom element
0̂. Given a poset P , we define the bounded extension
P̂ := P ∪ {0̂, 1̂},
where new elements 0̂ and 1̂ are adjoined (even if P already has a bottom or top
element).
A poset P is said to be a meet semilattice if every pair of elements x, y ∈ P
has a meet x ∧ y, i.e. an element less than or equal to both x and y that is greater
than all other such elements. A poset P is said to be a join semilattice if every
pair of elements x, y ∈ P has a join x ∨ y, i.e. a unique element greater than or
equal to both x and y that is less than all other such elements. If P is both a join
semilattice and a meet semilattice then P is said to be a lattice. It is a basic fact of
lattice theory that any finite meet (join) semilattice with a top (bottom) element
is a lattice.
The dual of a poset P is the poset P ∗ on the same underlying set with the
order relation reversed. Topologically there is no difference between a poset and its
dual since Δ(P ) and Δ(P ∗ ) are identical simplicial complexes. The direct product
P × Q of two posets P and Q is the poset whose underlying set is the cartesian
product {(p, q) : p ∈ P, q ∈ Q} and whose order relation is given by
(p1 , q1 ) ≤P ×Q (p2 , q2 ) if p1 ≤P p2 and q1 ≤Q q2 .
Define the join of two simplicial complexes Δ and Γ on disjoint vertex sets to
be the simplicial complex given by
(1.1.1) Δ ∗ Γ := {A ∪ B : A ∈ Δ, B ∈ Γ}.
The join (or ordinal sum) P ∗ Q of posets P and Q is the poset whose underlying
set is the disjoint union of P and Q and whose order relation is given by x < y if
LECTURE 1. BASIC DEFINITIONS, RESULTS, AND EXAMPLES 505
-2
1 1 2 1
-1 -1 -1 -1
where μ(·) is the classical Möbius function of number theory, which is defined on
the set of positive integers by
(−1)k if n is the product of k distinct primes
μ(n) =
0 if n is divisible by a square.
This example is the reason for the name Möbius function of a poset.
The combinatorial significance of the Möbius function was first demonstrated
by Rota in 1964 in his Steele-prize winning paper [149]. The Möbius function of a
poset is used in enumerative combinatorics to obtain inversion formulas.
Proposition 1.2.5 (Möbius inversion). Let P be a poset and let f, g : P → C.
Then
g(y) = f (x)
x≤y
if and only if
f (y) = μ(x, y) g(x).
x≤y
where β̃i (Δ) is the ith reduced Betti number of Δ, i.e., the rank, as an abelian
group, of the ith reduced homology of Δ over Z.
The Möbius function of a poset plays a fundamental role in the theory of
hyperplane arrangements and the homology of a poset plays a fundamental role in
the theory of subspace arrangements. We discuss the connection with arrangements
in the next section.
LECTURE 1. BASIC DEFINITIONS, RESULTS, AND EXAMPLES 507
3 4 5 8 6 7
2 6
1 5
7 1 2 3 4
8
^
0
A L (A)
x 2=x 3 123
x 1=x 3
x 1=x 2
1/2/3
A2 L(A2)=Π3
x1 = 0 012
x1 = -x 2 x1 = x2
0/1/2
B
B2 L(B2)=Π
2
025/17̄9/346̄8̄.
It maps to the subspace
{x ∈ R9 : x2 = x5 = 0, x1 = −x7 = x9 , x3 = x4 = −x6 = −x8 },
in L(Bn ). The type B braid arrangement B2 and the type B partition lattice ΠB
2
are shown in Figure 1.3.3.
510 WACHS, POSET TOPOLOGY
Examples 1.3.1 and 1.3.3 are referred to as type A examples, and Examples 1.3.2
and 1.3.4 are referred to as type B examples because of their connection with
Coxeter groups. Indeed, associated with every finite Coxeter group (i.e., finite
group generated by Euclidean reflections) is a simplicial complex called its Coxeter
complex. The order complex of the Boolean algebra Bn is the Coxeter complex
of the symmetric group Sn , which is the type A Coxeter group, and the order
complex of the face lattice of the cross-polytope Cn is the Coxeter complex of the
hyperoctahedral group, which is the type B Coxeter group. (The use of notation is
unfortunate here; Bn is type A and Cn is type B.)
Also associated with every finite Coxeter group is a hyperplane arrangement,
called its Coxeter arrangement, which consists of all its reflecting hyperplanes. The
group generated by the reflections about hyperplanes in the Coxeter arrangement is
the Coxeter group. The braid arrangement is the Coxeter arrangement of the sym-
metric group (type A Coxeter group) and type B braid arrangement is the Coxeter
arrangement of the hyperoctahedral group (type B Coxeter group). Coxeter groups
are discussed further in Section 3.3. See the chapters in this volume by Fomin and
Reading [71] and Stanley [176] for further discussion of Coxeter arrangements.
The types A and B partition lattices belong to another family of well-studied
lattices, namely the Dowling lattices. We will not define Dowling lattices, but we
will occasionally refer to them; see [83] for the definition. A broad class of Dowling
lattices arise as intersection lattices of complex hyperplane arrangements Am,n
consisting of hyperplanes of the forms zj = 0, where j = 1, . . . , n, and zj = ω h zi ,
2πi
where ω is the mth primitive root of unity e m , 1 ≤ i < j ≤ n, and h ∈ [m]. This
class includes the types A and B partition lattices.
We now state Zaslavsky’s seminal result.
Theorem 1.3.5 (Zaslavsky [215]). Suppose A is a hyperplane arrangement in
Rn . Let r(A) be the number of regions into which A divides Rn and let b(A) be the
number of these regions that are bounded. Then
(1.3.1) r(A) = |μ(0̂, x)|
x∈L(A)
and
(1.3.2) b(A) = |μ(L(A) ∪ 1̂)|.
The arrangement of Figure 1.3.1 has a total of 10 regions with 2 of them
bounded. One can use the values of the Möbius function given in Figure 1.2.1 to
confirm (1.3.1) and (1.3.2) for the arrangement of Figure 1.3.1. Note that if A is a
central arrangement, L(A) ∪ 1̂ has an artificial top element above the top element
of L(A). In other words L(A) ∪ 1̂ has exactly one coatom. It is easy to see that
posets with only one coatom have Möbius invariant 0. Since central arrangements
clearly have no bounded regions, (1.3.2) is trivial for central arrangements.
Exercise 1.3.6. Suppose we have a hyperplane H of Rn which is generic with
respect to a central hyperplane arrangement A in Rn . This means that dim(H ∩
X) = dim(X) − 1 for all X ∈ L(A). Let AH = {H ∩ K : K ∈ A}. This is a
hyperplane arrangement induced in H ∼ = Rn−1 . Show that the number of bounded
regions of AH is independent of the choice of generic hyperplane H (see [42]).
The next major development in the combinatorial theory of hyperplane arrange-
ments is a 1980 formula of Orlik and Solomon [131], which can be viewed as a
LECTURE 1. BASIC DEFINITIONS, RESULTS, AND EXAMPLES 511
for all i.
There is a striking common generalization of the Zaslavsky formula (1.3.1) and
the Orlik-Solomon formula, obtained by Goresky and MacPherson in 1988, which
involves subspace arrangements. A real subspace arrangement is a finite collection
of (affine) subspaces in Rn . Real hyperplane arrangements and complex hyper-
plane arrangements are both examples of real subspace arrangements. Indeed,
hyperplanes in Cn can be viewed as codimension 2 subspaces of R2n . Again the
intersection semilattice L(A) is defined to be the semilattice of nonempty intersec-
tions of subspaces in the subspace arrangement A.
Theorem 1.3.8 (Goresky and MacPherson [80]). Let A be a subspace arrangement
in Rn . The reduced integral cohomology of the complement MA := Rn −∪A is given
by the group isomorphism
H̃ i (MA ; Z) ∼
= H̃n−dim x−2−i ((0̂, x); Z),
x∈L(A)\{0̂}
for all i.
To see that the Goresky-MacPherson formula reduces to the Zaslavsky formula
and to the Orlik-Solomon formula, one needs to understand the homology of the
intersection lattice of a central hyperplane arrangement. The intersection lattice
belongs to a well-understood class of lattices called geometric lattices. A fundamen-
tal result due to Folkman [70] states that the proper part of any geometric lattice
L has vanishing reduced homology in every dimension except the top dimension
(i.e. dimension equal to l(L) − 2). In fact, the homotopy type is that of a wedge of
spheres of top dimension. The intersection lattice of an affine hyperplane arrange-
ment belongs to a more general class of lattices called geometric semilattices, which
were introduced and studied by Wachs and Walker [205]. The proper part of a
geometric semilattice also has the homotopy type of a wedge of spheres of top di-
mension. Topology of geometric (semi)lattices is discussed further in Sections 3.2.3
and 4.2.
Exercise 1.3.9. Use Folkman’s result to show that the Goresky-MacPherson for-
mula reduces to both the Zaslavsky formula and the Orlik-Solomon formula.
The intersection lattice of a hyperplane arrangement determines more than the
additive group structure of the integral cohomology of the complement. Orlik and
512 WACHS, POSET TOPOLOGY
Solomon show that it determines the ring structure as well. Ziegler [218] showed
that, in general, for subspace arrangements the combinatorial data (intersection
lattice and dimension information) does not determine ring structure. However
in certain special cases the combinatorial data does determine the cohomology
algebra, see [69], [214], [58]. In Section 5.4 we discuss some stronger versions of
the Goresky-MacPherson formula, namely a homotopy version due to Ziegler and
Živaljević [222], and an equivariant version due to Sundaram and Welker [186].
For further reading on hyperplane arrangements, see the chapter by Stanley in
this volume [176] and the text by Orlik and Terao [132]. Further information on
subspace arrangements can be found in Björner [28] and Ziegler [216].
eg., [55], [213], [73]. This topic is discussed in greater depth in Forman’s chapter
of this volume [74]. Applications of graph complexes in knot theory and group the-
ory are discussed in Section 5.2. There are also connections between the topology
of graph complexes and commutative algebra, which are explored in the work of
Reiner and Roberts [140] and Dong [61]. A direct application of poset topology in
a different complexity theory problem is discussed in Section 3.2.4.
Representability questions in lattice theory deal with whether an arbitrary
lattice can be represented as a sublattice, subposet or interval in a given class of
lattices. We briefly discuss three examples that have connections to poset topology.
A result of Pudlák and Tuma [136] states that every lattice is isomorphic to a
sublattice of some partition lattice Πn . This implies that every lattice can be repre-
sented as the intersection lattice of a subspace arrangement embedded in the braid
arrangement. There is another representability result that is much easier to prove;
namely that every meet semilattice can be represented as the intersection semilat-
tice of some subspace arrangement, see [216]. From either of these representability
results, we see that, in contrast to the situation with hyperplane arrangements,
where the topology of the proper part of the intersection semilattice is rather spe-
cial (a wedge of spheres), any topology is possible for the intersection semilattice
of a general subspace arrangement. Indeed, given any simplicial complex Δ, there
is a linear subspace arrangement A such that L(A) is homeomorphic to Δ; namely
A is the linear subspace arrangement whose intersection lattice L(A) is isomorphic
to the face lattice L(Δ).
An open representability question is whether every lattice can be represented
as an interval in the lattice of subgroups of some group ordered by inclusion. An
approach to obtaining a negative answer to this question, proposed by Shareshian
[155], is to establish restrictions on the topology of intervals in the subgroup lattice.
Conjecture 1.4.2 (Shareshian [155]). Let G be a finite group. Then every open
interval in the lattice of subgroups of G has the homotopy type of a wedge of spheres.
This conjecture was shown to hold for solvable groups by Kratzer and Thévenaz
[114] (see Theorem 3.1.13 which strengthens the Kratzer-Thévenaz result). Further
discussion of connections between poset topology and group theory can be found
in Section 5.2
Our last example deals with the order dimension of a poset P , which is defined
to be the smallest integer n such that P can be represented as an induced subposet
of a product of n chains. Order dimension is an important and extensively studied
poset invariant, see [191]. Reiner and Welker give a lower bound on order dimension
of a lattice in terms of its homology.
Theorem 1.4.3 (Reiner and Welker [143]). Let L be a lattice and let d be the
largest dimension for which the reduced integral simplicial homology of the proper
part of L is nonvanishing. Then the order dimension of L is at least d + 2.
the poset. For each poset P and integer j, define the chain space
Cj (P ; k) := k-module freely generated by j-chains of P,
where k is a field or the ring of integers.
The boundary map ∂j : Cj (P ; k) → Cj−1 (P ; k) is defined by
j+1
∂j (x1 < · · · < xj+1 ) = (−1)i (x1 < · · · < x̂i < · · · < xj+1 ),
i=1
where the ˆ· denotes deletion. We have that ∂j−1 ∂j = 0, which makes (Cj (P ; k), ∂j )
an algebraic complex. Define the cycle space Zj (P ; k) := ker ∂j and the boundary
space Bj (P ; k) := im∂j+1 . Homology of the poset P in dimension j is defined by
H̃j (P ; k) := Zj (P ; k)/Bj (P ; k).
The coboundary map δj : Cj (P ; k) → Cj+1 (P ; k) is defined by
(1.5.1) δj (α), β = α, ∂j+1 (β)
where α ∈ Cj (P ; k), β ∈ Cj+1 (P ; k), and ·, · is the bilinear form on ⊕j≥−1 Cj (P ; k)
for which the chains of P form an orthonormal basis. This is equivalent to saying
δj (x1 < · · · < xj ) =
j+1
(−1)i (x1 < · · · < xi−1 < x < xi < · · · < xj ),
i=1 x∈(xi−1 ,xi )
for all chains x1 < · · · < xj , where x0 is the bottom element of P̂ and xj+1
is the top element of P̂ . Define the cocycle space to be Z j (P ; k) := ker δj and
the coboundary space to be B j (P ; k) := imδj−1 . Cohomology of the poset P in
dimension j is defined to be
H̃ j (P ; k) := Z j (P ; k)/B j (P ; k).
When k is a field, H̃ j (P ; k) and H̃j (P ; k) are isomorphic vector spaces. The
jth (reduced) Betti number of P is given by
β̃j (P ) := dim H̃j (P ; C),
which is the same as the rank of the free part of H̃j (P ; Z).
We will work primarily with homology over C and Z. For x < y in P , we write
H̃j (x, y) for the complex homology of the open interval (x, y) of P , and β̃j (x, y) for
the jth Betti number of the open interval (x, y). When x = y, define H̃j (x, y) to
be C and β̃j (x, y) to be 1 if j = −2, and to be 0 for all other j.
Many of the posets that arise have the homotopy type of a wedge of spheres.
We review a basic fact pertaining to wedges of spheres and a partial converse.
Theorem 1.5.1. Suppose Δ has the homotopy type of a wedge of spheres of various
dimensions, where ri is the number of spheres of dimension i. Then for each i =
0, 1, . . . , dim Δ,
(1.5.2) H̃i (Δ; Z) ∼
= H̃ i (Δ; Z) ∼
= Zri .
Theorem 1.5.2. If Δ is simply connected and has vanishing reduced integral ho-
mology in all dimensions but dimension n, where homology is free of rank r, then
Δ has the homotopy type of a wedge of r spheres of dimension n.
LECTURE 1. BASIC DEFINITIONS, RESULTS, AND EXAMPLES 515
x1 xm
a
c
The first tool that we mention for computing homology of posets and simplicial
complexes is a very efficient computer software package called “SimplicialHomol-
ogy”, developed by Dumas, Heckenbach, Sauders, and Welker [63]. One can run it
interactively or download the source file at the web site:
http://www.cis.udel.edu/∼dumas/Homology,
where a manual can also be found. This package has been responsible for many
of the more recent conjectures in the field. Its output was also part of the proofs
of (at least) three results on integral homology appearing in the literature; see
[156, 157, 202].
A∪B/C∪D A∪B∪C
A/B/C/ D A/B/C/ D
Figure 1.6.2.
3 5 1
2 4
Figure 1.6.3.
are merged into one block A ∪ B ∪ C. Type I intervals have 4 elements and type II
intervals have 5 elements.
The two types of intervals induce two types of cohomology relations, Type
I and Type II cohomology relations on maximal chains. It is convenient to use
binary trees on leaf set [n] to describe these relations. A maximal chain of Π̄n is
just a sequence of merges of pairs of blocks. The binary tree given in Figure 1.6.3
corresponds to the sequence of merges:
(1) merge blocks {3} and {5}
(2) merge blocks {2} and {4}
(3) merge blocks {2, 4} and {1}.
This corresponds to the maximal chain
1/2/35/4 <· 1/24/35 <· 124/35
of Π̄5 The internal nodes of the tree represent the merges, and the leaf sets of the
left and right subtrees of the internal nodes are the blocks that are merged. The
sequence of merges follows the postorder traversal of the internal nodes, i.e. first
traverse the left subtree in postorder, then the right subtree in postorder, then the
root.
Given a binary tree T on leaf set [n], let c(T ) be the maximal chain of Πn
obtained by the procedure described above. Although not all maximal chains can
be obtained in this way, it can be seen that every maximal chain is equal, modulo
the cohomology relations of Type I, to ±c(T ) for some T . So the set
{c(T ) : T is a binary tree on leaf set [n]}
LECTURE 1. BASIC DEFINITIONS, RESULTS, AND EXAMPLES 517
generates top cohomology H̃ n−3 (Π̄n ; k). The Type I cohomology relations induce
the following relations
(1.6.1) c(· · · (A ∧ B) · · · ) = (−1)|A||B| c(· · · (B ∧ A) · · · ),
where X ∧Y denotes the binary tree whose left subtree is X and whose right subtree
is Y , and |X| denotes the number of internal nodes of X. The Type II cohomology
relations induce the following relations
(1.6.2) c(· · · (A ∧ (B ∧ C)) · · · ) + (−1)|C| c(· · · ((A ∧ B) ∧ C) · · · )
+ (−1)|A||B| c(· · · (B ∧ (A ∧ C)) · · · ) = 0.
Exercise 1.6.1. Show that the Type I cohomology relations yield (1.6.1) and the
Type II cohomology relations yield (1.6.2).
The relations (1.6.1) and (1.6.2) resemble the relations satisfied by the bracket
operation of a Lie algebra. The Type I relation (1.6.1) corresponds to the anticom-
muting relation and the Type II relation (1.6.2) corresponds to the Jacobi relation.
Indeed there is a well-known connection between the top homology of the partition
lattice and the free Lie algebra which involves representations of the symmetric
group. In the next lecture, we discuss representation theory.
Theorem 1.6.2 (Stanley [169], Klyachko [110], Joyal [106]). The representation
of the symmetric group Sn on H̃n−3 (Π̄n ; C) is isomorphic to the representation of
Sn on the multilinear component of the free Lie algebra over C on n generators
tensored with the sign representation.
This result follows from a formula of Stanley for the representation of the sym-
metric group on homology of the partition lattice (Theorem 4.4.7) and an earlier
similar formula of Klyachko for the free Lie algebra. The first purely combinatorial
proof was obtained by Barcelo [14]. The presentation of top cohomology discussed
above appeared in an alternative combinatorial proof of Wachs [199]. It also ap-
peared in the proof of a superalgebra version of this result obtained by Hanlon and
Wachs [91]. A k-analog of the Lie superalgebra result was also obtained by Hanlon
and Wachs [91]. A type B version (Example 1.3.4) was obtained by Bergeron [16]
and a generalization to Dowling lattices was obtained by Gottlieb and Wachs [83].
518 WACHS, POSET TOPOLOGY
LECTURE 2
Group Actions on Posets
In this lecture we give a crash course on the representation theory of the sym-
metric group and then discuss some representations on homology that are induced
by symmetric group actions on posets. For further details on the representation
theory of the symmetric group and symmetric functions, we refer the reader to the
following excellent standard references [76, 126, 151, 174].
There are various reasons that we are interested in understanding how a group
acts on the homology of a poset. One is that this can be a useful tool in computing
the homology of the poset. Another is that interesting representations often arise.
We limit our discussion to the symmetric group, but point out there are often
interesting analogous results for other groups such as the hyperoctahedral group,
wreath product groups, and the general linear group.
8 2 4 1 1 2 4 7
7 5 3 6
9 3 5 8
6 9
Let Tλ be the set of Young tableaux of shape λ and let M λ be the complex
vector space generated by elements of Tλ . The symmetric group Sn acts on M λ by
permuting entries of the Young tableaux. That is, for transposition σ = (i, j) ∈ Sn ,
the tableau σT is obtained from T by switching entries i and j. For example,
8 2 4 1 8 3 4 1
7 5 7 5
(2, 3) = .
9 3 9 2
6 6
The representation that we have described is clearly the left regular representation
of Sn . One can also let Sn act as the right regular representation on M λ . That is
for transposition σ = (i, j) ∈ Sn and T ∈ Tλ , the tableau T σ is obtained from T
by switching the contents of the ith and jth cell under some fixed ordering of the
cells of λ.
We will say that two tableaux in Tλ are row-equivalent if they have the same
sequence of row sets. For example, the tableaux
8 2 4 1 1 2 4 8
7 5 5 7
and
9 3 3 9
6 6
522 WACHS, POSET TOPOLOGY
For example if
1 2 1 2 3 2 2 1 3 1
T = then eT = − + − .
3 3 1 3 2
Since the left and right action of Sn on Tλ commute, we have
(2.2.1) πeT = eπT ,
for all T ∈ Tλ and π ∈ Sn . We can now define the Specht module S λ to be the
subspace of M λ given by
S λ := eT : T ∈ Tλ .
It follows from (2.2.1) that S λ is an Sn -submodule of M λ (under the left action).
Theorem 2.2.1. The Specht modules S λ for all λ n form a complete set of
irreducible Sn -modules.
A polytabloid eT is said to be a standard polytabloid if T is a standard Young
tableau. We will see shortly that the standard polytabloids of shape λ form a basis
for the Specht module S λ .
Now we give the quotient characterization. The row relations are defined for
all T ∈ Tλ and σ ∈ Rλ by
(2.2.2) rσ (T ) := T σ − T.
For all i, j such that 1 ≤ j ≤ λi , let Ci,j (λ) be the set of cells in columns j
through λi of row i and in columns 1 through j of row i + 1. Let Gi,j (λ) be the
subgroup of Sn consisting of permutations σ that fix all entries of the cells that
are not in Ci,j (λ) under the right action of σ on tableaux of shape λ. The Garnir
relations are defined for all i, j such that 1 ≤ j ≤ λi and for all T ∈ Tλ by
(2.2.3) gi,j (T ) := T σ.
σ∈Gi,j (λ)
For example if
7 1 5 10
3 4 2
T = and (i, j) = (1, 2)
9 8
11 6
LECTURE 2. GROUP ACTIONS ON POSETS 523
then the entries 1, 5, 10, 3, 4 are permuted while the remaining entries are fixed. So
7 1 3 4 7 1 3 4 7 1 3 5 7 1 3 5
5 10 2 10 5 2 4 10 2 10 4 2
g1,2 (T ) = + + + + ...
9 8 9 8 9 8 9 8
11 6 11 6 11 6 11 6
Again, since the left and right action of Sn on Tλ commute, we have
πrσ (T ) = rσ (πT )
πgi,j (T ) = gi,j (πT ),
for all π ∈ Sn . Consequently, the subspace U λ of M λ generated by the row
relations (2.2.2) and the Garnir relations (2.2.3) is an Sn -submodule of M λ .
Theorem 2.2.2. For all λ n,
Sλ ∼
=Sn M λ /U λ .
Now we can view the Specht module S λ as the module generated by tableaux
of shape λ subject to the row and Garnir relations.
Exercise 2.2.3. Prove Theorem 2.2.2 by first showing that,
(a) U λ ⊆ ker ψ, where ψ : M λ → S λ is defined by ψ(T ) = eT ,
(b) the standard polytabloids eT are linearly independent,
(c) the standard tableaux span M λ /U λ .
We have the following consequence of Exercise 2.2.3.
Corollary 2.2.4. The standard polytabloids of shape λ form a basis for S λ . The
standard tableaux of shape λ form a basis for M λ /U λ . Consequently dim S λ is equal
to the number of standard tableaux of shape λ.
There is a remarkable formula for the number of standard tableaux of a fixed
shape λ.
Theorem 2.2.5 (Frame-Robinson-Thrall hook length formula). For all λ n,
n!
dim S λ = ,
x∈λ hx
where the product is taken over all cells x in the Young diagram λ, and hx is the
number of cells in the hook formed by x, which consists of x, the cells that are below
x in the same column, and the cells to the right of x in the same row.
One can generalize Specht modules to skew shapes. By removing a smaller
skew diagram μ from the northwest corner of a skew diagram λ, one gets a skew
diagram denoted by λ/μ. For example if λ = (4, 3, 3) and μ = (2, 1) then
λ/μ = .
Skew Specht modules S λ/μ are defined analogously to “straight” Specht modules.
There is a submodule characterization and a quotient characterization. Theo-
rem 2.2.2 and Corollary 2.2.4 hold in the skew setting. There is a classical combina-
torial rule for decomposing Specht modules of skew shape into irreducible straight
shape Specht modules called the Littlewood-Richardson rule, which we will not
present here.
524 WACHS, POSET TOPOLOGY
Example 2.2.6. Some important classes of skew and straight Specht modules are
listed below.
··
λ/μ = S λ/μ = regular representation.
λ = .. S λ = sign representation
.
is the only skew hook with 11 cells and descent set {4, 5, 6, 7, 10}.
The Specht modules of skew hook shape are called Foulkes representations.
Note that for any skew hook H with n cells, the set of standard tableaux of shape
H corresponds bijectively to the set of permutations in Sn with descent set des(H).
(The descent set des(σ) of a permutation σ ∈ Sn is the set of all i ∈ [n − 1] such
LECTURE 2. GROUP ACTIONS ON POSETS 525
that σ(i) > σ(i + 1).) Indeed, by listing the entries of cells 1 through n, one gets
a permutation with descent set des(H). Hence by Corollary 2.2.4 for skew shapes,
the dimension of the Foulkes representation S H is the number of permutations in
Sn with descent set des(H).
A descent of a standard Young tableau is an entry i that is in a higher row
than i + 1. By applying the Littlewood-Richardson rule mentioned above, one gets
the following decomposition of the Foulkes representation into irreducibles,
(2.2.4) SH = cH,λ S λ ,
λn
where cH,λ is the number of standard Young tableaux of shape λ and descent set
des(H).
Exercise 2.2.8. Use (2.2.4) to show that the regular representation of Sn decom-
poses into Foulkes representations as follows:
CSn ∼ =Sn SH ,
H∈SHn
S =S •S .
Exercise 2.2.10. Let λ n.
(a) Show that
S λ ↓Sn ∼ Sμ
Sn−1 =Sn−1
μ
S ↓S8 ∼
=S7 S ⊕S ⊕S .
S7
S •S ∼
=S8 S ⊕S ⊕S .
There is an important generalization of Exercise 2.2.10 (b) known as Pieri’s
rule.
Theorem 2.2.11 (Pieri’s rule). Let m, n ∈ Z+ . If λ n then
S λ • S (m) ∼
=Sm+n Sμ,
μ
summed over all partitions μ of m + n such that μ contains λ and the skew shape
μ/λ has at most one cell in each column. Similarly
S λ • S (1 ) ∼
m
=Sm+n S μ,
μ
summed over all partitions μ of m + n such that μ contains λ and the skew shape
μ/λ has at most one cell in each row.
The conjugate of a partition λ is the partition λ whose Young diagram is the
transpose of that of λ.
Theorem 2.2.12. For all partitions λ n,
S λ ⊗ sgnn ∼
=Sn S λ ,
where the tensor product is an inner tensor product. This also holds for skew
diagrams.
Exercise 2.2.13. Let V be a representation of Sn . Show that
S
(V ⊗ sgnn ) ↑Sn+1 ∼ S
=Sn+1 V ↑Sn+1 ⊗ sgnn+1 ,
n n
and
(V ⊗ sgnn ) ↓S ∼ Sn
Sn−1 =Sn−1 V ↓Sn−1 ⊗ sgnn−1 .
n
We will usually suppress the C from our notation H̃i (P ; C) (resp., H̃ i (P ; C))
and write H̃i (P ) (resp., H̃ i (P )) instead when viewing (co)homology as a G-module.
There are several important bases for the vector space Λn . We mention just
two of them here; the basis of power sum symmetric functions and the basis of
Schur functions. These bases are indexed by partitions of n. For n ≥ 1, let
pn = xni
i≥1
Skew shaped Schur functions sD are defined analogously for all skew diagrams D.
While it is obvious that the power sum symmetric functions are symmetric
functions, it is not obvious that the Schur functions defined this way are.
Theorem 2.4.1. The sets {pλ : λ n} and {sλ : λ n} form bases for Λn .
Moreover, {sλ : λ n} is an integral basis, i.e., a basis for the Z-module ΛnZ of
homogeneous symmetric functions of degree n with integer coefficients.
The Schur function s(n) is known as the complete homogeneous symmetric
function of degree n and is denoted by hn . The Schur function s(1n ) is known as
the elementary symmetric function of degree n and is denoted by en . There is an
important involution ω : Λn → Λn defined by
ω(sλ ) = sλ ,
where λ is the conjugate of λ. Clearly
ω(hn ) = en .
For the power sum symmetric functions we have
(2.4.1) ω(pλ ) = (−1)|λ|−l(λ) pλ .
Exercise 2.4.2. Prove
hn = (1 − xi )−1 ,
n≥0 i≥1
530 WACHS, POSET TOPOLOGY
and
en = (1 + xi ),
n≥0 i≥1
where h0 = e0 = 1.
The Frobenius characteristic ch(V ) of a representation V of Sn is the symmetric
function given by
1
ch(V ) := χV (μ) pμ ,
zμ
μn
where zμ := 1m1 m1 !2m2 m2 ! . . . nmn mn ! for μ = 1m1 2m2 . . . nmn and χV (μ) is the
character χV (σ) for σ ∈ Sn of conjugacy type μ. Some basic facts on Frobenius
characteristic are compiled in the next result.
Theorem 2.4.3.
(a) For all (skew or straight) shapes λ,
ch(S λ ) = sλ .
(b) For all representations V of Sn ,
ω(chV ) = ch(V ⊗ sgnn ).
(c) For all representations U, V of Sn ,
ch(U ⊕ V ) = ch(U ) + ch(V )
(d) For all representations U of Sm and V of Sn ,
ch(U • V ) = ch(U ) ch(V ).
The direct sum n≥0 G(Sn ) of representation groups is a ring under the induc-
tion product. It follows from Theorem
2.4.3 that the Frobenius characteristic map
is an isomorphism from the ring n≥0 G(Sn ) to the ring of symmetric functions
over Z.
Definition 2.4.4. Let f ∈ Λ and let g be a formal power series with positive
integer coefficients. Choose any ordering of the monomials of g, where a monomial
appears in the ordering mi times if its coefficient is mi . For example, if g =
3y1 y22 + 2y2 y3 + . . . then the monomials can be arranged as
(y1 y22 , y1 y22 , y1 y22 , y2 y3 , y2 y3 , . . . ).
Pad the sequence of monomials with zero’s if g has a finite number of terms. Define
the plethysm of f and g, denoted f [g], to be the formal power series obtained from
f by replacing the indeterminate xi with the ith monomial of g for each i. Since
f is a symmetricfunction, the
chosen order of the monomials doesn’t matter. For
example, if f = n≥0 en = i≥1 (1 + xi ) and g is as above then
f [g] = (1 + y1 y22 )(1 + y1 y22 )(1 + y1 y22 )(1 + y2 y3 )(1 + y2 y3 ) · · · .
The following proposition is immediate.
Proposition 2.4.5. Suppose f, g ∈ Λ and h is a formal power series with positive
integer coefficients. Then
• If f has positive integer coefficients then f [pn ] = pn [f ] is obtained by
replacing each indeterminate xi of f by xni .
LECTURE 2. GROUP ACTIONS ON POSETS 531
where W ⊗m is the vector space W ⊗m with S [G] action given by
m
where T is any set of positive integers, Par(T, b) is the set of partitions of length b,
all of whose parts are in T , the λi are the parts of λ, and the zi are (commuting)
indeterminates.
The right hand side can be decomposed into Schur functions by using the following
symmetric function identity of Littlewood [123, p.238]:
|λ|−r(λ)
(1 − xi xj ) (1 − xi )−1 = (−1) 2 sλ ,
i≤j i≥1 λ=λ
where r(λ) is the rank of λ, i.e., the size of the main diagonal (or Durfee square)
of the Young diagram for λ. From this we conclude that
|λ|−r(λ)
(−1)k−1 H̃k (Mn ) ∼=Sn (−1) 2 S λ .
k≥−1 λ: λn
λ = λ
With additional work involving long exact sequences of relative homology, Bouc
obtains the following beautiful refinement.
Theorem 2.4.10 (Bouc [47]). For all n ≥ 1 and k ∈ Z,
(2.4.5) H̃k−1 (Mn ) ∼
=Sn S λ.
λ: λn
λ = λ
r(λ) = |λ| − 2k
From Bouc’s formula one obtains a formula for the Betti number in dimension
k − 1 as the number of standard Young tableaux of self-conjugate shape and rank
n − 2k (which can be computed from the hook-length formula, Theorem 2.2.5).
This result provides an excellent illustration of the use of representation theory in
the computation of Betti numbers.
We sketch a proof of Theorem 2.4.10 due to Dong and Wachs [62], which
involves a technique called discrete Hodge theory. Let Δ be a G-simplicial complex.
The combinatorial Laplacian Λk : Ck (Δ) → Ck (Δ) is defined by
Λk = δk−1 ∂k + ∂k+1 δk .
A basic result of discrete Hodge theory is that
(2.4.6) ∼G kerΛk .
H̃k (Δ) =
The key observation of [62] is that when one applies the Laplacian Λk−1 to an
oriented simplex γ ∈ Ck−1 (Mn ), one gets
Λk−1 (γ) = Tn · γ,
534 WACHS, POSET TOPOLOGY
where Tn = 1≤i<j≤n (i, j) ∈ CSn and (i, j) denotes a transposition. It is then
shown that Tn acts on the Specht module S λ as multiplication by the scalar cλ
defined by
r
αi + 1 βi + 1
cλ = − ,
i=1
2 2
where r = r(λ), αi = λi − i, and βi = λi − i. That is, αi is the number of cells to
the right of and in the same row as the ith cell of the diagonal of λ and βi is the
number of cells below and in the same column as the ith cell of the diagonal. The
array (α1 , . . . , αr | β1 , . . . , βr ) is known as Frobenius notation for λ. Note that λ is
uniquely determined by its Frobenius notation.
Next we decompose Ck−1 (Mn ) into Specht modules by using (2.4.4) and an-
other symmetric function identity of Littlewood, cf., [126, I 5 Ex. 9b], namely,
(1 − xi xj ) = (−1)|ν|/2 sν ,
i≤j ν∈B
The matching complex and chessboard complex have arisen in various areas of
mathematics, such as group theory (see Section 5.2), commutative algebra [140]
and discrete geometry [223]. These complexes are discussed further in Lecture 5.
See [202] for a survey article on the topology of matching complexes, chessboard
complexes and general bounded degree graph complexes.
536 WACHS, POSET TOPOLOGY
LECTURE 3
Shellability and Edge Labelings
537
538 WACHS, POSET TOPOLOGY
1 1 2 1 2
4
3
Figure 3.1.2
Exercise 3.1.1. Verify that the third and fifth simplicial complexes below are not
shellable, while the others are.
Theorem 3.1.2 (Brugesser and Mani [52]). The boundary complex of a convex
polytope is shellable.
A geometric construction called a line shelling is used to prove this result. See
Ziegler’s book [220] on polytopes for a description of this basic construction.
Shellability is not a topological property. By this we mean that shellability of a
complex is not determined by the topology of its geometric realization; a shellable
simplicial complex can be homeomorphic to a nonshellable simplicial complex. In
fact, there exist nonshellable triangulations of the 3-ball and the 3-sphere; see
[120],[220, Chapter 8], [221], [125].
Shellability does have strong topological consequences, however, as is shown by
the following result. By a wedge ∨ni=1 Xi of n mutually disjoint connected topological
LECTURE 3. SHELLABILITY AND EDGE LABELINGS 539
spaces Xi , we mean the space obtained by selecting a base point for each Xi and
then identifying all the base points with each other.
Theorem 3.1.3 (Björner and Wachs [40]). A shellable simplicial complex has the
homotopy type of a wedge of spheres (in varying dimensions), where for each i, the
number of i-spheres is the number of i-facets whose entire boundary is contained in
the union of the earlier facets. Such facets are usually called homology facets.
Proof idea. Let Δ be a shellable simplicial complex. We first observe that any
shelling of Δ can be rearranged to produce a shelling in which the homology facets
come last. So Δ has a shelling F1 , F2 , . . . Fk in which F1 , . . . , Fj are not homology
facets and Fj+1 , . . . , Fk are, where 1 ≤ j ≤ k. The basic idea of the proof is that as
we attach the first j facets, we construct a contractible simplicial complex at each
step. The remaining homology facets attach as spheres since the entire boundary
of the facet is identified with a point.
Corollary 3.1.4. If Δ is shellable then for all i,
(3.1.1) H̃i (Δ; Z) ∼
= H̃ i (Δ; Z) ∼
= Zri ,
where ri is the number of homology i-facets of Δ.
The homology facets yield more than just the Betti numbers; they form a basis
for cohomology. We discuss the connection with cohomology further in the context
of lexicographic shellability in the next section.
There are certain operations on complexes and posets that preserve shellability.
For a simplicial complex Δ and F ∈ Δ, define the link to be the subcomplex given
by
lkΔ F := {G ∈ Δ : G ∪ F ∈ Δ, G ∩ F = ∅}.
Theorem 3.1.5 ([40]). The link of every face of a shellable complex is shellable.
Define the suspension of a simplicial complex Δ to be
susp(Δ) := Δ ∗ {{a}, {b}, ∅},
where ∗ denotes the join, and a = b are not vertices of Δ.
Theorem 3.1.6 ([40]). The join of two simplicial complexes is shellable if and
only if each complex is shellable. In particular, susp(Δ) is shellable if and only if
Δ is shellable.
Define the k-skeleton of a simplicial complex Δ to be the subcomplex consisting
of all faces of dimension k or less.
Theorem 3.1.7 ([41]). The k-skeleton of a shellable simplicial complex is shellable
for all k ≥ 0.
A simplicial complex Δ is said to be r-connected (for r ≥ 0) if it is nonempty
and connected and its jth homotopy group πj (Δ) is trivial for all j = 1, . . . , r.
So 0-connected is the same as connected and 1-connected is the same as simply
connected. A nonempty simplicial complex Δ is said to be r-acyclic if its jth
reduced integral homology group H̃j (Δ) is trivial for all j = 0, 1, . . . , r. We say that
X is (−1)-connected and (−1)-acyclic when Δ is nonempty. It is also convenient
to say that every simplicial complex is r-connected and r-acyclic for all r ≤ −2.
It is a basic fact of homotopy theory that r-connected implies r-acyclic and
that the converse holds only for simply connected complexes. Another basic fact
540 WACHS, POSET TOPOLOGY
is extendably shellable. The conjecture was settled in the negative by Ziegler [221]
who showed that there are simple and simplicial polytopes whose boundary complex
is not extendably shellable. The conjecture is, however, true for the simplex since
every ordering of the facets of the boundary complex is a shelling. Hence the
k = d − 1 case of the following conjecture holds.
Conjecture 3.1.14 (Simon [162]). For all k ≤ d, the k-skeleton of the d-simplex
is extendably shellable.
Simon’s conjecture was shown to be true for k ≤ 2 by Björner and Eriksson
[31], who extended the conjecture to all matroid complexes. Recently Hall [84]
showed that the boundary complex of the 12-dimensional cross-polytope is a coun-
terexample to the extended conjecture of Björner and Eriksson.
A simplicial complex Δ is said to be minimally nonshellable (called an ob-
struction to shellability in [200]) if Δ is not shellable, but every proper induced
subcomplex of Δ is shellable. For example, the complex consisting of two disjoint
1-simplexes is a minimally nonshellable complex (see Exercise 3.1.1). Billera and
Myers [20] showed that this is the only minimally nonshellable order complex. It
is also the only 1-dimensional minimally nonshellable simplicial complex. It was
shown by Wachs [200] that the number of vertices in any 2-dimensional minimally
nonshellable simplicial complex is either 5, 6 or 7; hence there are only finitely many
such complexes.
Conjecture 3.1.15 (Wachs). There are only finitely many d-dimensional mini-
mally nonshellable simplicial complexes (obstructions to shellability) for each d.
3 1
2 1
1
2 2
1 3
3
3 2
2
1
We say that c is decreasing if the associated word λ(c) is weakly increasing. We can
order the maximal chains lexicographically by using the lexicographic order on the
corresponding words. Any edge labeling λ of P restricts to an edge labeling of any
closed interval [x, y] of P . So we may refer to increasing and decreasing maximal
chains of [x, y], and lexicographic order of maximal chains of [x, y].
Definition 3.2.1. Let P be a bounded poset. An edge-lexicographical labeling (EL-
labeling, for short) of P is an edge labeling such that in each closed interval [x, y]
of P , there is a unique increasing maximal chain, which lexicographically precedes
all other maximal chains of [x, y].
An example of an EL-labeling of a poset is given in Figure 3.2.1. The leftmost
chain, which has associated word 123, is the only increasing maximal chain of the
interval [0̂, 1̂]. It is also lexicographically less than all other maximal chains. One
needs to check each interval to verify that the labeling is indeed an EL-labeling.
A bounded poset that admits an EL-labeling is said to be edge-lexicographic
shellable (EL-shellable, for short). The following theorem justifies the name.
Theorem 3.2.2 (Björner [21], Björner and Wachs [40]). Suppose P is a bounded
poset with an EL-labeling. Then the lexicographic order of the maximal chains of
P is a shelling of Δ(P ). Moreover, the corresponding order of the maximal chains
of P̄ is a shelling of Δ(P̄ ).
Exercise 3.2.3. Prove Theorem 3.2.2.
It is for nonbounded posets that shellability has interesting topological conse-
quences since the order complex of a bounded poset is a just a cone.
Theorem 3.2.4 (Björner and Wachs [40]). Suppose P is a poset for which P̂
admits an EL-labeling. Then P has the homotopy type of a wedge of spheres, where
the number of i-spheres is the number of decreasing maximal (i + 2)-chains of P̂ .
The decreasing maximal (i + 2)-chains, with 0̂ and 1̂ removed, form a basis for
cohomology H̃ i (P ; Z).
The first part of Theorem 3.2.4 is a consequence of Theorems 3.1.3 and 3.2.2.
Indeed, the proper parts of the decreasing chains are the homology facets of the
shelling of Δ(P ) induced by the lexicographic order of maximal chains of P̂ . To es-
tablish the second part, one needs only show that the proper parts of the decreasing
chains span cohomology. This is proved by showing that the cohomology relations
enable one to express a maximal chain with an ascent as the negative of a sum
LECTURE 3. SHELLABILITY AND EDGE LABELINGS 543
ascent descents
lexicographically
larger
123
3 2 1
12 13 23
2 2
1
3 1 3
1 2 3
1 2 3
For each k ≤ n, define the truncated Boolean algebra Bnk to be the subposet of
Bn given by
Bnk = {A ⊆ [n] : |A| ≥ k}.
Define an edge labeling λ of Bnk ∪ {0̂} as follows:
max A2 if A1 = 0̂ and |A2 | = k
λ(A1 , A2 ) =
a if A2 − A1 = {a}.
It is easy to check that this is an EL-labeling. The decreasing chains correspond to
permutations with descent set {k, k + 1, . . . , n − 1}. Hence dim H̃ n−k−1 (B̄nk ) equals
the number of permutations in Sn with descent set {k, . . . , n − 1}, which equals the
number of standard Young tableaux of hook shape k1n−k . An equivariant version of
this result is given by the following special case of a result of Solomon (the general
result is given in Section 3.4).
Theorem 3.2.7 (Solomon [163]). For all k ≤ n,
H̃n−k−1 (B̄nk ) ∼
n−k
=Sn S (k1 ) .
Combinatorial proof idea. We use a surjection from the set of tableaux of shape
k1n−k to the set of maximal chains of B̄nk , illustrated by the following example in
which n = 5 and k = 2.
3 1
2
→ ({3, 1} <· {3, 1, 2} <· {3, 1, 2, 4}).
4
5
The surjection determines a surjective Sn -homomorphism from the tableaux mod-
ule M λ to the chain space Cn−k−2 (B̄nk ; C). To show that this homomorphism in-
duces a surjective homomorphism from the quotient space S λ to the quotient space
H̃ n−k−2 (B̄nk ) = Cn−k−2 (B̄nk ; C)/B n−k−2 (B̄nk ; C), we observe that the row relations
map to 0 in Cn−k−2 (B̄nk ; C) and show that the Garnir relations map to coboundary
relations. Since the dimensions of the two vector spaces are equal, the homomor-
phism is an isomorphism. We demonstrate the fact that Garnir relations map to
cohomology relations on two examples and leave the general proof as an exercise.
The Garnir relation
3 1 3 1
2 + 4
4 2
maps to the sum of maximal chains of the proper part of the poset
1234
123 134
13
φ
LECTURE 3. SHELLABILITY AND EDGE LABELINGS 545
maps to the sum of maximal chains of the proper part of the poset
1234
123
13 12 23
and that the chains c̄σ , where σ ∈ Sn and σ(n) = n, form a basis for H̃ n−3 (Π̄n , Z).
From this basis one can immediately see that
(3.2.2) H̃n−3 (Πn ) ↓Sn = ∼S H̃ n−3 (Πn ) ↓Sn ∼
=S CSn−1 ,
Sn−1 n−1 Sn−1 n−1
a result obtained by Stanley [169] as a consequence of his formula for the full
representation of Sn on H̃n−3 (Π̄n ) (Theorem 4.4.7).
546 WACHS, POSET TOPOLOGY
3124
31/24
3/124 312/4
3/1/2/4
Next we describe a nice basis for homology of Π̄n that is dual to the decreasing
chain basis for λ2 . To split a permutation σ ∈ Sn at positions j1 < j2 < · · · < jk
in [n − 1] is to form the partition
σ(1), σ(2), . . . , σ(j1 ) / σ(j1 +1), σ(j1 +2), . . . , σ(j2 ) / . . . / σ(jk +1), σ(jk +2), . . . , σ(n)
of [n]. For each σ ∈ Sn , let Πσ be the induced subposet of the partition lattice
Πn consisting of partitions obtained by splitting the permutation σ at any set of
positions in [n − 1] . The subposet Π3124 of Π4 is shown in Figure 3.2.4. Each
poset Πσ is isomorphic to the subset lattice Bn−1 . Therefore Δ(Π̄σ ) is an (n − 3)-
sphere embedded in Δ(Π̄n ), and hence it determines a fundamental cycle ρσ ∈
H̃n−3 (Π̄n ; Z).
Theorem 3.2.8 (Wachs [198]). The set {ρσ : σ ∈ Sn , σ(n) = n} forms a basis
for H̃n−3 (Π̄n ; Z) dual to the decreasing chain basis {c̄σ : σ ∈ Sn , σ(n) = n} for
cohomology. Call the homology basis, the splitting basis.
Now we describe the decreasing chain basis for cohomology for the EL-labeling
λ1 and its dual basis for homology. Given any rooted nonplanar (i.e. children of a
node are unordered) tree T on node set [n], by removing any set of edges of T , one
forms a partition of [n] whose blocks are the node sets of the connected components
of the resulting graph. Let ΠT be the induced subposet of the partition lattice Πn
consisting of partitions obtained by removing edges of T ; see Figure 3.2.5. (If T is
a linear tree then ΠT is the same as Πσ , where σ is the permutation obtained by
reading the nodes of the tree from the root down.) Each poset ΠT is isomorphic
to the subset lattice Bn−1 . We let ρT be the fundamental cycle of the spherical
complex Δ(Π̄T ). Let T be an increasing tree on node set [n], i.e., a rooted nonplanar
tree on node set [n] in which each node i is greater than its parent p(i). We form
the chain cT in ΠT , from top down, by removing the edges {i, p(i)}, one at a time,
in increasing order of i. For the increasing tree T in Figure 3.2.5,
cT = (1/2/3/4 <· 3/24/1 <· 324/1 <· 1234).
We claim that the cT , where T is an increasing tree on node set [n], are the de-
creasing chains of λ1 .
LECTURE 3. SHELLABILITY AND EDGE LABELINGS 547
1234
1
3 / 124 123 / 4 324 / 1
3 / 4 / 12 3 / 24 / 1 32 / 4 / 1
3 4
1/2/3/4
Theorem 3.2.9. Let Tn be the set of increasing trees on node set [n]. The set
{ρT : T ∈ Tn } forms a basis for H̃n−3 (Π̄n ; Z) dual to the decreasing chain basis
{c̄T : T ∈ Tn } for cohomology. Call the homology basis, the tree splitting basis.
Exercise 3.2.10.
(a) Show λ1 is an EL-labeling whose decreasing maximal chains are of the
form cT , where T is an increasing tree.
(b) Prove Theorem 3.2.9
(c) Show λ2 is an EL-labeling whose decreasing maximal chains are of the
form cσ , where σ ∈ Sn is such that σ(n) = n.
(d) Prove Theorem 3.2.8.
Both of the EL-labelings and their corresponding bases are special cases of more
general constructions. The first EL-labeling λ1 and its bases are specializations of
constructions for geometric lattices due to Björner [21, 22]; this is discussed in
the next section. The second EL-labeling λ2 generalizes to all Dowling lattices;
see the next exercise. A geometric interpretation of the splitting basis, in which
the fundamental cycles correspond to bounded regions in an affine slice of the real
braid arrangement, is given by Björner and Wachs [42]. This leads to analogs of the
splitting basis for intersection lattices of other Coxeter arrangements, in particular
the type B partition lattice ΠB n . An analog of the splitting basis for all Dowling
lattices is given by Gottlieb and Wachs [83].
Exercise 3.2.11 (Gottlieb [81, Section 7.3]).
(a) Find an EL-labeling of the type B partition lattice ΠB n analogous to the
EL-labeling λ2 of Πn .
(b) Describe the decreasing chains in terms of signed permutations in Sn [Z2 ]
whose right-to-left maxima are positive, where for i = 1, 2, . . . n, we say
that σ(i) is a right-to-left maxima of σ if |σ(i)| > |σ(j)| for all j = i . . . n.
(c) Show that the number of decreasing chains is (2n − 1)!! := 1 · 3 · · · (2n − 1),
thereby recovering the well-known fact that Π̄B n has the homotopy type
of a wedge of (2n − 1)!!-spheres of dimension n − 2.
A partition π ∈ Πn is said to be noncrossing if for all a < b < c < d, whenever
a, c are in a block B of π and b, d are in a block B of π then B = B . Let NCn be
the induced subposet of Πn consisting of noncrossing partitions. This poset, known
as the noncrossing partition lattice, was first introduced by Kreweras [115] who
548 WACHS, POSET TOPOLOGY
a1 , a2 , . . . , ak of the atoms of the geometric lattice L. Then label each edge x <· y of
the Hasse diagram with the smallest i for which x ∨ ai = y. Note that if the atoms
of the subset lattice Bn are ordered {1}, {2}, . . . , {n}, then the geometric lattice
EL-labeling is precisely the labeling given in Section 3.2.1.
Exercise 3.2.14.
(a) Show that the edge labeling for geometric lattices described above is an
EL-labeling.
(b) Find an ordering of the atoms of Πn such that the induced geometric
lattice EL-labeling is equivalent to the EL-labeling λ1 of Section 3.2.2.
(c) Is λ2 of Section 3.2.2 equivalent to a geometric lattice EL-labeling for
some ordering of the atoms?
(d) Show that every semimodular lattice has an EL-labeling.
The decreasing chains of the geometric lattice EL-labeling have a very nice
characterization, due to Björner [22], which is described in the language of matroid
theory. A set A of atoms in a geometric lattice L is said to be independent if
r(∨A) = |A|. A set of atoms that is minimally dependent (i.e., every proper subset
is independent) is called a circuit. An independent set of atoms that can be obtained
from a circuit by removing its smallest element (with respect to the fixed ordering
a1 , a2 , . . . , ak of the atoms of L) is called a broken circuit. A maximal independent
set of atoms is said to be an NBC base if contains no broken circuits. There is a
natural bijection from the NBC bases of L to the decreasing chains of L. Indeed,
given any NBC base A = {ai1 , . . . , air }, where 1 ≤ i1 < i2 < · · · < ir ≤ k, construct
the maximal chain
cA := (0̂ < air < air ∨ air−1 < · · · < air ∨ air−1 ∨ · · · ∨ ai1 = 1̂).
It is not difficult to check that the label sequence of cA is (ir , ir−1 , . . . , i1 ), which
is decreasing, and that the map A → cA is a bijection from the NBC bases of L to
the decreasing chains of L. We conclude that the set {c̄A : A is an NBC base of L}
is a basis for top cohomology of L.
Björner [22] also constructs a basis for homology of L indexed by the NBC
bases, which is dual to the cohomology basis described above. Any independent
set of atoms in a geometric lattice generates (by taking joins) a Boolean algebra
embedded in the geometric lattice. Given any independent set of atoms A, let LA
be the Boolean algebra generated by A and let ρA be the fundamental cycle of L̄A .
Theorem 3.2.15 (Björner [22]). Fix an ordering of the set of atoms of a geometric
lattice L. The set
{ρA : A is an NBC base of L}
is a basis for top homology of L̄, which is dual to the decreasing chain basis
{c̄A : A is an NBC base of L}
for top cohomology.
Exercise 3.2.16. Prove Theorem 3.2.15.
Exercise 3.2.17. Show that the tree splitting basis for homology of the partition
lattice is an example of a Björner NBC basis.
For further reading on the homology of geometric lattices see Björner’s book
chapter [26].
550 WACHS, POSET TOPOLOGY
123456
123/4/5/6
Björner and Welker [45] prove that all open intervals of Πn,k have the homotopy
type of a wedge of spheres of varying dimensions. Since homotopy type of a wedge of
spheres (of top dimension) is the main topological consequence of pure shellability,
this result led Björner and Wachs to consider shellability for nonpure complexes.
We now present the nonpure EL-labeling of Πn,k obtained in [40] . First linearly
order the label set {1̄ < 2̄ < · · · < n̄ < 1 < 2 < · · · < n}. Now label the edge π <· τ
as follows:
⎧
⎪
⎨max B if a new block B is formed from singleton blocks
λ(π, τ ) = a if a nonsingleton block is merged with a singleton {a}
⎪
⎩
max B1 ∪ B2 if two nonsingletons B1 and B2 are merged.
13 11 4
13 11 4
12
12
6
6
10 7 1
10 7 1
T = 2 T =
15 14 3
15 14 3
9
9
8
8
5
5
Figure 3.2.7.
Exercise 3.2.19.
(a) [40] Show that the set of decreasing chains with respect to λ is the set
{cT : T is a reverse standard tableau of k-broken skew hook shape,
with n cells and n in the corner of the leftmost hook}.
(b) [40] Show that the Betti numbers are given by
t
n−1 ji − 1
β̃n−3−t(k−2) (Π̄n,k ) =
j1 − 1, j2 , . . . , jt i=1 k − 1
j1 + j2 + . . . + jt = n
ji ≥ k
where t ≥ 1, and
β̃i (Π̄n,k ) = 0
if i is not of the form n − 3 − t(k − 2) for any t ≥ 1.
(c) [184] Prove that
H̃n−3−t(k−2) (Π̄n,k ) ↓Sn ∼ SD
Sn−1 =Sn−1
D
LECTURE 3. SHELLABILITY AND EDGE LABELINGS 553
summed over all k-broken skew hook diagrams D of size n in which the
corner of the leftmost hook is removed. This is equivalent to
t−1
H̃n−3−t(k−2) (Π̄n,k ) ↓Sn ∼ jt −k
S1 • S (k−1) •
ji −k
S (k1 )
,
Sn−1 =
j1 + j2 + . . . + jt = n i=1
ji ≥ k
where • and denote induction product.
The decreasing chains given in Exercise 3.2.19(a) were also used by Sundaram
and Wachs [184] to obtain a formula for the unrestricted representation of Sn
on H̃n−3−t(k−2) (Π̄n,k ), involving composition product of representations. In or-
der to transfer the representation of the symmetric group on homology of the k-
equal partition poset to representations of the symmetric group on the homology
of the complement MAn,k := Rn − ∪An,k of the real k-equal arrangement An,k and
the complement MACn,k := Cn − ∪AC C
n,k of the complex k-equal arrangement An,k ,
Sundaram and Welker derived an equivariant version of the Goresky-MacPherson
formula; see Theorem 5.4.2. By computing the multiplicity of the trivial represen-
tation in the homology of the complement, they obtain Betti numbers for the orbit
C
spaces MAn,k /Sn and MA n,k
/Sn . For instance, they recover the following result
of Arnol’d [3]:
C 1 if i = 2k − 3
(3.2.3) β̃i (MA n,k
/Sn ) =
0 otherwise.
C
The orbit space MA n,k
/Sn is homeomorphic to the space of monic polynomials of
degree n whose roots have multiplicity at most k − 1.
Type B and D analogs of the k-equal arrangement and the k-equal partition
lattice were studied by Björner and Sagan [36]. Gottlieb [82] extended this study
to Dowling lattices.
2
3
1
2 1
3 2
1 3
chain-edge labeling that each maximal chain r of [0̂, x] determines a unique restric-
tion of λ to ME([x, y]). This enables one to talk about increasing and decreasing
maximal chains and lexicographic order of maximal chains in the rooted interval
[x, y]r .
Definition 3.3.1. Let P be a bounded poset. A chain-lexicographic labeling (CL-
labeling, for short) of P is a chain-edge labeling such that in each closed rooted
interval [x, y]r of P , there is a unique strictly increasing maximal chain, which
lexicographically precedes all other maximal chains of [x, y]r . A poset that admits
a CL-labeling is said to be CL-shellable.
The chain-edge labeling of Figure 3.3.1 is a CL-labeling. The unique increasing
maximal chain of the poset is the leftmost maximal chain.
It is easy to see that EL-shellability implies CL-shellability. All the conse-
quences of EL-shellability discussed in Section 3.2 are also consequences of the more
general CL-shellability. It is unknown whether CL-shellability and EL-shellability
are equivalent notions.
We now present the original example that motivated this more technical version
of lexicographic shellability.
Definition 3.3.2. A Coxeter system (W, S) consists of a a group W together with
a set of generators S such that the following relations form a presentation of W :
• s2 = e, for all s ∈ S
• (st)ms,t = e, where ms,t ≥ 2, for certain s = t ∈ S.
The group W is said to be a Coxeter group.
Finite Coxeter groups can be characterized as finite reflection groups, i.e, fi-
nite groups generated by linear reflections in Euclidean space. Coxeter groups are
an important class of groups, which have fascinating connections to many areas
of mathematics, including combinatorics; see the chapter by Fomin and Reading
in this volume [71] and the recent book of Björner and Brenti [30]. The finite
irreducible Coxeter groups have been completely classified. There are four infinite
families and six exceptional irreducible Coxeter groups. The most basic family con-
sists of the type A Coxeter groups, which are the symmetric groups Sn with the
adjacent transpositions (i, i + 1), i = 1, . . . , n − 1, forming the generating set. This
is the reflection group of the (type A) braid arrangement. It is also the group of
symmetries of the n-simplex. The hyperoctahedral groups Sn [Z2 ] with generators
given by signed adjacent transpositions (1, −1) and (i, i + 1), i = 1, . . . , n − 1, form
LECTURE 3. SHELLABILITY AND EDGE LABELINGS 555
the type B family. This is the reflection group of the type B braid arrangement. It
is also the symmetry group of the n-cube and the n-cross-polytope.
Let (W, S) be a Coxeter system. Every σ ∈ W can be expressed as a product,
σ = s 1 . . . sk ,
where si ∈ S. The word s1 . . . sk is said to be a reduced expression for σ if its length
k is minimum among all words whose product is σ. The length l(σ) of σ is defined
to be the length of a reduced expression for σ.
Bruhat order on W is a partial order relation on W that is defined via the
covering relation, σ <·τ if
• τ = tσ, for some t ∈ T := {αsα−1 : s ∈ S, α ∈ W }
• l(τ ) = l(σ) + 1.
For any subset J of the generating set S, there is an induced subposet of Bruhat
order on W , called the quotient by J, defined as follows
W J := {σ ∈ W : sσ > σ for all s ∈ J}.
Note W = W ∅ .
Bruhat order describes the inclusion relationships of the Schubert subvarieties
of a flag manifold. It was conjectured by de Concini and Stanley in the late 1970’s
that any open interval of of Bruhat order on a quotient of a Coxeter group is
homeomorphic to a d-sphere or a d-ball, where d is the length of the interval.
This result was needed by de Concini and Lakshmibai [57] in their work on Cohen-
Macaulayness of homogeneous coordinate rings of certain generalized Schubert vari-
eties. In an attempt to prove the conjecture by establishing EL-shellability, Björner
and Wachs instead came up with the notion of CL-shellability and constructed a
CL-labeling of the dual Bruhat poset (call this a dual CL-labeling). This labeling
relied on the following well-known characterization of Bruhat order.
Theorem 3.3.3 (Subword characterization of Bruhat order). Let (W, S) be a Cox-
eter system. Then σ < τ in Bruhat order on W if and only if for any reduced
expression w for τ there is a reduced expression for σ that is a subword of w.
We describe the dual CL-labeling of intervals [σ, τ ] of W J (this works for infinite
W as well as finite W ). Fix a reduced expression w for τ . It follows easily from
the subword characterization that as we travel down a maximal chain, we delete
(unique) letters of w one at a time until we reach a reduced word for σ. Label the
edges of each maximal chain from top down with the position of the letter in w
that is crossed out. This is illustrated in Figure 3.3.2 on the full interval [e, 321] of
Bruhat order on S3 , where si denotes the adjacent transposition (i, i + 1).
Theorem 3.3.4 (Björner and Wachs [37]). The dual chain-edge labeling described
above is a dual CL-labeling of closed intervals of Bruhat order on W J . The number
of decreasing chains (from top to bottom) is 1 if J = ∅ and is at most 1 otherwise.
It follows from Theorem 3.3.4 that every open interval (σ, τ ) of W J has the
homotopy type of a (l(τ ) − l(σ) − 2)-sphere or is contractible. A stronger topo-
logical consequence can be obtained by using the Danaraj and Klee result (Theo-
rem 3.1.11). Indeed, it can be shown that every closed interval of length 2 in W
has exactly 4 elements and that every closed interval of length 2 in W J has at most
4 elements.
556 WACHS, POSET TOPOLOGY
s1s2s1=321
3
1
s2s1=312 2 s1s2=231
3
2 1
s1=213 s2=132
1
3 2
e=123
Besides for quotients, there are other interesting classes of induced subposets of
Bruhat order such as descent classes, studied by Björner and Wachs [39], and the
subposet of involutions, which arose in algebraic geometry work of Richardson and
Springer [145]. There is a natural notion of descent set for general Coxeter groups
which generalizes the descent set of a permutation. A descent class of a Coxeter
system (W, S) is the set of all elements whose descent set is some fixed subset of
S. Björner and Wachs used the chain-edge labeling described above to show that
every finite interval of any descent class of any Coxeter group is dual CL-shellable.
This chain-edge labeling does not work for the subposet of involutions because the
maximal chains of the subposet are not maximal in the full poset. Recently Incitti
found an EL-labeling that does work for the classical Weyl groups (i.e., the type
A, B or D Coxeter groups) and conjectured that his result extends to all intervals
of all (finite or infinite) Coxeter groups. More recently, Hultman [99] proved the
main consequence of this conjecture, namely that all intervals of Bruhat order on
the set of involutions of a Coxeter group are homeomorphic to spheres.
An alternative way to label the edges of Hasse diagram of W J is by labeling
the edge σ <· τ with the element t ∈ T such that τ = tσ. By imposing a certain
linear order on T , Edelman [67] showed that this edge labeling is an EL-labeling
for the symmetric group. Proctor [134] did the same for the classical Weyl groups.
Several years after the introduction of CL-shellability, Dyer [65] found a way to
linearly order T so that the edge labeling by elements of T is an EL-labeling for
all Coxeter groups and all quotients. Dyer’s linear order (called reflection order)
was recently used by Williams [212] to obtain an EL-labeling of the poset of cells
in a certain cell decomposition of Rietsch [146] of the totally nonnegative part of
an arbitrary flag variety (for any reductive linear algebraic group). This and the
fact that the poset is thin (cf., Theorem 3.1.12) led Williams to conjecture that
Rietsch’s cell decomposition is a regular cell decomposition of a ball (if true this
would improve a result of Lusztig [124] asserting that the totally nonnegative part
of the flag variety is contractible).
LECTURE 3. SHELLABILITY AND EDGE LABELINGS 557
There are two other natural partial order relations on a Coxeter group (W, S)
that are important and interesting. By replacing T by S in the definition of Bruhat
order, one gets the definition of weak order. By replacing ordinary length l by
absolute length al in the definition of Bruhat order, where al(σ) is the length of
the shortest factorization of σ in elements of T , one gets the definition of absolute
length order.
The Hasse diagram of weak order is the same as the Cayley graph of the group
W with respect to S, directed away from the identity. It is pure and bounded
with the same rank function as Bruhat order, ordinary length l. Its topology was
first studied by Björner [24] who showed that although it is not lexicographically
shellable, it has the homotopy type of a sphere.
The Hasse diagram of absolute length order W al is the same as the Cayley
graph of the group W with respect to T , directed away from the identity. It is pure
with rank function, absolute length al; its bottom element is e, but it has no unique
top element. In the symmetric group Sn , the maximal elements are the n-cycles.
Interest in the absolute length order is a fairly recent development, which arose in
work on the braid group [48], [49], [17]. It was shown by Brady [48] that if W is
the symmetric group Sn , then every interval [e, c] of W al , where c in an n-cycle,
is isomorphic to the noncrossing partition lattice N C n discussed in Section 3.2.2.
This observation (and connections to finite type Artin groups) led Bessis [17] and
Brady and Watt [49] to define the noncrossing partition poset for any finite Coxeter
group to be the interval [e, c] of W al , where c is a Coxeter element of (W, S). A
Coxeter element is a product of all the elements of S in some order. In Sn the
Coxeter elements are the n-cycles. Reiner’s type B noncrossing partition lattice as
well as the classical (type A) noncrossing partition lattice are recovered from this
definition.
Exercise 3.3.6 (Brady [48]). Show that the interval [e, c], where c is an n-cycle,
in absolute length order of the symmetric group is isomorphic to the noncrossing
partition lattice NCn .
Although the topology of the interval (e, c), where c is a Coxeter element, is
known to have the homotopy type of a wedge of spheres via the Athanasiadis-
Brady-Watt proof of EL-shellability [6], little is known about the topology of the
full absolute length poset, W al , even for the symmetric group.
Problem 3.3.7 (Reiner [2, Problem 3.1]). What can be said about the topology of
W al − {e}, where W is an arbitrary finite Coxeter group? Is W al lexicographically
shellable, for types A and B? It is known that for type D, the poset W al is not
shellable.
In the next exercise we see other examples of posets that admit CL-labelings.
Exercise 3.3.8. Let Wn be the poset of finite words over alphabet [n], ordered by
the subword relation. We have 34 < 23244 in Wn . Let Nn be the induced subposet
of normal words, where a word is said to be normal if no two adjacent letters are
equal. For example, 2324 is normal while 23244 is not.
(a) (Björner and Wachs [38]) Find a dual CL-labeling of each interval [u, v] in
Nn , and show that (u, v) is homeomorphic to an (l(v) − l(u) − 2)-sphere.
(b) (Björner [25]) Find a dual CL-labeling of each interval [u, v] in Wn .
558 WACHS, POSET TOPOLOGY
Equivalently,
(3.4.2) C|R|−1 (PR ) ∼
=G H̃|T |−1 (PT ).
T ⊆R
In [169] Theorem 3.4.9 is proved for a more general class of posets called Cohen-
Macaulay posets. These posets are discussed in the next section. Equation (3.4.1) is
a consequence of the Hopf trace formula and the fact that pure shellability, or more
generally Cohen-Macaulayness, is preserved by rank-selection. Equation (3.4.2)
follows by the principle of inclusion-exclusion or Möbius inversion on the subset
lattice.
Note that Theorems 3.4.4 and 3.4.9 imply the fact that the regular representa-
tion of Sn decomposes into a direct sum of Foulkes modules, cf., Exercise 2.2.8.
Exercise 3.4.10. Prove Theorem 3.4.4 by using Theorem 3.4.9.
As was mentioned in the introduction to this lecture, shellability theory is inti-
mately connected with the enumeration of faces of polyhedral complexes, a central
topic in geometric combinatorics; see the books of Stanley [170] and Ziegler [220].
The descent set of a maximal chain is a specialization of a more general con-
cept in shellability theory known as the restriction of a facet, which is just the
smallest new face that is added to the complex when the facet is added. When
facets of pure shellable simplicial complexes are enumerated according to the size
of their restriction, an important combinatorial invariant of the simplicial complex,
known as the h-vector, is computed. For any (d − 1)-dimensional simplicial com-
plex Δ (shellable or not), the h-vector (h0 (Δ), h1 (Δ), . . . , hd (Δ)) and the f -vector
(f−1 (Δ), f0 (Δ), . . . , fd−1 (Δ)) determine each other,
d
d
hi xd−i = fi−1 (x − 1)d−i ,
i=0 i=0
LECTURE 3. SHELLABILITY AND EDGE LABELINGS 561
but the h-vector is usually more convenient for expressing relations such as the up-
per bound conjecture, a symmetry relation known as the Dehn-Sommerville equa-
tions, the celebrated Billera-Lee [19] and Stanley [168] characterization of the f-
vector of a simplicial polytope, and the conjectured extension to convex polytopes
and homology spheres. Although shellability has played an important role in the
study of f -vectors and h-vectors, much fancier tools from commutative algebra and
algebraic geometry have come into play. See the books of Stanley [170] and Ziegler
[220] for basic treatments of this material, the survey article of Stanley [175] for
important recent developments, and the paper of Swartz [187] for even more recent
developments.
Refinements of the f -vector and h-vector called the flag f -vector and the flag
h-vector, respectively, (defined for all pure posets and the more general balanced
complexes) have been extensively studied, beginning with the Bayer-Billera [15]
analog of the Dehn-Sommerville equations; see [170] for further information. For
pure lexicographically shellable posets the entries of the flag h-vector have a simple
combinatorial interpretation as the number of maximal chains with fixed descent
set.
For nonpure complexes, two-parameter generalizations of the f-vector and h-
vector are defined in [40]. The f-triangle and h-triangle also determine each other.
The f-triangle counts faces of a simplicial complex according to the maximum size
of a facet containing the face and the size of the face. For shellable complexes, it is
shown in [40] that the h-triangle counts facets according to the size of the restriction
set and the size of the facet. Duval [64] shows that the entries of the h-triangle
are nonnegative for a more general class of complexes than the shellable complexes;
namely the sequentially Cohen-Macaulay complexes, which are discussed in the
next lecture. It is pointed out by Herzog, Reiner, and Welker [96] that the h-
triangle of a sequentially Cohen-Macaulay complex Δ encodes the multigraded Betti
numbers appearing in the minimal free resolution of the Stanley-Reisner ideal of
the Alexander dual of Δ. Stanley-Reisner rings and ideals are discussed briefly in
the next lecture and Alexander duality is discussed in the last lecture.
Descent sets and the more general restriction sets also play an important role
in direct sum decompositions of Stanley-Reisner rings of shellable complexes; due
to Kind and Kleinschmidt [109], Garsia [77], and Björner and Wachs [41, Section
12].
562 WACHS, POSET TOPOLOGY
LECTURE 4
Recursive Techniques
{i1 , i2 , . . . , ij } ∈
/ Δ. The Stanley-Reisner construction is a two-way bridge used
to obtain topological and enumerative properties of the simplicial complex from
properties of the ring and vice versa. A simplicial complex is Cohen-Macaulay if
and only if its Stanley-Reisner ring is a Cohen-Macaulay ring (see [170] for the
definition of Cohen-Macaulay ring). The equivalence of the characterization given
in Definition 4.1.1 and the ring theoretic characterization is due to Reisner [144].
It follows from Theorem 4.1.2 that any triangulation of a d-sphere is Cohen-
Macaulay. The ring theoretic consequence of this fact played an essential role in
Stanley’s celebrated proof of the Upper Bound Conjecture for spheres. For n > d,
define the cyclic polytope C(n, d) to be the convex hull of any n distinct points on
the moment curve {(t, t2 , . . . , td ) ∈ Rd : t ∈ R} (the face poset of the polytope is
independent of the choice of points). The boundary complex of the cyclic polytope
is a simplicial complex. The upper bound conjecture for spheres asserts that the
boundary complex of the cyclic polytope achieves the maximum number of faces
of each dimension, over all simplicial complexes on n vertices that triangulate a
d-sphere. See [170] for Stanley’s proof of this conjecture and other very important
uses of the Stanley-Reisner ring and commutative algebra in the enumeration of
faces of simplicial complexes.
The Stanley-Reisner bridge can also be crossed in the opposite direction obtain-
ing ring theoretic information from the topology and combinatorics of the simplicial
complex, as exemplified by the commutative algebra results mentioned at the end
of the last lecture. Another example is a fundamental result of Eagon and Reiner
[66] which states that a square free monomial ideal has a linear resolution if and
only if it is the Stanley-Reisner ideal IΔ of a simplicial complex Δ whose Alexan-
der dual is Cohen-Macaulay. We will not define linear resolution, but Alexander
duality is discussed in Section 5.1. A nonpure generalization of this result involving
sequential Cohen-Macaulayness (discussed below) appears in papers of Herzog and
Hibi [95] and Herzog, Reiner and Welker [96]. For further reading on the extensive
connections between simplicial topology and commutative algebra, see Stanley’s
classic book [170] and the recent book of Miller and Sturmfels [129].
Exercise 4.1.3 (Walker [206]). Show that if lkΔ F is empty, 0-dimensional, or
connected for all F ∈ Δ, then Δ is pure. Consequently, Cohen-Macaulay simplicial
complexes are pure.
The main tool for establishing Cohen-Macaulayness is shellability. Indeed, it
follows from Theorem 3.1.5 and Corollary 3.1.4 that pure shellability implies the
Cohen-Macaulay property over any k. From Exercise 4.1.3 we see that this impli-
cation does not hold for nonpure shellability. In order to extend the implication to
the nonpure setting, Stanley [170] introduced nonpure versions of Cohen-Macaulay
for complexes and rings, called sequentially Cohen-Macaulay, and showed that all
shellable complexes are sequentially Cohen-Macaulay. Duval [64] and Wachs [201]
found similar simpler characterizations of Stanley’s sequential Cohen-Macaulayness
for simplicial complexes. Here we take Wachs’ characterization as the definition.
Definition 4.1.4. Let Δ be a simplicial complex. For each m = 1, 2, . . . , dim(Δ),
let Δm be the subcomplex of Δ generated by facets of dimension at least m. The
complex Δ is said to be sequentially acyclic over k if H̃i (Δm ; k) = 0 for all i < m.
We say that Δ is sequentially Cohen-Macaulay over k if lkΔ F is sequentially acyclic
over k for all F ∈ Δ.
LECTURE 4. RECURSIVE TECHNIQUES 565
only to compute its Möbius invariant. Indeed, it follows from the Phillip Hall
Theorem (Proposition 1.2.6) and the Euler-Poincaré formula (Theorem 1.2.8) that
if the homology of P vanishes below the top dimension then
By applying the recursive definition of Möbius function we get the recursive formula
observed by Sundaram in [177],
(4.1.2) β̃l(P )−1 (P \ {0̂}) = (−1)l(P )+r(x) β̃r(x)−2 (0̂, x),
x∈P
where P is a Cohen-Macaulay poset with a bottom element 0̂ and r(x) is the rank
of x.
Since general sequentially Cohen-Macaulay posets can have homology in mul-
tiple dimensions, (4.1.1) and (4.1.2) do not hold in the nonpure setting. How-
ever, if the sequentially Cohen-Macaulay poset has a property known as semipure,
then the following generalization of (4.1.2) shows that Möbius function can still
be used to compute its Betti numbers. A poset P is said to be semipure if
P≤y := {x ∈ P : x ≤ y} is pure for all y ∈ P . The proper part of the poset
given in Figure 3.2.1 is semipure. Also the face poset of any simplicial complex is
semipure.
pure EL-shellable
=⇒ EL-shellable
pure CL-shellable
=⇒ CL-shellable
pure shellable
=⇒ shellable
homotopy Cohen-Macaulay
=⇒ sequentially homotopy
Cohen-Macaulay
Cohen-Macaulay
over Z =⇒ sequentially Cohen-Macaulay
over Z
Cohen-Macaulay over k =⇒ sequentially Cohen-Macaulay over k
It is known that all the implications but
(pure) EL-shellable =⇒ (pure) CL-shellable
are strict. Whether or not there are CL-shellable posets that are not EL-shellable is
an open question. It is also unknown whether or not CL-shellability and dual CL-
shellability are equivalent. Examples of shellable posets that are not CL-shellable
were obtained by Vince and Wachs [196] and Walker [208].
There are other important properties of simplicial complexes which have recur-
sive formulations such as vertex decomposability (introduced by Provan and Billera
[135] for pure simplicial complexes and extended to nonpure simplicial complexes
by Björner and Wachs [41]) and constructible complexes (introduced by Hochster
[97] and extended to the nonpure case by Jonsson [105]). There are no special
poset versions of these tools, however, as there are for shellability.
(a) (b)
Figure 4.2.1
Theorem 4.2.2 (Björner and Wachs [38, 40]). A bounded poset P is CL-shellable
if and only if P admits a recursive atom ordering.
Proof idea. We use a recursive procedure to obtain a recursive atom ordering
from a CL-labeling and vice versa. Given a CL-labeling λ, order the atoms of P in
increasing order of the labels λ(c, 0̂ <· a), where c is any maximal chain containing
the atom a (the label is independent of the maximal chain c since 0̂ <· a is a bottom
edge). Then recursively use the restriction of the CL-labeling to each interval [a, 1̂]
to obtain a recursive atom ordering of [a, 1̂].
Conversely, given a recursive atom ordering of P , label the bottom edge of
each maximum chain with the position of the atom in the atom ordering. Then
recursively use the recursive atom ordering of [a, 1̂] to obtain an appropriate chain-
edge labeling of [a, 1̂], for each atom a.
A bounded poset P is said to be semimodular if for all u, v ∈ P that cover some
x ∈ P , there is an element y ∈ P that covers both u and v. Semimodular posets are
not necessarily shellable. A poset is said to be totally semimodular if every closed
interval is semimodular. Note that semimodular lattices are totally semimodular.
In particular, the boolean algebra and the partition lattice are totally semimodular.
Theorem 4.2.3 (Björner and Wachs [38]). Every ordering of the atoms of a totally
semimodular poset is a recursive atom ordering.
Theorem 4.2.4 (Björner and Wachs [38]). An ordering of the facets of a simplicial
complex Δ is a shelling if and only if the ordering is a recursive coatom ordering
of the face lattice L(Δ).
Exercise 4.2.5. Prove
(a) Theorem 4.2.3.
(b) Theorem 4.2.4
Definition 4.2.6 (Wachs and Walker [205]). A meet semilattice P with rank
function r is said to be a geometric semilattice if
(i) every interval is a geometric lattice
(ii) for all x ∈ P and subset A of atoms whose join exists, if r(x) < r( A)
then there is an a ∈ A such that a ≤ x and a ∨ x exists.
Examples of geometric semilattices include
• geometric lattices
LECTURE 4. RECURSIVE TECHNIQUES 569
(b) Use formula (4.1.2) to show that the top Betti number β̃k−1 (I¯n,n ) is equal
to dn , the number of derangements of n elements. Consequently, I¯n,n has
the homotopy type of a wedge of dn spheres of dimension k − 1 .
(c) Let Nn,k be the induced subposet of Nn consisting of words of length
at most k. Use Exercise 3.3.8, Theorem 3.4.1, and formula (4.1.2) to
show that N̄n,k has the homotopy type of a wedge of (n − 1)k spheres of
dimension k − 1.
(d) Let Wn,k be the induced subposet of Wn consisting of words of length
at most k. Show that W̄n,k also has the homotopy type of a wedge of
(n − 1)k spheres of dimension k − 1.
Remark 4.3.3. The injective word poset and normal word poset of Example 4.3.1
were introduced by Farmer [68] who showed that Īn,k and N̄n,k have the homology
of a wedge of (k − 1)-spheres. Björner and Wachs [38] recovered these results and
strengthened them to homotopy by establishing shellability. The Betti number
computations of Exercise 4.3.2 (b) and (c) are due to Reiner and Webb [142] and
Farmer [68], respectively.
Recall that in Example 3.2.2, we used EL-shellability to show that the proper
part of the partition lattice Πn has the homotopy type of a wedge of (n−1)! spheres
of dimension n − 3. In the next two examples we use the techniques of Sections 4.1
and 4.2 to obtain analogous results for the even and odd block size partition posets.
Example 4.3.4. Let Πeven2n be the subposet of Π2n consisting of partitions whose
block sizes are even. The even block size partition poset is pure but lacks a bot-
tom element. The upper intervals [x, 1̂] are all partition lattices, which are totally
semimodular; so every atom ordering of [x, 1̂] is a recursive atom ordering. Asso-
2n ∪ {0̂} by listing the elements of each block
ciate a word with each atom of Πeven
in increasing order and then listing the blocks in lexicographic order. We claim
that lexicographic order on the words corresponds to a recursive atom ordering of
2n ∪ {0̂}. The verification is left as an exercise.
Πeven
We next use (4.1.2) to compute the Betti numbers of Π̄even even
2n . Since Π2n is pure
and the upper intervals are isomorphic to partition lattices, to compute the unique
nonvanishing Betti number β̃n−2 (Π̄even
2n ), we apply (4.1.2) to the dual of the poset.
Let β2n denote the Betti number β̃n−2 (Π̄even
2n ). By (4.1.2) we have
β2n = (−1)b(x)+n β̃b(x)−3 (x, 1̂)
x
= (−1)b(x)+n (b(x) − 1)!
x
n
= (−1)r+n (r − 1)! |{x ∈ Πeven
2n : b(x) = r}|,
r=1
where b(x) denotes the number of blocks of x. Note that
1 2n
|{x ∈ Π2n : b(x) = r}| =
even
,
r! 2j1 , 2j2 , . . . , 2jr
(j1 ,j2 ,...,jr )n
The exponential generating function for the Betti numbers is thus given by,
u2n r1 2n u2n
β2n = (−1) (−1)n
(2n)! r 2j1 , 2j2 , . . . , 2jr (2n)!
n≥1 r≥1 n≥1 (j1 ,j2 ,...,jr )n
1 u2j r
= (−1)r (−1)j
r (2j)!
r≥1 j≥1
(4.3.1) = − ln(cos u).
By taking derivatives we get,
u2n−1
β2n = tan u.
(2n − 1)!
n≥1
2n−1
u
So β2n is equal to the coefficient of (2n−1)! in the Taylor series expansion of tan u. It
is well known that this coefficient is equal to the Euler number E2n−1 , where Em is
defined to be the number of alternating permutations in Sm , i.e., permutations with
descents at all the even positions and ascents at all the odd positions. We conclude
2n has the homotopy type of a wedge of E2n−1 spheres of dimension n − 2.
that Π̄even
There is an alternative way to arrive at β2n = E2n−1 . Replace the last line of
(4.3.1) with
u2j
− ln (−1)j
(2j)!
j≥0
and then take derivatives of both sides of the equation. This results in
u2n−1 u2j u2j−1
β2n (−1)j = (−1)j−1 .
(2n − 1)! (2j)! (2j − 1)!
n≥1 j≥0 j≥0
Remark 4.3.7. The result on the Möbius invariant of the even block size partition
lattice derived in Example 4.3.4 first appeared in in the 1976 MIT thesis of Garrett
Sylvestor on Ising ferromagnets [188]. Stanley [167] extended this result to the d-
divisible partition lattice of Exercise 4.3.6 (d). The Möbius invariant results derived
in Example 4.3.5 and Exercise 4.3.6 (e) are also due to Stanley. The recursive atom
ordering for the d-divisible partition lattice (Example 4.3.4 and Exercise 4.3.6 (a))
is due to Wachs (see [150]), as is an EL-labeling of this poset [198]. The recursive
atom ordering for the 1 mod d partition lattice (Example 4.3.5 and Exercise 4.3.6
(b)) is due to Björner (see [54] and [38]). The general recursive atom ordering of
Πknd+k
mod d
∪ {0̂} of Exercise 4.3.6 (c) is due to Wachs [201].
In the next example, we demonstrate the full power of Theorem 4.1.12 by
finding the Betti numbers for a nonpure shellable poset.
Example 4.3.8. For n ≥ k ≥ 3, let Π≥k n be the subposet of Πn consisting of
partitions whose block sizes are at least k. It was shown by Björner and Wachs
[40] that the poset Π≥k
n ∪ {0̂} admits a recursive atom ordering similar to that of
2n ∪ {0̂}. The following computation of Betti numbers appears in [201]. The
Πeven
dual of Π≥k
n is semipure and the upper intervals are partition lattices; so we apply
(4.1.3) to the dual. First note that
b(x)
|Bi |
(4.3.3) m(x) = − 1,
i=1
k
We have,
1 n
|{x ∈ Π≥n
n : b(x) = r, m(x) = m − 1}| = .
r! j1 , j2 , . . . , jr
(j1 , j2 , . . . , jr ) n
ji ≥ k ∀i
i=1 ji /k = m
r
Problem 4.3.10. Linusson [121] computed the Möbius invariant of the poset
Π≥k
n ∪ {0̂} in order to compute lower bounds for the complexity of a problem
similar to the k-equal problem of Section 3.2.4; namely that of determining whether
a given list of real numbers has the property that the number of occurrences of each
entry is at least k. It was shown by Björner and Lovász [33] that Betti number
computations give better bounds than Möbius function computations. Can the
Betti number computation of Example 4.3.8 be used to improve Linusson’s lower
bound for the complexity of the “at least k problem”?
Exercise 4.3.11 (Wachs [201]). In this exercise, we generalize the results of Ex-
amples 4.3.4 and 4.3.5 to the nonpure case of Exercise 4.3.6 . For positive integers
n, d, k, with k ≤ d, let k0 = k/gcd(k, d) and d0 = d/gcd(k, d). Show that
nd+k uid+k
mod d md0 +1 u
(−1)m βm−2 (Π̄knd+k )t =f ti/k0 d0 +1 ,
(nd + k)! (id + k)!
m≥1 i≥0
n≥0
For any Cohen-Macaulay G-poset P with bottom element 0̂, Whitney homology
of P is defined for each integer r as follows,
WHr (P ) = H̃r−2 (0̂, x),
x∈Pr
Exercise 4.4.2.
(a) Prove Theorem 4.4.1.
(b) Show that if P is the face poset of a simplicial complex then (4.4.1) reduces
to the Hopf trace formula.
Example 4.4.3 (Reiner and Webb [142]). Consider the injective word poset In,k of
Example 4.3.1. Let G be the symmetric group Sn , which acts on In,k in the obvious
way. Since In,k is Cohen-Macaulay, we can apply (4.4.2). Let x ∈ In,k be a word
of length r. Clearly Gx is isomorphic to the Young subgroup (S1 )×r × Sn−r , since
the letters of x must be fixed and the letters outside of x may be freely permuted.
Gx acts trivially on the letters outside of x. The interval (0̂, x) is isomorphic to
the proper part of the Boolean algebra Br ; so its top homology is 1-dimensional.
Hence,
∼G S (1) ⊗ · · · ⊗ S (1) ⊗S (n−r)
H̃r−2 (0̂, x) = x
r
LECTURE 4. RECURSIVE TECHNIQUES 575
Exercise 4.4.6 (Sundaram [177]). Show that by applying (4.4.2) to the Boolean
algebra Bn , one obtains the well-known symmetric function identity,
n
(−1)i ei hn−i = 0.
i=0
Since Δ(Πn \ {1̂}) is contractible for all n > 1 and is {∅} when n = 1, it follows
that
h1 = (−1)r chH̃r−2 (Π̄r+1 ) hi .
r≥0 i≥1
Since h1 is the plethystic identity,
[−1]
(4.4.6) (−1)r−1 chH̃r−3 (Π̄r ) = hi
r≥1 i≥1
1
= μ(d) log(1 + pd )
d
d≥1
where [−1] denotes plethystic inverse, μ is the number theoretic Möbius function,
and pd is the power sum symmetric function. The last equation follows from a
formula of Cadogan [53], which is also derived in [177]. By extracting the degree
n term, we have
1 n/d
(4.4.7) chH̃n−3 (Π̄n ) = (−1)n−n/d μ(d) pd .
n
d|n
LECTURE 4. RECURSIVE TECHNIQUES 577
By (2.4.1) and Theorem 2.4.3 (b), this together with (4.4.7) implies (4.4.4), as
desired.
The representation e2πi/n ↑S n
Cn is a well-studied representation called the Lie
representation because it is isomorphic to the representation of the symmetric group
on the multilinear component of the free Lie algebra on n generators, cf., Theo-
rem 1.6.2.
Exercise 4.4.8 (Sundaram [177]). Use the Whitney homology technique to prove
the following results of Calderbank, Hanlon and Robinson [54].
(a) (equivariant version of Exercise 4.3.6 (e))
[−1]
(−1)n chH̃n−2 (Π̄1nd+1
mod d
)= hid+1 .
n≥0 i≥0
Example 4.4.10 (equivariant version of Example 4.3.8 [201]). We will use Theo-
rem 4.4.9 and (2.4.3) to obtain the following formula for the two parameter gener-
ating function for the homology of Π̄≥k
n ,
[−1]
(4.4.11) (−1)m−1 chH̃m−2 (Π̄≥k
n ) u n m
t = h i h i u i ki
t .
m≥1 i≥1 i≥k
n≥k
where T = {k, k + 1, . . . }, m(λ) = i λi /k − 1 (recall (4.3.3) here), and the
remaining notation is defined in Example 2.4.9 . By setting zi = ui ti/k in (2.4.3),
we obtain
chWHr,m−1 ((Π≥k ∗
n ) )u t
n m
= ch H̃r−2 (x, 1̂) u|λ| t i λi /k
m,n λ∈Par(T,r+1) x∈Π(λ)
h i u i t k .
i
= chH̃r−2 (Πr+1 )
i≥k
The following result generalizes (4.4.8) and (4.4.11). Its proof is similar to that
of (4.4.11) described above.
We remark that the restricted block size partition poset ΠSn ∪ {0̂} of Theo-
rem 4.4.11 is the intersection lattice of a subspace arrangement, which is discussed
further in Section 5.4.
(−1)m chH̃m−1 (Π̄jnd+j
mod d
) und+j tmd0 +1
m≥1
n≥0
[−1]
ji d0 +1
= hid0 +1 hid+j uid+j t 0 ,
i≥0 i≥0
j d
where j0 = and d0 = .
gcd(j,d) gcd(j,d)
Problem 4.4.13. Do the results of this section on restricted block size partition
lattices have nice generalizations to Dowling lattices or intersection lattices of Cox-
eter arrangements?1
For other restricted block size partition posets with very interesting equivariant
homology, see the work of Sundaram [182].
123456
The theorem is proved by first showing that for all α, β ∈ Adn , if cdα ∈ Πdβ then
α ≤ β in lexicographic order. This is used to establish linear independence of both
{ρdσ : σ ∈ Adn } and {cdσ : σ ∈ Adn }. The result then follows from Exercise 4.3.6 (d).
The “splitting basis” given in Theorem 4.5.1 is used in [198] to give a combi-
natorial proof of the following result of Calderbank, Hanlon and Robinson, which
was first conjectured by Stanley.
Theorem 4.5.2 (Calderbank, Hanlon and Robinson [54]). Let Hn,d be the skew
hook of size nd − 1 and descent set {d, 2d, . . . , (n − 1)d} (cf. Section 2.2). Then
H̃n−2 (Π̄0 mod d ) ↓Snd ∼
nd Snd−1 =S S Hn,d
nd−1
To prove this, one first observes that the row permutations leave cT invariant;
then one shows that the Garnir relations map to cohomology relations.
There is a dual version of polytabloid defined for each tableaux T by
e∗T := sgn(β) T βα.
α∈Rλ β∈Cλ
(This is actually closer to the traditional notion of polytabloid than the one we
gave in Section 2.2.) For each skew or straight shape λ, it is known that
∼S S λ .
e∗ : T ∈ Tλ =
T n
All the results of this subsection were generalized to the restricted block size
partition posets ΠSn of Theorem 4.4.11 by Browdy and Wachs [50, 51]. The nonpu-
rity of ΠSn in the general case significantly increases the complexity of the results.
The “at least k” partition poset Π≥k n is an example of such a nonpure poset.
4 6
5
2 7 3 4 8
2
6
1 3 1 5 9
(a) (b)
Figure 4.5.2
Let Tnd+1
d
be the set of rooted planar (d + 1)-ary trees on leaf set [nd + 1] (i.e.,
rooted trees in which each internal node has exactly d + 1 children that are ordered
from left to right). For any node x of T ∈ Tnd+1d
, let m(x) be the smallest leaf in
the tree rooted at x. A tree T in Tnd+1 is said to be a d-brush if for each node y of
d
T , the m-values of the children of y increase from left to right, and the child with
the largest m-value is a leaf. An example of a 1-brush is given in Figure 4.5.2 (a)
and of a 2-brush is given in Figure 4.5.2 (b). Note that every 1-brush looks like a
comb, which is the reason for the terminology “brush”.
Exercise 4.5.5.
(a) Show that the number of 1-brushes on leaf set [n + 1] is n!.
(b) Show that the number of 2-brushes on leaf set [2n + 1] is (2n − 1)!!2 .
Recall from Section 1.6 that the postorder traversal of a binary tree on leaf set
[n] yields a maximal chain of Π̄n . Now we consider a more general construction,
which associates a maximal chain cT of Π̄1nd+1mod d
to each tree T in Tnd+1d
. Each
internal node y of T corresponds to a merge of d + 1 blocks that are the leaf sets of
the trees rooted at the d+ 1 children of y. Postorder traversal of internal nodes of T
yields a sequence of merges, which corresponds to a maximal chain cT of Π̄1nd+1 mod d
.
For example if T is the tree of Figure 4.5.2 (b) then
cT = 159/2/7/3/4/8/6 <· 15927/3/4/8/6 <· 15927/348/6
Theorem 4.5.6 (Hanlon and Wachs [91]). Let Bnd+1 d
be the set of d-brushes in
Tnd+1 . The set {cT : T ∈ Bnd+1 } forms a basis for H̃ n−2 (Π̄1nd+1
d d mod d
; Z).
Exercise 4.5.7 (Hanlon and Wachs [91]). Prove Theorem 4.5.6 by showing
(a) The set {cT : T ∈ Bnd+1
d
} spans H̃ n−2 (Π̄1nd+1
mod d
; Z).
(b) |Bnd+1 | = |μ(Πnd+1 )|.
d 1 mod d
18 12
7
8
13
15 1 9
11
6 10
4
2
14
19
16
3
17 5
Figure 4.5.3
trees. Note also that a 2-clique tree is an ordinary tree. An example of a 3-clique
tree, which we refer to as a triangle tree, is given in Figure 4.5.3.
Given a (d + 1)-clique tree T on node set [nd + 1], one obtains a partition
in Π1nd+1
mod d
by choosing any set of (d + 1)-cliques of T and removing the edges
of each clique in the set. The blocks of the partition are the node sets of the
connected components of the resulting graph. We say that the partition is obtained
by splitting the (d + 1)-clique tree T at the chosen set of (d + 1)-cliques. For
example, the partition obtained by splitting the triangle tree at the shaded triangles
in Figure 4.5.4 is
19, 16, 6 / 8, 15, 18, 12, 13 / 10, 2, 1, 9, 7 / 3 / 17, 5, 4, 11, 14.
Now let ΠT be the subposet of Π1nd+1
mod d
consisting of partitions obtained by splitting
T . Clearly ΠT is isomorphic to the subset lattice Bn . Therefore Δ(Π̄T ) is an (n−2)-
sphere which determines a fundamental cycle ρT .
Theorem 4.5.8 (Shareshian and Wachs [159]). The set of fundamental cycles ρT
such that T is a (d + 1)-clique tree on node set [nd + 1], spans H̃n−2 (Π̄1nd+1
mod d
; Z).
This is proved by identifying a set S of (d+1)-clique trees, called increasing (d+
1)-clique trees, and establishing a bijection and unitriangular relationship between
(d + 1)-clique trees and d-brushes. This shows that {ρT : T ∈ S} is a basis for
H̃n−2 (Π̄1nd+1
mod d
; Z). The increasing 2-clique trees are the increasing trees discussed
in Section 3.2.2 and the homology basis is the tree splitting basis.
Next we discuss an application of Theorem 4.5.8 that led to the discovery of the
clique tree splitting basis in the first place. Let NPM2n be the poset of nonempty
graphs on node set [2n] that don’t contain a perfect matching (i.e., a subgraph
in which each of the 2n vertices has degree 1), ordered by inclusion of edge sets.
Linusson, Shareshian, and Welker [122] show, using discrete Morse theory, that
LECTURE 4. RECURSIVE TECHNIQUES 583
18 12
7
8
13
15 1 9
11
6 10
4
2
14
19
16
3
17 5
Figure 4.5.4
NPM2n has the homotopy type of a wedge of (2n − 1)!!2 spheres of dimension
3n − 4. It was in this work that the increasing triangle trees first arose as the
critical elements of a Morse matching on NPM2n (increasing triangle trees are just
called trees of triangles in [122]). See the chapter by Forman [74] in this volume to
learn about discrete Morse theory and critical elements. Recall that the property
of containing (or not containing) a perfect matching is an example of a monotone
graph property (see Section 1.4). When we discuss Alexander duality in the next
lecture, we will see how the homology of the poset of graphs that have a monotone
graph property is related to the homology of the poset of graphs that don’t have
the property.
Theorem 4.5.8 and discrete Morse theory play essential roles in the proof of the
following equivariant version of the Linusson-Shareshian-Welker result. Indeed, dis-
crete Morse theory is used to show that there is a sign twisted S2n−1 -isomorphism
between the homology of NPM2n and the cohomology of another poset called the
factor critical graph poset. Discrete Morse theory is also used to show that the co-
homology of the factor critical graph poset is generated by certain maximal chains
naturally indexed by triangle trees. This gives a natural map from the triangle tree
generators ρT of homology of Π12n−1 mod 2
to the maximal chains of the factor critical
graph poset that are indexed by the triangle trees. Relations on the triangle tree
generators of H̃n−3 (Π̄12n−1
mod 2
) are derived, which are shown to map to coboundary
relations in the factor critical graph poset.
Theorem 4.5.9 (Shareshian and Wachs [159]). For all n ≥ 1,
H̃3n−4 (NPM2n ) ↓S ∼
S2n−1 =S2n−1 H̃n−3 (Π̄2n−1 ) ⊗ sgn2n−1 .
2n 1 mod 2
Another graph poset, recently studied by Jonsson [105], curiously has the same
bottom nonvanishing homology as NPM2n and Π̄12n−1 mod 2
(up to sign twists). A graph
is said to be d-edge-connected if removal of any set of at most d − 1 edges leaves
584 WACHS, POSET TOPOLOGY
and
H̃i (NECddn+1 ) = 0
d+1
if i < 2 n − 2.
The conjecture is true for d = 1, 2. Indeed, Jonsson’s homology result and
Theorem 4.5.10 comprise the d = 2 case. The conjecture for d = 1 says that that
the poset of disconnected graphs on node set [n + 1] has homology isomorphic, as
an Sn+1 -module, to that of the partition lattice. A proof of this well-known result
is discussed in Example 5.2.8.
There is another interesting poset with the same homotopy type and Snd+1 -
equivariant homology as that of Π̄1nd+1 mod d
, worth mentioning here. This poset is
the proper part of the poset Tnd+2 of homeomorphically irreducible trees on leaf
2 mod d
set [nd + 2] in which each internal node has degree congruent to 2 mod d. By
homeomorphically irreducible we mean nonrooted and no node has degree 2. The
order relation is as follows: T1 < T2 if T1 can be obtained from T2 by contracting
internal edges. So the bottom element of Tnd+2 2 mod d
is the star tree (the tree with only
one internal node), and the maximal elements are trees in which each internal node
has degree exactly d + 2. The d = 1 case of the tree poset Tnd+2 2 mod d
has arisen in
various areas such as algebraic geometry, homotopy theory, geometric group theory,
mathematical physics and mathematical biology; see eg., [46, 197, 148, 211, 147,
1, 18, 138] and the references contained therein. Vogtmann [197] showed that
the tree poset in the d = 1 case is homotopy Cohen-Macaulay. There is a natural
action of Snd+2 on Tnd+2
2 mod d
whose representation on the homology of Tnd+2 2 mod d
−{0̂}
was computed by Robinson and Whitehouse [148] in the d = 1 case. Hanlon [87]
introduced the general tree poset Tnd+2
2 mod d
, proved that it is Cohen-Macaulay, and
generalized the Robinson-Whitehouse result.
Theorem 4.5.12 (Robinson and Whitehouse, d = 1 [148], Hanlon [87]). For all
d, n ≥ 1,
) ∼
S
H̃n−2 (T̄nd+2
2 mod d
=Snd+2 H̃n−2 (Π̄1nd+1
mod d
) ↑Snd+2
nd+1
−H̃n−1 (Π̄1nd+2
mod d
).
LECTURE 4. RECURSIVE TECHNIQUES 585
The d = 1 case of the Snd+2 -module given in Theorem 4.5.12 has come to be
known as the Whitehouse module. It has occurred in a variety of diverse contexts
such as homotopy theory [148, 211], cyclic Lie operads [79, 111], homology of
partition posets [181, 182], knot theory and graph complexes [8], and hyperplane
arrangements and Lie algebra homology [90].
From Theorem 4.5.12, one can show that (see [183]),
For example, if Q is the poset of connected graphs on node set [n] and R is the
poset of disconnected graphs on node set [n] then
∼S H̃ n ⊗n
H̃i (Q̄) = n ( 2 )−i−3 (R̄) ⊗ sgnn .
Exercise 5.1.3. Given a simplicial complex Δ on vertex set V , its Alexander
dual Δ∨ is defined to be the simplicial complex on V consisting of complements of
nonfaces of Δ, i.e.,
Δ∨ = {V − F : F ⊆ V and F ∈
/ Δ}.
How are the homology of Δ and its Alexander dual related?
Let P be a G-poset and let Q be an H-poset. Then the join P ∗ Q and the
product P × Q are (G × H)-posets with respective (G × H) actions given by
gx if x ∈ P
(g, h)x = ,
hx if x ∈ Q
587
588 WACHS, POSET TOPOLOGY
and
(g, h)(p, q) = (gp, hq).
We have the following poset version of the Künneth theorem of algebraic topology.
Theorem 5.1.4. Let P be a G poset and let Q be an H-poset. Then
H̃r (P ∗ Q) ∼
=G×H H̃i (P ) ⊗ H̃r−i−1 (Q),
i
for all r.
A result of Quillen [137] states that if P and Q have bottom elements 0̂P and
0̂Q , respectively, then there is a (G × H)-homeomorphism
P × Q \ {(0̂P , 0̂Q )} ∼
=G×H P \ {0̂P } ∗ Q \ {0̂Q }.
Walker [206, 207] proves the similar result that if P and Q have top elements as
well as bottom elements then there is a (G × H)-homeomorphism
P ×Q∼
=G×H P̄ ∗ Q̄ ∗ A2 ,
where A2 is a two element antichain on which the trivial group acts. Theorem 5.1.4
and Quillen’s and Walker’s results yield,
Theorem 5.1.5. Let P be a G-poset with a bottom element 0̂P and let Q be an
H-poset with a bottom element 0̂Q . Then for all r,
H̃r (P × Q \ {(0̂P , 0̂Q )} ∼
=G×H H̃i (P \ {0̂P }) ⊗ H̃r−i−1 (Q \ {0̂Q }).
i
Exercise 5.1.6. Let P and Q be posets with bottom elements. Show that P × Q
is Cohen-Macaulay if and only if P and Q are Cohen-Macaulay. This result also
holds in the nonpure case but is more difficult to prove; see [44].
The products in Theorem 5.1.5 are known as reduced products. There is a
similar formula for the homology of ordinary direct products which follows from
the Künneth theorem and another (G × H)-homeomorphism
P ×Q∼
= P × Q
of Quillen [137] and Walker [207].
Theorem 5.1.7. Let P be a G-poset and let Q be an H poset. Then for all r,
Hr (P × Q) ∼
=G×H Hi (P ) ⊗ Hr−i (Q).
i
Next we consider the n-fold product P ×n of a G-poset P . The group that acts
on P ×n is the wreath product Sn [G], defined in Section 2.4. The action on P ×n is
given by
(g1 , g2 , . . . , gn ; σ)(p1 , p2 , . . . , pn ) = (g1 pσ−1 (1) , g2 pσ−1 (2) , . . . , gn pσ−1 (n) ).
LECTURE 5. POSET OPERATIONS AND MAPS 589
and
H̃(P, λ, 2) := H̃(P, i, mi , 2).
i:mi >0
Theorem 5.1.9 (Sundaram and Welker [185]). Let P be a G-poset with a bottom
element 0̂. Then for all r,
H̃r (P ×n − {(0̂, . . . , 0̂)}) ∼
S [G]
=Sn [G] H̃(P − {0̂}, λ, 1) ↑Snλ [G] .
λ r+n+1
l(λ) = n
We remark that we have stated Theorems 5.1.8 and 5.1.9 in a form different
from the original given in [185] but completely equivalent.
590 WACHS, POSET TOPOLOGY
Example 5.1.10 (The Boolean algebra Bn ). We apply the second part of The-
orem 5.1.9 to Bn = B1×n . The trivial group S1 acts on B1 and this induces an
action of Sn [S1 ] = Sn on Bn , which is precisely the action given in (2.3.1). Clearly
H̃i−2 (B̄1 ) = 0 unless i = 1, in which case H−1 (B̄1 ) is the trivial representation of
S1 . It follows that
m m
H̃(B1 , 1, m, 2) = S (1 )
[S (1) ] = S (1 )
,
and H̃(B1 , i, m, 2) = 0 for i = 1. Hence
n
S (1 )
if λ = (1n )
H̃(B1 , λ, 2) =
0 otherwise.
It follows that the only nonzero term in the decomposition given in Theorem 5.1.9
is the term corresponding to λ = (1n ) and r = n − 2. Hence we recover the fact
that H̃r (B̄n ) is the sign representation for r = n − 2 and is 0 otherwise.
Exercise 5.1.11. The face lattice Cn of the n-cross-polytope with its top element
removed can be expressed as a product,
Cn \ 1̂ = (C1 \ 1̂)×n .
Since S2 acts on C1 \ {1̂}, the product induces an action of the hyperoctahedral
group Sn [S2 ] on Cn \ 1̂.
(a) Show that this action is the action given in Example 2.3.3.
(b) Use Theorem 5.1.9 to prove that
n
(1 ) (12 )
∼S [S ] S [S ] if r = n − 1
H̃r (C̄n ) = n 2
0 otherwise.
(c) Use Theorem 5.1.9 to obtain a generalization of the formula in (b) for the
×n
poset Xm , where Xm is the poset consisting of a bottom element 0̂ and
m atoms.
Example 5.1.12 ([185]). In this example we compute the homology of lower
intervals in the partition lattice Πn . Each lower interval is isomorphic to a product
of smaller partition lattices,
[0̂, x] ∼
= X Π×m
i
i
,
i
where mi is equal to the number of blocks of x of size i. The stabilizer (Sn )x of
x under the action of Sn on Πn is isomorphic to Xi Smi [Si ]. The (Sn )x -poset
[0̂, x] is isomorphic to the Xi Smi [Si ]-poset Xi Π×m
i
i
. We use the second parts
of Theorems 5.1.5 and 5.1.9 to compute the representation of Xi Smi [Si ] on the
homology of (0̂, x). This yields the following formula of Lehrer and Solomon [119]
for the only nonvanishing homology,
H̃r(x)−2 (0̂, x) ∼
m2
=Xi Smi [Si ] S (m1 ) [H̃−2 (Π̄1 )]⊗S (1 ) [H̃−1 (Π̄2 )]⊗S (m3 ) [H̃0 (Π̄3 )]⊗· · · .
By summing over all set partitions of rank r and taking Frobenius characteristic
one gets,
ch H̃r−2 (0̂, x) = degree n term in (−1)r hn−r (−1)m−1 chH̃m−3 (Π̄m ) .
x ∈ Πn m≥1
r(x) = r
LECTURE 5. POSET OPERATIONS AND MAPS 591
Exercise 5.1.13 ([177, 178]). Use Theorems 5.1.5 and 5.1.9 to compute the rep-
resentation of Xi Smi [Si ] on Hi (0̂, x), where mi is equal to the number of blocks
of x of size i and (0̂, x) is an interval in the
(a) d-divisible partition lattice Π0ndmod d ,
(b) 1 mod d partition poset Π1nd+1 mod d
.
Computations such as those of Example 5.1.12 and Exercise 5.1.13 are used in
applications of equivariant versions of the Orlik-Solomon formula and the Goresky-
MacPherson formula. This is discussed further in Section 5.4.
The examples above don’t adequately demonstrate the power of the product
theorems because in these examples homology occurs in only one dimension. A
demonstration of the full power can be found in [178], [184] and [201], where
representations on the homology of intervals of the k-equal partition lattice Πn,k ,
the at least k partition lattice Π≥k j mod d
n , the general j mod d partition poset Πnd+j ,
and the other restricted block size partition posets of Section 4.4, are computed.
Let’s look at Quillen’s original example [137]. For a finite group G, let Sp (G)
be the poset of nontrivial p-subgroups of G ordered by inclusion and let Ap (G)
be the induced subposet of nontrivial elementary abelian p-subgroups of G. The
poset Sp (G) and its order complex were studied by Brown in his work on group
cohomology. Quillen proposed the smaller poset Ap (G) as a way of studying the
homotopy invariants of the Brown complex. He used the Quillen fiber lemma
to establish homotopy equivalence between the Brown complex and the Quillen
complex. Both posets are G-posets, where G acts by conjugation.
Theorem 5.2.3 (Quillen [137]). For any finite group G and prime p,
(5.2.1) Ap (G) Sp (G),
and
(5.2.2) H̃j (Ap (G)) ∼
=G H̃j (Sp (G)) ∀j.
Proof. The inclusion map
i : Ap (G) → Sp (G)
is a G-poset map. Let us show that the fiber i−1 (Sp (G)≤H ) is contractible for all
H ∈ Sp (G) so that we can apply the Quillen fiber lemma. Note that
i−1 (Sp (G)≤H ) = Ap (H).
Since H is a nontrivial p-group, it has a nontrivial center. Let B be the subgroup
of the center consisting of all elements of order 1 or p. Consider the poset map
f : Ap (H) → {BA : A ∈ Ap (H)} defined by f (A) = BA. The fibers of this map
are contractible since they all have maximum elements. So by the Quillen fiber
lemma
Ap (H) {BA : A ∈ Ap (H)}.
Since {BA : A ∈ Ap (H)} has minimum element B, it is contractible. So Ap (H)
and thus i−1 (Sp (G)≤H ) is contractible. Hence by the Quillen fiber lemma, (5.2.1)
holds and by the equivariant homology version, (5.2.2) holds.
For certain finite groups G, the homology of the Quillen complex Δ(Ap (G)) is
well-understood; namely for groups of Lie type. Quillen [137] shows that for groups
G of Lie type in characteristic p, the Quillen complex Δ(Ap (G)) is homotopic to
a simplicial complex known as the building for G, which has the homotopy type
of a wedge of spheres of a single dimension. Webb [209] shows that the unique
nonvanishing G-equivariant homology of Ap (G) is the same as that of the building,
which is known as the Steinberg representation. For general finite groups, the
Quillen complex is not nearly as well-behaved, nor well-understood. In fact, it
was only recently shown by Shareshian [156] that for certain primes the integral
homology of the Quillen complex of the symmetric group has torsion.
The following long-standing conjecture of Quillen imparts a sense of the signif-
icance of poset topology in group theory.
Conjecture 5.2.4 (Quillen Conjecture). For any finite group G and prime p, the
poset Ap (G) is contractible if and only if G has a nontrivial normal p-subgroup.
The necessity of contractibility was proved by Quillen; the sufficiency is still
open. However significant progress has been made by Aschbacher and Smith [4].
In order to gain understanding of the Quillen complex for the symmetric group,
Bouc [47] considered the induced subposet Tn of S2 (Sn ) consisting of nontrivial
LECTURE 5. POSET OPERATIONS AND MAPS 593
(5.2.3) Tn Tn Mn
and for all i,
(5.2.4) H̃i (Tn ) ∼
=Sn H̃i (Tn ) ∼
=Sn H̃i (Mn ).
In addition to computing the representation of the symmetric group on the homol-
ogy of the matching complex, Bouc discovered torsion in the integral homology of
the matching complex. See [157], [202] and [105] for further results on torsion in
the matching complex.
Exercise 5.2.5. Prove (5.2.3) and (5.2.4).
The usefulness of the matching complex in understanding the topology of the
Quillen and Brown complexes was recently demonstrated by Ksontini [116, 117].
He used simple connectivity of Mn for n ≥ 8, which was proved by Bouc, to establish
simple connectivity of S2 (Sn ) for n ≥ 8. It was also shown by Ksontini [116, 117,
118], Shareshian [156], and Shareshian and Wachs [158] that a hypergraph version
of the matching complex is useful in studying the topology of Δ(Sp (Sn )) when
p ≥ 3.
Exercise 5.2.6. Let x̄ be a closure operator on P , i.e. x ≤ x̄ and x̄¯ = x̄ for all
x ∈ P . Define cl(P ) := {x ∈ P : x̄ = x}.
(a) Use the Quillen fiber lemma to prove:
P cl(P ).
(b) Use (a) to show that
W̄(n, k) N̄ (n, k).
(We already know this and (c) from Exercise 4.3.2.)
(c) Derive an equivariant homology version of the homotopy result in (a) and
use it to prove
H̃i (W̄(n, k)) ∼
=Sn H̃i (N̄ (n, k)) ∀i.
Exercise 5.2.7 (Homology version of Rota’s crosscut theorem). Let L be a lattice
and let M be the subposet of L̄ consisting of non-0̂ meets of coatoms. Prove:
(a) L̄ M .
(b) If G is a group acting on L then for all i, H̃i (L̄) ∼
=G H̃i (M ).
(c) Let Γ(L) be the simplicial complex whose vertex set is the set of coatoms
of L and whose faces are sets of coatoms whose meet is not 0̂ (this is
known as the cross-cut complex of L). Show L̄ Γ(L) and if G is a group
acting on L then for all i, H̃i (L̄) ∼
=G H̃i (Γ(L)).
The next two examples are connected with the combinatorics of knot spaces
and arose in the work of Vassiliev [193, 194, 195].
594 WACHS, POSET TOPOLOGY
Example 5.2.8. Let NCGn be the poset of disconnected graphs on node set [n]
ordered by inclusion of edge sets. Let
f : NCGn → Π̄n
be the poset map such that f (G) is the partition of [n] whose blocks are the node
sets of the connected components of G. The fibers of f are given by
Note that the representation on the right side of (c) is the Whitehouse module
discussed in Section 4.5.2. We describe the proof method of Babson, Björner,
Linusson, Shareshian, and Welker. Define the poset map
2
f : NCGn → Bn−1 × Πn−1
by letting
f (G) = (S, π),
LECTURE 5. POSET OPERATIONS AND MAPS 595
2 4
6
1 5
3
where S is the set of nodes joined to node n by an edge, and π is the partition
whose blocks correspond to the connected components of G \ {n}. For example, if
G is the not 2-connected graph in Figure 5.2.1 then
f (G) = ({2, 4, 6}, 123/45/6).
Some of the fibers have maximum elements; so one sees immediately that they
are contractible. For example,
f −1 ((0̂, ({2, 4, 6}, 123/45/6)]) = (0̂, G],
where G is the graph in Figure 5.2.1. But it is not so easy to see that the fibers with-
out maximum elements are contractible. For example, the fiber
f −1 ((0̂, ({1, 2}, 123456)]) has no maximum. To show that such fibers are con-
tractible, Babson, Björner, Linusson, Shareshian, and Welker used discrete Morse
theory, which had just been introduced by Forman [72]. In fact, this was the first
of many applications of discrete Morse theory in topological combinatorics.
Now by the Quillen fiber lemma, we have
2
NCGn Bn−1 × Πn−1 .
Since the product of shellable bounded posets is shellable (Theorem 3.1.10), and
the proper part of a shellable poset is shellable (Corollary 3.1.9), Bn−1 × Πn−1 has
the homotopy type of a wedge of (2n − 5)-spheres. The number of spheres is the
absolute value of the Möbius invariant, which by Proposition 1.2.1 and (3.2.1) is
(n − 2)!. Hence Part (a) of the theorem holds.
For Part (b), we use the equivariant homology version of the Quillen fiber
lemma. Clearly the map f commutes with the permutations that fix n. Hence,
2
H̃2n−5 (NCGn ) ↓S ∼
Sn−1 =Sn−1 H̃2n−5 (Bn−1 × Πn−1 ).
n
Shareshian [154] uses discrete Morse theory to determine the homotopy type of
2
CGn directly without resorting to Theorem 5.2.11, and to obtain bases for homology
2
of CGn , thereby solving another problem of Vassiliev [195]. The only value of k
other than k = 1, 2, n−2, for which anything significant is known about the topology
of the not k-connected graph complex is k = 3. Discrete Morse theory is used to
obtain the following result.
3
Theorem 5.2.13 (Jonsson [104]). NCGn has the homotopy type of a wedge of
(n − 3)(n − 2)!/2 spheres of dimension 2n − 4.
Problem 5.2.14. What can be said about the representation of Sn on the homol-
3
ogy of NCGn ?
k
Problem 5.2.15. ([8]) What can be said about the topology of NCGn for 3 < k <
k
n − 2? When does NCGn fail to be a wedge of spheres? When does its integral
n−2
homology have torsion? Recall that the integral homology of NCGn has torsion
(Exercise 5.2.12 (b)).
We will refer to a poset map f : P → Q such that for all q ∈ Q the fiber
Δ(f −1 (Q≤q )) is l(f −1 (Q<q ))-connected as being well-connected. Note that the
connectivity condition implies that each fiber f −1 (Q≤q ) is nonempty.
Example 5.3.2. Let f : P → Q be the poset map depicted in Figure 5.3.1. For the
two maximal elements of Q the fiber Δ(f −1 (Q≤q )) is a 1-sphere. For the bottom
element of Q the fiber Δ(f −1 (Q≤q )) is a 0-sphere, and Δ(Q>q ) is a 0-sphere too.
So in either case Δ(f −1 (Q≤q )) ∗ Δ(Q>q ) is homeomorphic to a 1-sphere. Hence
the simplicial complex on the right side of (5.3.1) has a 1-sphere attached to each
element of Q. Thus Theorem 5.3.1 determines Δ(P ) to have the homotopy type
of a wedge of three 1-spheres. One can see this directly by observing that Δ(P ) is
homeomorphic to two 1-spheres intersecting in two points.
LECTURE 5. POSET OPERATIONS AND MAPS 597
1 2 3 4
f
P Q
5 6
In [43], a version of the general fiber theorem for homology over the integers
or over any field is also given, as are equivariant homotopy and homology versions.
We state only the equivariant homology version here.
Theorem 5.3.3 (Björner, Wachs, Welker [43]). Let f : P → Q be a G-poset map.
If f −1 (Q≤q ) is l(f −1 (Q<y ))-acyclic for all q ∈ Q then
(5.3.3)
r #G
⏐
H̃r (P ) ∼
=G H̃r (Q) ⊕ H̃i (f −1 (Q≤q )) ⊗ H̃r−i−1 (Q>q ) ⏐ .
Gq
q∈Q/G i=−1
The general fiber theorems are proved using techniques of [210, 222] involving
diagrams of spaces and spectral sequences, which were developed to study subspace
arrangements. In Section 5.4, we discuss the connection between the general fiber
theorems and subspace arrangements.
Example 5.3.4. Theorem 5.3.3 can be applied to the well-connected poset map
f : P → Q given in Example 5.3.2. Let G be the cyclic group S2 whose non-
identity element acts by (1, 2)(3, 4) on P and trivially on Q. The map f is clearly
a G-poset map. For each q ∈ Q, we have Gq = S2 . If q is the bottom element
of Q then the fiber f −1 (Q≤q ) is S2 -homeomorphic to a 0-sphere on which S2
acts trivially. The same is true for Q>q . It follows that the representation of S2
on H̃0 (f −1 (Q≤q )) ⊗ H̃0 (Q>q ) is the trivial representation S (2) . If q is one of the
maximal elements of Q then the fiber f −1 (Q≤q ) is S2 -homeomorphic to a circle
with (1, 2)(3, 4) acting by reflecting the circle about the line spanned by a pair
of antipodal points. Hence the representation of S2 on H̃1 (f −1 (Q≤q )) is the sign
2
representation S (1 ) . Since Q>q is the empty simplicial complex, the representation
of S2 on H̃−1 (Q>q ) is the trivial representation. It follows that the representation
2
of S2 on H̃1 (f −1 (Q≤q )) ⊗ H̃−1 (Q>q ) is S (1 ) . We conclude from (5.3.3) that the
2 2
S2 -module H̃1 (P ) decomposes into S (2) ⊕ S (1 ) ⊕ S (1 ) and that H̃i (P ) = 0 for
i = 1.
598 WACHS, POSET TOPOLOGY
1-2-3
Figure 5.3.2. P3
Example 5.3.5. Consider the action of the hyperoctahedral group Sn [Z2 ] on the
type B partition lattice ΠB
n . Define the bar-erasing map
(5.3.4) n → Π̄{0,1,...,n}
f : Π̄B
by letting f (π) be the partition obtained by erasing the bars from the barred
partition π. For example
f (0 4 7 /1 5̄ 6 9̄ /2 3̄) = 0 4 7 /1 5 6 9 /2 3
It is clear that f is a Sn [Z2 ]-poset map. We establish well-connectedness in the
following exercise.
Exercise 5.3.6 (Wachs [204]). Let Pn be the subposet of ΠB n consisting of barred
partitions whose zero-block is a singleton, where the zero-block is the block that
contains 0. The poset P3 with the zero-block suppressed is given in Figure 5.3.2.
Show the following.
(a) The fibers of the bar-erasing map have the form
where S (d−1)
is the unit (d − 1)-sphere in R . Recall that L(A) denotes the inter-
d
section lattice of A (ordered by reverse inclusion). Let L̄(A) denote L(A) \ {0̂, 1̂}
if A is essential (i.e., ∩A = {0}), and L(A) \ {0̂} otherwise.
Theorem 5.4.1 (Ziegler & Živaljević [222]). Let A be a linear subspace arrange-
ment. Then
! "
(5.4.1) VAo Δ(L̄(A)) ∨ suspdim x (Δ(0̂, x)) : x ∈ L̄(A) ,
where the wedge is formed by identifying each vertex x in Δ(L̄(A)) with any point
in suspdim x (Δ(0̂, x)). Consequently, for each i,
(5.4.2) H̃i (VAo ; Z) ∼
= H̃i−dim x ((0̂, x); Z).
x∈L(A)\{0̂}
contained in VAo . Note that Δ(FH (X)) is a triangulation of the (dim X − 1)-sphere
X ∩ S(d−1) and Δ(FH (A)) is a triangulation of VAo .
Now let
f : FH (A) → L̄(A)∗
be defined by &
f (τ ) = Xi .
i:Xi ⊇τ
Clearly f is order preserving. We claim that f is well-connected. Observe that for
all X ∈ L̄(A),
f −1 ((L̄(A)∗ )≤X ) = FH (X).
So f −1 ((L̄(A)∗ )≤X ) is a (dim X − 1)-sphere, which is (dim X − 2)-connected. Since
f −1 ((L̄(A)∗ )<X ) has length at most dim X − 2, f is indeed well-connected, and
thus (5.4.1) follows from Theorem 5.3.1.
The reason we refer to Theorem 5.4.1 as a homotopy version of the Goresky-
MacPherson formula is that the Goresky-MacPherson formula (for linear subspace
arrangements) follows from (5.4.2) by Alexander duality. Indeed, one can simply
apply Theorem 5.1.1 to the subposet FH (A) of F (H). A homotopy version of the
Goresky-MacPherson formula for general affine subspace arrangements is also given
in [222].
Now let G be a finite subgroup of the orthogonal group Od that maps subspaces
in A to subspaces in A. We say that A is a G-arrangement. Clearly G acts as a
group of poset maps on L(A) and as a group of homomorphisms on H̃i (VAo ) and on
H̃i (MA ). For each x ∈ L(A), let Sx be the (dim x − 1)-sphere x ∩ S(d−1) . Each g
in the stabilizer Gx acts as an orientation preserving or reversing homeomorphism
on Sx . Define
1 if g is orientation preserving
sgnx (g) :=
−1 if g is orientation reversing.
By viewing g as an element of GL(x), we have sgnx (g) = det(g). Note that
H̃dim x−1 (Sx ) is a one dimensional representation of Gx whose character is given by
sgnx .
Theorem 5.4.2 (Sundaram and Welker [186]). Let A be a G-arrangement of linear
subspaces in Rd . Then for all i,
(5.4.3) H̃i (VAo ) ∼
=G (H̃i−dim x (0̂, x) ⊗ sgnx ) ↑G
Gx .
x∈L(A)\{0̂}/G
Consequently,
(5.4.4)H̃ i (MA ) ∼
=G (H̃d−2−i−dim x (0̂, x) ⊗ sgnx ⊗ sgn0̂ ) ↑G
Gx .
x∈L(A)\{0̂}/G
Exercise 5.4.3. Use Theorems 5.3.3 and 5.1.1 to prove Theorem 5.4.2.
In [186] Theorem 5.4.2 is applied to subspace arrangements whose intersection
lattices are the k-equal partition lattice discussed in Section 3.2.4 and the d-divisible
partition lattice discussed in Exercises 4.3.6 and 4.4.8. By computing the multiplic-
ity of the trivial representation in H̃ i (MA ), Sundaram and Welker obtain the Betti
number formula of Arnol’d given in (3.2.3) and an analogous formula for the space
LECTURE 5. POSET OPERATIONS AND MAPS 601
of monic polynomials of degree n, with at least one root multiplicity not divisible
by d. For another recent approach to studying these Betti numbers, see [113].
In [201], Theorem 5.4.2 is applied to the subspace arrangement whose intersec-
tion lattice is the restricted block size partition lattice ΠSn ∪ {0̂} of Theorem 4.4.11.
Since each interval (0̂, x) of ΠSn is the product of smaller ΠSm , one uses Theo-
rem 4.4.11, and the product formulas of Section 5.1 to compute the homology
of (0̂, x) (see Example 5.1.12 and Exercise 5.1.13). This yields,
Theorem 5.4.4 (Wachs [201]). Suppose S ⊆ {2, 3, . . . } is such that S and {s −
min S : s ∈ S} are closed under addition (eg., S = {k, k + 1, . . . } or more generally
S = {kd, (k + 1)d, . . . }). For n ∈ S, let MnS be the manifold
{z := (z1 , . . . , zn ) ∈ Cn : for some i the number of occurrences of zi is not in S}.
Then
[−1]
chH̃ m (MnS ) un (−t)2n−m−1 = h i ti hi hi ui tφ(i) ,
m∈Z i≥1 i≥1 i∈S
n∈S
where φ(i) := max{j ∈ Z+ : i − (j − 1) min S ∈ S}. (Our notation reflects the fact
that plethysm is associative.)
We leave it as an open problem to use this formula to compute the multiplicity
of the trivial representation in H̃ m (MnS ), and thereby obtain a formula (which
generalizes the above mentioned Sundaram-Welker formula) for the Betti numbers
of the space of monic polynomials of degree n with at least one root multiplicity
not in S.
work as a more general example of a Tits coset complex. One can easily see that
r
Mm,n is the (r, r, . . . , r)-inflation of Mm,n .
Exercise 5.5.6 (Wachs [203]). Use Theorems 5.5.2, 5.5.4, and 5.5.5 to obtain a
r
colored version of Theorem 5.5.5, i.e. a generalization to Mm,n , where r ≥ 1.
For open problems on the topology of chessboard complexes, matching com-
plexes and related complexes, see [202] and [157].
604 WACHS, POSET TOPOLOGY
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Convex Polytopes:
Extremal Constructions
and f -Vector Shapes
Günter M. Ziegler
IAS/Park City Mathematics Series
Volume 14, 2004
Convex Polytopes:
Extremal Constructions
and f -Vector Shapes
Günter M. Ziegler
Introduction
These lecture notes treat some current aspects of two closely interrelated top-
ics from the theory of convex polytopes: the shapes of f -vectors, and extremal
constructions.
The study of f -vectors has had huge successes in the last forty years. The
most fundamental one is undoubtedly the “g-theorem,” conjectured by McMullen
in 1971 and proved by Billera & Lee and Stanley in 1980, which characterizes the f -
vectors of simplicial and of simple polytopes combinatorially. See also Section 5.2
of Forman’s article in this volume, where h-vectors are discussed in connection
with the Charney–Davis conjecture. Nevertheless, on some fundamental problems
embarassingly little progress was made; one notable such problem concerns the
shapes of f -vectors of 4-polytopes.
A number of striking and fascinating polytope constructions have been pro-
posed and analyzed over the years. In particular, the Billera–Lee construction
produces “all possible f -vectors” of simplicial polytopes. Less visible progress was
made outside the range of simple or simplicial polytopes — where our measure of
progress is that new polytopes “with interesting f -vectors” should be produced.
Thus, still “it seems that overall, we are short of examples. The methods for
coming up with useful examples in mathematics (or counterexamples to commonly
believed conjectures) are even less clear than the methods for proving mathematical
statements” (Gil Kalai, 2000).
These lecture notes are meant to display a fruitful interplay of these two ar-
eas of study: The discussion of f -vector shapes suggests the notion of “extremal”
polytopes, that is, of polytopes with “extremal f -vector shapes.” Our choice of con-
structions to be discussed here is guided by this: We will be looking at constructions
that produce interesting f -vector shapes.
2007
c Günter M. Ziegler
619
620 GÜNTER M. ZIEGLER, CONVEX POLYTOPES
After treating 3-polytopes in the first lecture and the f -vector shapes of very
high-dimensional polytopes in the second one, we will start to analyze the case of
4-dimensional polytopes in detail. Thus the third lecture will explain a surprisingly
simple construction for 2-simple 2-simplicial 4-polytopes, which have symmetric f -
vectors. Lecture four sketches the geometry of the cone of f -vectors for 4-polytopes,
and thus identifies the existence/construction of 4-polytopes of high “fatness” as a
key problem. In this direction, the last lecture presents a very recent construction
of “projected products of polygons,” whose fatness reaches 9 − ε. This shows that,
on the topic of f -vectors of 4-polytopes, there is a narrowing gap between “the
constraints we know” and “the examples we can construct.”
All the polytopes considered in these lecture notes are convex. A d-polytope is
a d-dimensional polytope; thus the 3-dimensional polytopes to be discussed in this
lecture are plainly 3-polytopes.*
How many 3-dimensional polytopes “do we know”? When pressed for examples,
we will perhaps start with the platonic solids: the regular tetrahedron, cube and
octahedron, icosahedron and dodecahedron.
The classes of stacked and cyclic polytopes are of great importance for high-
dimensional polytope theory because of their extremal f -vectors (according to the
lower bound theorem and the upper bound theorem): Stacked polytopes arise from
a simplex by repeatedly stacking pyramids onto the facets (cf. Lecture 2); cyclic
polytopes are constructed as the convex hull of n > d points on a curve of order d.
However, neither of these constructions produces particularly impressive objects in
dimension 3 (compare Figure 1.2, and Exercise 1.8).
The same must be said about pyramids and bipyramids over n-gons (n ≥ 3)
— see Figure 1.3.
*We assume that the readers are familiar with the basic terminology and discrete geometric
concepts; see e.g. [79, Lect. 0] or [40].
621
622 GÜNTER M. ZIEGLER, CONVEX POLYTOPES
Figure 1.3. The pyramid and the bipyramid over a regular 10-gon
f2
15
14
13 f2 ≤ 2f0 − 4
12 (equality:
simplicial
11 polytopes)
10
9
8
7
6
5 f0 ≤ 2f2 − 4
4 (equality:
simple polytopes)
4 5 6 7 8 9 10 11 12 13 14 15 f0
which is tight exactly for the f -vectors of simplicial polytopes, and its dual, f0 ≤
2f2 − 4, which in the case of equality characterizes the f -vectors of simple 3-
polytopes.
For the centennial of Steinitz’ lemma, in 2006, let’s strive for a characterization
of the cone spanned by the f -vectors of 4-dimensional polytopes, cone(F4 ). As we
will see at the beginning of Lecture 4, this is a much more modest goal than a
characterization of F4 , which is not the set of all integral points in a convex set: It
has “concavities” and even “holes.”
Steinitz’ lemma, as graphed in Figure 1.5, also shows that all (f -vectors of)
convex 3-polytopes lie between the extremes of simple and of simplicial polytopes.
And indeed, there seems to be the misconception that an analogous statement
should be true in higher dimensions as well — it isn’t. As we will see, there are
additional interesting extreme cases in dimension 4, which are by far not as well
understood as the simple and simplicial cases.
For any 3-polytope that is not a simplex, we may compute the “slope”
f2 − 4
φ(P ) :=
f0 − 4
it generates in the graph of Figure 1.5, with respect to the apex (4, 4) of the cone,
which corresponds to a simplex. This slope satisfies
1
2 ≤ φ(P ) ≤ 2,
where the lower bound characterizes simple polytopes, while the upper bound is
tight for simplicial polytopes. Another interpretation of the parameter φ is that it
is a homogeneous coordinate for the cone, where the denominator f0 − 4 measures
the “size” of the f -vector. (φ is homogeneous, so it yields 00 for the f -vector of a
simplex, which is the apex of the cone. Compare Exercise 1.5.)
LECTURE 1. CONSTRUCTING 3-DIMENSIONAL POLYTOPES 625
Steinitz type proofs. Such proofs (of which Steinitz gave details on one in [74], and
three are given in the Steinitz–Rademacher book [75] that appeared after Steinitz’
death), are based on the following principle. Any planar 3-connected graph can be
“reduced” to the complete graph K4 by local operations, which yields a sequence
G = G0 → G1 → G2 → . . . → GN −1 → GN = K4 .
of 3-connected planar graphs.
This reduction sequence should then be reversed: Starting with a simplex Δ3
(with graph K4 ) we build up a sequence of polytopes,
P = P0 ← P1 ← P2 ← . . . ← PN −1 ← PN = Δ3 ,
where Pi is a 3-polytope with graph Gi , again by simple/local construction steps.
Such a proof is presented in detail in [79, Lect. 4], so there is no need to do this
here. We just mention that a number of interesting extensions and corollaries may
be derived from Steinitz type proofs. Indeed, Barnette & Grünbaum [9] proved that
in the construction of the polytope P , the shape of one face of the polytope may
be prescribed. For example, some hexagon face may be required to be a regular
hexagon, which imposes a non-trivial additional constraint. Similarly, Barnette [7]
proved with a Steinitz type argument that a “shadow boundary” may be prescribed:
P may be constructed in such a way that from some view-point outside the polytope,
the edges that bound the visible part of the surface of the polytope correspond to a
prescribed simple cycle in the graph of the polytope (which need not be induced).
Ê
Equivalently, we may construct P ⊂ 3 so that the image π(P ) of P under the
Ê Ê
orthogonal projection π : 3 → 2 is a polygon whose edges are given exactly by
the edges of P that realize the prescribed cycle. Indeed, the edges must be “strictly
preserved” by the projection, in the terminology that we will develop and use in
Lecture 5.
Figure 1.7. A Tutte drawing of the icosahedron graph, and the corresponding
Maxwell–Cremona lifting
be derived from a variational principle (that is, an energy functional). In this line of
work, Bobenko & Springborn [17] have quite recently discovered an explicit, elegant
and quite general variational principle for the construction of circle patterns with
prescribed intersection angles. In the following, we prove the Steinitz theorem based
on their functional — taking advantage of all the simplifications that occur in their
proof and formulas if one wants to “just” get the orthogonal circle patterns needed
for the Steinitz theorem. (See also Springborn [68] for an additional discussion of
uniqueness.)
In our presentation of the proof, we first explain how any edge-tangent repre-
sentation of a polytope P induces a circle pattern on the sphere, which in turn yields
LECTURE 1. CONSTRUCTING 3-DIMENSIONAL POLYTOPES 629
a planar circle pattern, and the combinatorics of the planar circle pattern yields a
quad graph (a planar graph whose faces are quadrilaterals), which has G(P ) as a
subdivided subgraph. This yields steps (1) to (4) in the following scheme:
construct
facet and connect
vertex horizon stereographic circle take
circles projection centers subgraph
Our plan is to then reverse this four-step process, in order to construct an edge-
tangent polytope from the given graph G. In step (5), the quad graph is derived
directly from the graph G = G(P ), by superposing the graph with its dual. Then,
in step (6), we construct the rectangular circle pattern with the combinatorics of
the quad graph, and then proceed to construct P from it.
The steps (5), (7), and (8) are quite straightforward: The key, non-trivial step
is (6), the construction of the (unique) rectangular circle pattern, which we achieve
via the “euclidean Bobenko–Springborn functional.”
Proof. We start with a detailed description of the four-step process from edge-
tangent polytopes to planar 3-connected graphs, via circle packings and quad
graphs.
Ê
(1). Assume that P ⊂ 3 is a 3-polytope whose edges are tangent to the unit
Ê
sphere S 2 ⊂ 3 . Then the facet planes of P intersect the unit sphere S 2 in circles
that we call the facet circles: We get one circle for each facet, and the circles are
disjoint, but they touch exactly if the corresponding facets are adjacent. We also
get a second set of circles which we call the vertex horizon circles: Each such circle
is the boundary of the spherical cap consisting of all the points on the sphere that
are “visible” from the respective vertex. We get one vertex horizon circle for each
vertex, and the circles are disjoint, but they touch exactly if the corresponding
vertices are adjacent.
Moreover, at each edge tangency point, the two touching facet circles and
the two touching vertex horizon circles intersect orthogonally; see Figure 1.9 for
an example. (The vertex horizon circles of P are the facet circles of the dual
polytope P ∗ , whose edges have the same tangency points as the edges of P ; the
facet circles for P are also the vertex horizon circles for P ∗ ; corresponding edges
e ⊂ P and e∗ ⊂ P ∗ intersect orthogonally at the respective tangency point.)
(2). We perform a stereographic projection to the plane, using one of the edge
tangency points p0 as the projection center, and mapping all the facet and vertex
horizon circles to the equator plane corresponding to the projection point. In the
resulting planar figure, the two facet circles through p0 yield two parallel lines (and
after a rotation we may assume that these are horizontal); the two vertex horizon
circles through p0 also yield two parallel lines, orthogonal to the first two (and thus
630 GÜNTER M. ZIEGLER, CONVEX POLYTOPES
vertical). So we get a planar pattern that consists of four lines bounding an axis-
parallel rectangle, and circles that touch resp. intersect orthogonally in the plane.
This is the rectangular circle pattern.
If the faces adjacent to the edge f through p0 are an h1 -gon and an h2 -gon,
then we get h1 − 2 resp. h2 − 2 circles along the horizontal edges of the rectangle.
Similarly, if the end vertices of f have degrees v1 and v2 , then we get v1 − 2 resp.
v2 −2 circles along the vertical edges of the rectangle. The example that one obtains
from the cube (Figure 1.9) is displayed in Figure 1.10.
p0
Figure 1.9. The facet circles and the vertex horizon circles (dashed) for an
edge-tangent representation of a regular cube.
(3). Any rectangular circle pattern yields a quad graph drawing as follows: The
vertex set consists of the centers of all the circles, with four additional vertices
“far out” representing the four lines that bound the rectangles (as in Figure 1.11).
We obtain drawings of both G and G∗ by connecting the centers of touching facet
circles resp. vertex horizon circles. This includes one horizontal edge f of G “going
through infinity,” while dual graph G∗ has the corresponding edge f ∗ going through
infinity vertically.
From the rectangular circle pattern, we obtain a decomposition of a rectangle
into quadrilaterals by connecting the centers of adjacent facet circles, and the cen-
ters of adjacent vertex horizon circles. See the example of Figure 1.11, where the
rectangle is shaded. The graph of this rectangle decomposition is the quad graph:
Its vertices correspond to (the centers of) the facet circles that don’t contain p0 ,
the vertex horizon circles that don’t contain p0 , and intersection points of edges e
and e∗ of G and G∗ , other than the edges f, f ∗ that contain p0 .
(4). In particular, the graph G may be derived from the quad graph, by “deleting
the dashed edges.”
LECTURE 1. CONSTRUCTING 3-DIMENSIONAL POLYTOPES 631
f∗
Figure 1.11. The quad graph for the cube, generated from Figure 1.10: The
white vertices are given by the facet circle centers, while the black vertices
correspond to the vertex horizon circles; the dashed edges connect the centers
of adjacent facet circles, and the straight edges correspond to adjacent vertex
horizon circles.
This ends the description of the passage from an edge tangent polytope to the
planar graph drawing. Now we start the way back: Another four-step process leads
us from graphs via quad graphs and circle patterns to edge-tangent 3-polytopes.
632 GÜNTER M. ZIEGLER, CONVEX POLYTOPES
(5). The quad graph may be derived from knowledge of the graph G alone, plainly
by overlaying G and G∗ . For our cube example, the result may look like the drawing
given in Figure 1.12.
f∗
Figure 1.12. The quad graph for the cube, generated from an overlay of the
cube graph (black edges) layed out with the edge f “at infinity” and the dual
graph (dashed edges), with the dual edge f ∗ “at infinity.” The shaded part
defines the restricted quad graph.
The input for the next step will be the restricted quad graph: It is obtained
from the full quad graph by deleting everything that is adjacent to the original
edges f and f ∗ . Its bounded faces are quadrilaterals (quads for short), with two
black and two dashed edges each. Each quad has
• a black vertex and a white vertex
(the black vertex, where the two black edges meet, corresponds to the center of
a face circle; the white one, where the two dashed edges meet, corresponds to
the center of a horizon circle),
• and two more vertices where a black and a dashed edge meet
(they correspond to edge tangency points).
For the following, we use I0 as an indexing set for the black and white vertices in
the restricted quad graph. It is in bijection with the vertices of G and of G∗ , except
for the vertices of the edges f and f ∗ , which yield lines rather than circles. That
is, we have
I0 := V (G − f ) ∪ V (G∗ − f ∗ ).
The following step, which takes us from combinatorics (a graph drawing) to geom-
etry (a circle pattern), is the crucial one.
(6). In the “correct” realization of the restricted quad graph, which would yield a
circle packing, each quad is drawn as a kite in which
• the two black edges have the same length
(radius ri of the corresponding vertex horizon circle),
• the two dashed edges have the same length
(radius rj of the corresponding facet circle),
• and there are two right angles between black and dashed edges
(where facet and vertex horizon circles intersect).
LECTURE 1. CONSTRUCTING 3-DIMENSIONAL POLYTOPES 633
ri rj
i ϕij ϕji
j
Figure 1.13. A kite, with radii ri = eρi , rj = eρj , and angles ϕij and ϕji
r
j
(1.1) 2 arctan = Φi for all vertices i ∈ I0 ,
ri
j:i j
where the right-hand-sides are given by
π if i is on the boundary,
Φi :=
2π if i is in the interior.
In the equation whose right hand side is Φi , the sum on the left hand side is taken
over all vertices j ∈ I0 that are opposite to i in one of the kites. (If i is a white
vertex, then j will be black, and vice versa.)
Indeed, if (1.1) is satisfied, then we can easily construct the kites and piece
them together to get a flat rectangle and the circle packing. Badly enough, (1.1) is
a non-linear system of equations, which we have to solve in positive variables ri > 0.
We want to know that this has a solution, which is unique up to multiplying all the
ri s with the same factor, and which can be computed efficiently. Luckily, we can do
this, since the system is solved by minimizing an explicit and easy-to-write-down
“energy” functional which will turn out to be convex, with a unique minimum. For
this, we first do a change of variables,
ρi := log ri .
Then we normalize by the condition i ri = 1, that is,
ρi = 0.
i
634 GÜNTER M. ZIEGLER, CONVEX POLYTOPES
Furthermore, we define
f (x) := arctan(ex ).
This auxiliary function is graphed in Figure 1.14. Note that f (−x) = π
2 − f (x).
arctan(ex )
1.5
1.0
0.5
x
−4 −2 2 4
Figure 1.14. f (x) = arctan ex
We differentiate f ,
1 1
f (x) = 2x
ex = ,
1+e 2 cosh x
Ê
which yields f (−x) = f (x) > 0 for all x ∈ . We also integrate f , and define
x
F (x) := f (t)dt.
−∞
(1.4) BS(ρ) := F (ρj −ρi ) + F (ρi −ρj ) − π2 (ρi + ρj ) + Φi ρ i ,
i j i ∈ I0
where the first sum is over all unordered pairs {i, j} of vertices i, j ∈ I0 that are
opposite in one of the kites. The claim is now that
(A) the critical points of BS(ρ) are exactly
the solutions to our system (1.3),
(B) the functional is convex: Restricted to i ρi = 0 it is strictly positive definite,
so the critical point is unique if it exists, and
(C) the functional gets large if any of the differences ρi − ρj gets large: Thus
the functional must have a critical point (a minimum) — the solution we are
looking for.
For (A), a simple computation yields the gradient of BS(ρ):
∂BS(ρ)
= Φi − 2f (ρj − ρi ).
∂ρi
i j
LECTURE 1. CONSTRUCTING 3-DIMENSIONAL POLYTOPES 635
Thus the critical points of BS(ρ) are exactly the solutions to (1.3).
For (B), we compute the Hessian (the matrix of second derivatives) for BS(ρ),
and find that
xT BS(ρ) x = 2 f (ρj − ρi ) (xj − xi )2 .
i j
We know that f (ρj − ρi ) > 0, so this quadratic form can vanish only if all the
differences xj − xi vanish for “adjacent” i, j ∈ I0 (that is, for black/white vertices
that share a kite). But the graph we
consider is connected, so this implies that all
variables xi are equal. Restricted to i xi = 0 this yields that all xi vanish, so the
Hessian is positive definite on the restriction hyperplane, and the solution we are
striving for is unique if it exists.
To prove the existence claim (C), we have to find that BS(ρ) grows large if any
difference of variables ρk − ρi gets large. With the same argument we just used
this implies that some difference of “adjacent” variables will become large. Then
also F (ρj −ρi ) + F (ρi −ρj ) ≥ π2 |ρj − ρi | gets large, but it will grow only linearly in
|ρj − ρi |, and it is not obvious that the growing positive terms in (1.4) will “outrun”
the negative terms. This will require a careful “matching” between positive and
negative terms.
To achieve this, we use the existence of a coherent angle system, that is, an
assignment of angles ϕij , ϕji > 0 to the kites that satisfies the conditions
(1.5) ϕij + ϕji = π2 and 2ϕij = Φi .
j:i j
Any solution to (1.1) would give us a coherent angle system, but the existence of
such a coherent angle system is much weaker, far from solving the system (1.1): If
we have a coherent angle system, then we could construct kites from this — whose
angles would fit together at the black and white vertices, but whose side lengths
might not. (Compare Figure 1.15.)
For any coherent angle system, ε0 := min ϕk is a positive number.
k,
If there is a coherent angle system, then the minimum exists. Let’s assume for now
that a coherent angle system exists (this will be proved below). Then
BS(ρ) = F (ρj −ρi ) + F (ρi −ρj ) − π2 (ρi + ρj ) + Φi ρ i
i j i
(i) π
π − (ρi + ρj )
> 2 ρ i − ρ j 2 + Φi ρ i
i j i
π
(ii) π
= 2 ρi − ρj − 2 (ρi + ρj ) + 2(ϕij ρi + ϕji ρj )
i j i j
(iii)
= −π min{ρi , ρj } + 2(ϕij ρi + ϕji ρj )
i j i j
(iv)
≥ −π min{ρi , ρj } + π min{ρi , ρj } + 2 min{ϕji , ϕij }|ρi − ρj |
i j i j
= 2 min{ϕji , ϕij }|ρi − ρj | ≥ 2ε0 |ρi − ρj |.
i j i j
636 GÜNTER M. ZIEGLER, CONVEX POLYTOPES
π π π
7π
3 6 16
π 7π 16
3 16
π π π π
3 3 6 16 7π
π π 16 π
6 π π
3 16
6 5π 4
π π 16 π
6 6 3π 4
16
π 5π 3π
3 16 16
π
3 π 5π
6 3π
16
16
Figure 1.15. The assignment in this figure is a coherent angle system – but
not one that corresponds to a correct circle pattern.
(Note that the construction of the coherent angle system proceeds from the
plane graph without use of a straight edge drawing. In the figures further
down we draw the graphs with straight edges for simplicity, but this structure
is not used in the proof. Rather, it is produced by the proof.)
Here
• the estimate for (i) uses F (x) + F (−x) ≥ π2 |x|, which is (1.2).
• (ii) is obtained by substituting (1.5). We need the second term in the second
sum in (ii) since the sums over “i j” are sums over unordered pairs; there is
no extra summand for “j i.”
• (iii) follows from |x − y| − (x + y) = −2 min{x, y},
• For (iv), in the case ρj ≥ ρi we compute
2(ϕij ρi + ϕji ρj ) = πρi − 2ϕji ρi + 2ϕji ρj
= π min{ρi , ρj } + 2ϕji |ρi − ρj |
≥ π min{ρi , ρj } + 2 min{ϕji , ϕij }|ρi − ρj |,
and analogously for ρi ≥ ρj .
We are dealing with a connected quad graph. Thus if the norm of the vector ρ
gets large, while the sum of the ρi is zero, then also for two i, j ∈ I0 in the same
quadrilateral the difference |ρi − ρj | gets large. Thus by the computation above,
BS(ρ) > 2ε0 |ρi − ρj | gets large. This is sufficient to prove that the strictly convex
function BS(ρ) does have a (unique) minimum — the solution to our problem.
A coherent angle system exists. Finally, we have to verify the existence of a coherent
angle system. We will see here that via some simple network flow theory, this follows
from an expansion property in the “diagonal graph” D(G ∪ G∗ ). After that, we
will prove the expansion property.
Let G be a 3-connected planar graph, G∗ its dual, both of them again drawn
into the plane with dual edges f, f ∗ intersecting “at infinity.” Then the diagonal
graph D = D(G ∪ G∗ ) has the same vertex set as G ∪ G∗ . Its edges correspond to
the diagonals in the quad graph given by G ∪ G∗ .
Equivalently, the diagonal graph D has black vertices corresponding to the
vertices of G, and white vertices corresponding to the faces of G. The edges of D
correspond to the vertex–face incidences of G. See Figure 1.16 for an example.
The reduced diagonal graph D = (V , E ) is obtained from the diagonal graph
D = (V, E) by removing the two vertices of f , the two vertices of f ∗ , and the four
LECTURE 1. CONSTRUCTING 3-DIMENSIONAL POLYTOPES 637
edges that connect them, but none of the others. So indeed, D does have pending
edges (half-edges) which have lost one of their end-vertices.* See Figure 1.17 for an
example.
Figure 1.16. The diagonal graph D = D(G ∪ G∗ ), given by the fat edges,
where G is the graph of the cube
Figure 1.17. The fat edges in this figure display the reduced diagonal graph
D = D (G ∪ G∗ ) in the case where G is the graph of the cube, derived from
Figure 1.16. Note that the fat edges leaving the rectangle are included in D ,
their vertices at the other end are not. So in this example D has 10 vertices
and 20 edges, including 6 half-edges with only one end-vertex.
*I am sure you won’t be troubled too much by the fact that this is not a graph in the usual
technical sense, since it does have half-edges with only one end-point.
638 GÜNTER M. ZIEGLER, CONVEX POLYTOPES
The diagonal graph D = D(V, E) is a quad graph: All its faces, including the
“unbounded” face (if we draw it in the plane) are quadrilaterals. From this, we get
by double counting that 2|F | = |E| and thus |V | = 2|E| − 4 by Euler’s relation.
The reduced quad graph D = (V , E ) has |V | = |V |− 4 vertices and |E | = |E|− 4
edges. Hence we get |E | = 2|V |: The reduced quad graph has exactly double as
many edges as vertices.
The concept of a coherent angle system has a very nice interpretation in terms
of the restricted diagonal graph: Each vertex vi gets a weight of 2π, and this has
to be distributed to the edges e incident to vi such that
• each edge e incident to vi gets a positive part of the weight 2π of v,
• all of the weight 2π of vi is distributed to its incident edges, and
• the weights assigned to each edge sum to π.
Indeed, in such an assignment any half-edge clearly gets a weight of π from its
only end-vertex, which corresponds to a boundary vertex of the restricted quad
graph; thus the boundary vertex vi distributes a weight of exactly Φi = π to its
other incident edges, that is, to the (diagonals of the) kites it is incident to. The
vertices of D without an incident half-edge correspond to interior vertices vj of the
restricted quad graph, so they have a weight/angle of Φj = 2π to distribute to the
incident edges/kites.
The capacity of an (s, t)-cut in a network with upper and lower bounds is the
sum on the upper bounds of the forward arcs, minus the sum of the lower bounds
on the backward arcs across the cut. So in our example the cuts [{s}, V ∪ E ∪ {t}]
and [{s} ∪ V ∪ E , {t}] have capacity 2π|V | = π|E |. Could there be a cut of
smaller capacity? Any (s, t)-cut is of the form
V E
[0, 2π] [ε, ∞] [0, π]
s t
E1
s t
V2
E2
Figure 1.19. The dashed line indicates the cut [{s} ∪ V1 ∪ E1 , V2 ∪ E2 ∪ {t}]
in our network
for partitions V = V1 V2 and E = E1 E2 . Such a cut has finite capacity if
there are no arcs (v , e ) from V1 to E2 ; compare Figure 1.19. That is, we should
take E1 to include all the edges that are incident to a vertex in V1 .
The capacity of the cut [{s} ∪ V1 ∪ E1 , V2 ∪ E2 ∪ {t}] is
2π|V2 | + π|E1 | − ε|A(V2 , E1 )| = 2π|V | − 2π|V1 | + π|E1 | − ε|A(V2 , E1 )|,
where |A(V2 , E1 )| denotes the number of arcs from V2 to E1 . For small enough ε,
say ε = 1/|A(V , E )|, we have ε|A(V2 , E1 )| < 1. Thus the following “expansion
property” for the diagonal graph implies that all cuts [{s} ∪ V1 ∪ E1 , V2 ∪ E2 ∪ {t}]
have capacity larger than 2π|V |, except in the two trivial cases given as examples
above, where the capacity is exactly 2π|V |. Thus the maximal flow, of value 2π|V |,
exists; it is positive, and yields the coherent angle system.
640 GÜNTER M. ZIEGLER, CONVEX POLYTOPES
Expansion in the diagonal graph. It remains to verify the following: Let V1 ⊆ V be
a set of vertices in the reduced diagonal graph D (G ∪ G∗ ) = (V , E ), and assume
that E1 ⊆ E includes all edges of D that are incident to a vertex in V1 . Then
(1.6) |E1 | ≥ 2|V1 |,
with equality only in the trivial cases V1 = ∅ and V1 = V .
For this we may assume that the subgraph induced by V1 is connected, because
we can consider its components separately. We may also assume that |V1 | ≥ 2, so
V1 contains both a black and a white vertex.
Now let U be an open subset of the plane (or of S 2 ) whose boundary curves
separate V1 from the h + 1 components of the graph D \ V1 , as illustrated in
Figure 1.20. Topologically, U is an open disk with h ≥ 0 holes.
The diagonal graph yields a cell decomposition of U , consisting of f0 = |V1 |
vertices, f1int interior edges, f1bdy other (half-)edges, q quadrilateral faces, and b1 +
b2 + b3 boundary faces, where bi counts the faces with i vertices in I . In particular
the total number of edges is f1 = f1int + f1bdy = |E1 |,
For this we count the vertices v of D \ V1 which are adjacent to V1 , that is, such
that some quad in the full quad-graph D contains both v and a vertex from V1 .
Walking along the boundary curves of U , and exploring the quads that we traverse
that way, we see that there are not more than 2b1 + b2 such vertices v: We find
at most two new vertices in any quad that contains a boundary cell with 1 vertex
in V1 , and at most one new vertex in the quad of a boundary cell with 2 vertices
in V1 . The vertices found during the walk need not be all distinct, and some may
not even lie outside V1 (compare Figure 1.21). Thus we get only an inequality,
2b1 + b2 ≥ #{vertices of D \ V1 adjacent to V1 }.
In the boundary of each “hole” of U we will discover at least one vertex of D \ V1 .
In the outer face during our walk we even discover a cycle of D (see Figure 1.21).
Since D is bipartite, this cycle has even length. In the trivial case of V1 = V this
is exactly the 4-cycle C given by D \ D . If V1 = V , then the vertices we discover
either yield the cycle C plus additional vertices, or we find a different cycle. But
any cycle other than C must have at least 6 vertices: Indeed, it is an even cycle,
on which black and white vertices alternate. The black vertices on the cycle either
include both the vertices of f , or with respect to the original graph G they separate
a black vertex in V1 from a vertex of f ; from the 3-connectivity of G we thus get
that the cycle contains at least three black vertices, that is, at least 6 vertices in
total. The same holds for the white vertices, the dual graph G∗ , which is also
3-connected, and the vertices of f ∗ . Thus
#{vertices of D in the boundary of U} ≥ 4,
with equality only if V1 = V . This completes the proof for the expansion property,
and thus for the existence of a coherent angle system, and of the circle packing.
Figure 1.21. The cycle in the outer face to be discovered during the walk
along the boundary curve of U is drawn with fat edges; it is a 6-cycle. With
respect to G, which is drawn in thin black lines, the three vertices of the 6-cycle
separate a vertex of f from the two black vertices in V1 .
642 GÜNTER M. ZIEGLER, CONVEX POLYTOPES
(7), (8). Given a correct rectangular circle pattern, it is easy to reconstruct the
spherical circle pattern (via an inverse stereographic projection). From this, we
obtain the edge-tangent polytope: Its face planes are given by the facet circles (and
its vertices are given by the cone points for which the vertex horizon circles do
indeed appear on the horizon). Thus construction steps (7) and (8) are easy — the
hard part was (6).
Is this the perfect proof? I think it is really nice, but still one could dream of
a proof that avoids the stereographic projection, and produces the circle packing
directly from some functional on the sphere . . . .
Exercises
1.1. Show that each 3-polytope has a triangle face, or a simple vertex (a vertex
of degree 3), or both. Even stronger, show that the number of triangle faces
plus the number of simple vertices is at least eight, so there are at least four
triangle faces, or at least four simple vertices.
Hint: Use the Euler equation.
1.2. Prove that each 3-polytope has two faces with the same number of vertices.
Hint: Do not use the Euler equation.
1.3. Prove the Steinitz Lemma 1.1:
– Prove the “upper bound theorem” for dimension 3, that is, that f2 ≤ 2f0 − 4
(you may use Euler’s equation), and derive f0 ≤ 2f2 − 4 by duality.
– Compute the f -vectors of the pyramids over n-gons.
– How does (f0 , f2 ) change if you stack a pyramid onto a triangle 2-face, or if
you truncate a simple vertex?
1.4. If a 3-dimensional polytope has f1 = 23 edges, how many vertices/faces can it
have? Construct an example for each possible pair (f0 , f2 ).
1.5. Alternative homogeneous coordinates for the cone of f -vectors are given by the
“imbalance” σ := ff21−f−6 , where the self-dual term f1 − 6 measures the “size.”
0
is a long sequence of large numbers, which we may graph just like a continuous
function, and ask for its “shape.” Indeed, we might look at a shape function
ϕ : [0, 1] → Ê that is defined by ϕ(x) := fx(d−1) ; this is defined for any x = d−1 k
1
that is a multiple of d−1 , and these values are rather dense if d is large. We might
interpolate if we want. But what types of f -vector shape functions ϕ do we get
that way?
Figure 2.1 shows two “naive” views, of the shape of an f -vector, and — equiv-
alently — of the shape of a typical face lattice (displayed as a Hasse diagram, so
the sizes of rank levels are the fi -values).
d
d−1
0
−1
−1 d
0 d−1
Figure 2.1. A rough, “naive” picture of the shape of the face lattice, and the
f -vector, for a high-dimensional polytope
A very simple observation is that each vertex of a d-polytope has degree at least
d, so double counting yields f1 ≥ d2 f0 > f0 ; dually, we have fd−2 ≥ d2 fd−1 > fd−1 .
So in the first step, the f -sequence increases, in the last step it decreases. Does
this mean that the f -vector “first goes up, then comes down,” that it is unimodal,
with no “dip” in the middle?
643
644 GÜNTER M. ZIEGLER, CONVEX POLYTOPES
This is trivially true for d ≤ 5. It also is true for simplicial d-polytopes (the f -
vectors of simplicial polytopes indeed increase up to the middle, and they decrease
in the last quarter), but the available proof for this depends on the necessity part
of the g-theorem, so it is quite non-trivial; see [15].
To demonstrate our ignorance on such basic f -vector shape matters, here is a
suspiciously innocuous conjecture. Apparently no one has an idea for a proof, up
to now.
Conjecture 2.3 (Bárány). For any d-polytope, fk ≥ min{f0 , fd−1 }.
Bárány’s conjecture holds for d ≤ 6 [78]. However, not even
fk ≥ 1
10000 min{f0 , fd−1 }
is proven for large dimensions d ! We know so little . . .
Example 2.5 (Cross polytopes). For the d-dimensional cross polytope Cd∗ =
conv{±e1 , . . . , ±ed } we have
d
fk (Cd∗ ) = 2k+1 .
k+1
Again, approximating crudely and taking logarithms base 2, we get
log ϕ(x) ∼ −x log x − (1 − x) log(1 − x) + x.
The derivative
d 1 1
ϕ(x) ∼ − log x − + log(1 − x) + +1
dx ln 2 ln 2
vanishes at x = 23 : That’s where log ϕ(x) has its maximum, and where ϕ(x) has a
sharp peak (compare Figure 2.2).
Thus the f -vector of a d-dimensional cross polytope, for large d, has a sharp
peak at k = 23 d. By duality, this means that the f -vector of the d-cube peaks at
k = 13 d, for large d.
Example 2.6 (Cyclic polytopes). Let’s look at cyclic polytopes Cd (n) with many
vertices, n d. For simplicity, we assume that the dimension d is even.
Ê
A curve in d has degree d if no d + 1 points on the curve lie on a hyperplane.
The convex hull of any n > d points on such a curve is a cyclic polytope Cd (n).
Gale’s evenness criterion [30] gives a combinatorial description for the facets, which
is easy to visualize (see Figure 2.3): Any d points on a degree d curve span a
hyperplane H. If the d points are supposed to span a facet of the polytope, then
all the other n − d points must lie on the same side of H. Since the curve crosses H
only in these d points, this means that the d points split into d2 adjacent pairs. So,
if we number the points 1, 2, . . . , n along the curve, then the facets of their convex
hull (the cyclic polytope) are given by d2 pairs i, i + 1 mod n. The (k − 1)-faces
are given by the k-subsets of such a d-set: For k ≤ d2 any such subset will do (the
cyclic polytopes are neighborly), while for k > d2 the faces consist of k − d2 pairs,
and d − k singletons. Thus the (k − 1)-faces may be obtained by choosing d2 vertices
646 GÜNTER M. ZIEGLER, CONVEX POLYTOPES
fk /107
k
5 10 15 20 25
12
fk /10
k
5 10 15 20 25
fk /1018
5
k
5 10 15 20 25
Figure 2.2. The f -vector shapes of the 28-dimensional simplex Δ28 , the cross
∗ , and a cyclic polytope with 80 vertices C (80)
polytope C28 28
ij arbitrarily, and also taking ij + 1 for k − d2 of these (see Figure 2.4). Thus, with
a bit of an over-count, we get
= nk for k ≤ d2 ,
fk−1 (Cd (n)) d
∼ nd k−2 d for k > d2 .
2 2
LECTURE 2. SHAPES OF F -VECTORS 647
Figure 2.4. An estimate for the number of facets of Cd (n), for n d, with d
n
even: There are d/2 choices for the black points; with high probability, they
d
are non-adjacent; the 2
pairs can be completed by taking the gray points.
Clearly this peaks at x = 34 : We get the larger entries in the case k > d2 ,
d/2
and then the maximum is achieved when k−d/2 is maximal, that is, for k = 34 d.
Figure 2.2 gives a realistic impression of the f -vector shape of a cyclic polytope.
An explicit, exact formula for fk−1 (Cd (n)) is available (Exercise 2.3), but this
doesn’t answer all the questions. In particular, is it really true that the f -vector
is unimodal? As far as I know, the Unimodality Conjecture 2.1 has not been
established in full for the cyclic polytopes. It does hold for small n > d, and
certainly also if n d is sufficiently large compared to d (with the f -vector peak
at k = 3(d−1)
4 ), but in an intermediate range for n a challenge remains . . .
for m ≥ 0.
The product construction is dual to the “free sum” construction, P ⊕ Q: For
Ê Ê
this let x0 ∈ P ⊂ d and y0 ∈ Q ⊂ e be interior points, and take the convex hull
P ⊕ Q := conv P × {y0 } ∪ {x0 } × Q .
The proper faces of P ⊕ Q (that is, faces other than the polytope itself) arise as
joins of proper faces of P and of Q.
The product and the free sum construction are illustrated in Figure 2.5.
Q P
Since joins come up as faces of free sums, let’s briefly talk about joins.
Example 2.8 (Joins). Let again P and Q be polytopes of dimensions d and e.
Then the join P ∗ Q is obtained by positioning P and Q into skew affine subspaces,
and taking the convex hull. Thus the join is a polytope of dimension dim(P ∗ Q) =
dim P + dim Q + 1 = d + e + 1.
The faces of P ∗ Q are the joins of faces of P and faces of Q: This refers to all
faces, including the empty face and the polytope itself. The corresponding formula,
with f−1 (P ) = f−1 (Q) = 1, is
(2.2) fm (P ∗ Q) = fk (P ) f (Q),
k+=m−1
k,≥−1
Joins are illustrated in Figure 2.6. The dual construction to taking joins is the
join construction again.
Product and join are two distinct constructions, and they do yield different
polytopes, of different dimensions (by 1). However, in a birds’ eye view, asymptot-
ically, they do behave quite similarly, and indeed, their effects on f -vector shapes
are almost the same. Namely, the formulas (2.2) and (2.1) describe finite convolu-
tions, and the only difference is whether the entry f−1 = 1 is counted. For large
dimensions, and large f -vectors, this does not make much of a difference, and in
both cases we get a convolution of f -vector shapes. Thus, in particular, if the f -
vectors of P and of Q have sharp peaks, then the product or join will have a peak
as well:
(peak at x) ∗ (peak at y) −→ (peak at d+e d e
x + d+e y).
In particular, for d = e this yields
(peak at x) ∗ (peak at y) −→ (peak at x+y 2 ).
To see this, just compute that if the peak (or, just the largest f -vector entry) for
P1 is at x = kd and for P2 at y = e , then the peak for P1 ∗ P2 will be at
k+ k d e d e
d+e = d d+e + e d+e = x d+e + y d+e .
This also yields a convolution formula for the f -vector shape of P1 × P2 or P1 ∗ P2 ,
for large dimensions:
1
d
ϕ(x) = ϕ1 t d+e ϕ2 (1 − t) d+e
e
dt
0
Thus, by just taking products of sums of suitable cyclic polytopes and their duals,
we do get polytopes with f -vector peaks in the whole range between 14 and 34 .
f -vectors of two polytopes with peaks at 14 and at 34 , say a cyclic polytope and its
dual. And indeed, just as we can glue a pyramid onto a simplicial facet, we can
glue any polytope with a simplicial facet onto another one — after a projective
transformation, if needed [79, p. 274]. The f -vector effect of such a glueing is
essentially
f (P #P ) = f (P ) + f (P ) − f (Δd−1 );
if the f -vector components of P and of P are large, then the simplex may be
neglected, and we are essentially just “adding the f -vectors.”
We can even do this with cyclic polytopes: For example, Cd (n) is simplicial; its
dual, Cd (n)∗ is simple (without simplicial facets), but if we cut off (“truncate”) one
of the simple vertices, then a simplicial facet results. Write Cd (n) for the “dual
with a vertex cut off.”
Corollary 2.11 (Eckhoff [24]). The Unimodality Conjecture 2.1 fails for d-poly-
topes of dimensions d ≥ 8. In particular,
f (C8 (25)#C8 (25) ) = (7149, 28800, 46800, 46400, 46400, 46800, 28800, 7149).
This f -vector has a nice “1% dip” in the middle! We don’t know whether the
Unimodality Conjecture 2.1 is true for dimensions d = 6 or 7.
Exercises
2.1. For d = 3, 4, 5, . . . construct a d-polytope with 12 vertices and 13 facets. How
far do you get?
2.2. Show that f -vectors of 4-polytopes are unimodal.
2.3. Derive an exact formula for fd−1 (Cd (n)), and for fk (Cd (n)), for even n.
2.4. Compute fi (C8 (25)). How bad is the approximation given in Example 2.6?
2.5. Count and describe the 2-faces of a product of a pentagon and a heptagon,
P5 × P7 .
2.6. Compute f ((C10 )10 ), for the product of ten 10-gons. Where is the peak?
2.7. Estimate/compute d and N such that the “N -fold truncated d-cube” has a
non-unimodal f -vector.
2.8. If you stack “too often” onto C20 (200), then unimodality is restored. How
often?
LECTURE 3
2-Simple 2-Simplicial 4-Polytopes
*These were apparently introduced by Dr. Victor Schlegel, a highschool (Gymnasium) teacher from
Waren an der Müritz, in his paper [67] from 1883. The plates for the paper include a Schlegel
diagram (“Zellgewebe”) of a 4-cube, as well as two quite insufficient drawings representing the
24-cell. Classical, beautiful drawing may be found in Hilbert & Cohn-Vossen [41, p. 135].
653
654 GÜNTER M. ZIEGLER, CONVEX POLYTOPES
3.1. Examples
Let’s start with examples of well-known 4-polytopes — and for each of those let’s
look at a Schlegel diagram, and record the f -vector
(f0 , f1 , f2 , f3 ) = ( # vertices, # edges, # 2-faces (= ridges), # facets ).
A Schlegel diagram is a way to visualize a 4-polytope in terms of a 3-dimensional
complex. We can’t develop the theory of Schlegel diagrams here (see [39, Sect. 3.3]
and [79, Lect. 5]), but we can offer two interpretations, both in terms of dimensional
analogy.
• Assume that one face of a 3-polytope is transparent (a “window”), press your
nose to the window, and look inside: Then you will see all the other faces of
the polytope through the window. If you now close one eye (and thus lose the
spatial impression, or depth view), then you will see how the other faces tile the
window; you can see how they fit together, and thus the whole combinatorial
structure of the 3-polytope is projected into a 2-dimensional window. This is
the Schlegel diagram of a 3-polytope.
• Any 3-polytope can be projectively deformed in such a way that looking at it
from a suitable point, you see all faces except for one single face, which is on
the back. What you see is a polytopal complex which has the same shape as
the back face, but this is broken into all the many faces that you see on the
front side. What you see is the 2-dimensional Schlegel diagram of a 3-polytope.
The Schlegel diagram of a 4-polytope, analogously, is a 3-dimensional complex that
represents all the faces of the polytope, except for one facet (the window resp. back
facet). The whole combinatorial structure of the polytope may be read from such
a visualization. Thus, for example, one can tell whether the polytope is simple, or
simplicial, or cubical, etc.
The pictures of Schlegel diagrams as presented in the following are generated au-
tomatically in the polymake system by Gawilow & Joswig [31], with the javaview
back-end by Polthier et al. [58]. They have three limitations: They show only a
2-dimensional projection of an object that you should see rotating, 3-dimensionally,
on a screen; they depict only the edges, so in some examples it is hard to tell/imagine
where the faces and facet-boundaries go; and we don’t have color available here.
Nevertheless, I think they are impressive, and you should be able to “see” in them
what the (boundary complexes of) some 4-polytopes look like.
Example 3.1 (Simplex, cube, and cross polytope). Schlegel diagrams of the 4-
simplex, the 4-cube and the 4-dimensional cross polytope appear in Figure 3.1.
You should read off the f -vectors from this figure: f (Δ4 ) = (5, 10, 10, 5), f (C4 ) =
(16, 32, 24, 8), and f (C4∗ ) = (8, 24, 32, 16).
The simplex and cube are simple, so f1 = 2f0 , while the simplex and cross
polytope are simplical, so f2 = 2f3 .
Example 3.2 (A cubical 4-polytope with the graph of a 5-cube [43]). The con-
struction
P := conv((2Q × Q) ∪ (Q × 2Q)),
for a square such as Q = [−1, 1]2 , yields a 4-polytope whose Schlegel diagram is
displayed in Figure 3.2. This polytope is cubical : All its facets are combinatorially
equivalent to the 3-cube [−1, 1]3.
LECTURE 3. 2-SIMPLE 2-SIMPLICIAL 4-POLYTOPES 655
Figure 3.1. Schlegel diagrams for the 4-dimensional simplex, cube, and cross polytope
The f -vector (32, 80, 72, 24) may be derived from the figure, but indeed it may
also be deduced just from the information that this is a cubical 4-polytope with
the graph of a 5-cube. (The latter yields f0 and f1 , the “cubical” property implies
2f2 = 6f3 by double counting, and then there is the Euler–Poincaré equation [79,
Sect. 8.2], which for 4-polytopes reads f0 − f1 + f2 − f3 = 0. See also Exercise 3.2.)
The
5 first representation is more symmetric: It yields “by inspection” that all
2 = 10 vertices of this polytope are equivalent (under symmetries that permute
the coordinates), but that there are two types of facets, five simplices and five oc-
tahedra, which appear in vertex-disjoint pairs, “opposite to each other,” in parallel
hyperplanes. In particular, all the facets are simplicial, that is, all the 2-faces are
triangles, so the polytope is 2-simplicial.
The second representation has the advantage of being full-dimensional, and it
supplies us with a Schlegel diagram (using an octahedron facet as a “window”), as
displayed in Figure 3.3. In the figure we may see that the (ten, equivalent) vertex
figures are triangular prisms, so they are simple; thus in this 4-polytope, each edge
is in exactly three facets, so the polytope is 2-simple. So we have seen our first
example (other than the 4-simplex) of a 2-simple, 2-simplicial 4-polytope.
From the data given it is easy to compute the f -vector of the hypersimplex: It
is f = (10, 30, 30, 10).
LECTURE 3. 2-SIMPLE 2-SIMPLICIAL 4-POLYTOPES 657
Figure 3.4. Simplicial, simple, and 2s2s 4-polytopes in terms of their face
lattices: The shaded intervals, and all the other intervals between the same
rank levels, must be boolean.
note one interesting property that is specific for the 4-dimensional case, and which
also confirms the impression that 2s2s 4-polytopes form a “diagonal” case.
Proof. If P is 2-simplicial, then each 2-face has three edges. Thus the number of
incidences between 2-faces and edges, denoted f12 , is f12 = 3f2 . If it is 2-simple,
then each edge lies in three 2-faces, that is, the number of indicences is f12 = 3f1 .
Combination of the two conditions forces f1 = f2 . With this, Euler’s equation
yields f0 = f3 .
This proof may be rephrased in terms of the face lattice: For 4-polytopes the
2s2s conditions force the two middle rank levels of the face lattice to form a bipartite
cubic graph — which as any other regular bipartite graph has to have the same
number of vertices on each shore. You should identify this bipartite cubical graph
in the face lattice of the hypersimplex, as displayed in Figure 3.5, and thus verify
the 2s2s property for this face lattice. The symmetry of the f -vector (10, 30, 30, 10)
is explained by Lemma 3.5; nevertheless, the hypersimplex and its face lattice are
not self-dual: There are two types of facets, but only one symmetry class of vertices.
The fact that the dual of any 2s2s 4-polytope is again 2s2s (by definition), and
the symmetry property for the f -vector, might suggest that 2s2s polytopes live in
some sense “between” simple and simplicial. This is not true, as we will see in
the next lecture, when we locate their f -vectors in the cone of all f -vectors of 4-
polytopes. Indeed, the 2s2s polytopes are so interesting because they form a class
of extremal polytopes in terms of the flag vector: A 4-polytope is 2s2s if and only
if the valid inequality
For d ≥ 3, every deep vertex truncation polytope DVT(P ) has two types of
facets:
• deep vertex truncations DVT(F ) of the facets F of P , and
• the vertex figures P ∩ Hv = conv{pe : e v} of P .
Proposition 3.8 (Paffenholz & Ziegler [57]). If P is a simplicial 4-polytope, then
any deep vertex truncation DVT(P ) is 2-simple and 2-simplicial.
Proof. The two types of facets of DVT(P ) are the octahedra DVT(F ), for the
tetrahedron facets F of P , and the vertex figures of P , which are simplicial. Thus
DVT(P ) 2-simplicial.
Since all edges of P are reduced to points by deep vertex truncation, all the
edges of DVT(P ) are “new,” they arise by deep vertex truncation from the 2-faces
(that is, the ridges) of P . Each such ridge lies in two facets F1 , F2 of P , so the edge
we are looking at lies in two facets DVT(F1 ) and DVT(F2 ) of the first type, and
in one facet of the second type. Thus each edge of DVT(P ) lies in exactly three
facets, that is, DVT(P ) is 2-simple.
So we have that DVT(P ) is 2s2s for any simplicial 4-polytope P . . . if it exists.
And that’s the problem: In general it is not at all guaranteed that deep vertex
truncation can be performed. One would try to realize cyclic 4-polytopes in such
a way that deep vertex truncations can be performed, but it seems that this is not
possible. Similarly, if a sum Pm ⊕ Pn is realized “the obvious way,” with regular
polygons in orthogonal subspaces, then deep vertex truncation is not possible except
for very special cases (such as m 1
+ n1 ≥ 12 ): It is quite surprising that the sums
of polygons do have a realization such that deep vertex truncation is possible, as
proved by Paffenholz [55]. On the other hand, there does not seem to be a single
example of a simplicial polytope for which it has been proved that deep vertex
truncation is impossible for all realizations.
However, in special cases deep vertex truncation can indeed be performed. In
particular, any regular polytope admits a deep vertex truncation — just take the
LECTURE 3. 2-SIMPLE 2-SIMPLICIAL 4-POLYTOPES 661
edge midpoints for pe . From this we get the following three examples of 2s2s 4-
polytopes:
• Deep vertex truncation of a simplex, DVT(Δ4 ), yields the hypersimplex.
• Deep vertex truncation of the 4-dimensional cross polytope,
C4∗ = conv{±ei : 1 ≤ i ≤ 4} = {x ∈ Ê4 : |x1 | + |x2| + |x3 | + |x4 | ≤ 1},
yields Schläfli’s 24-cell (see Figure 3.7):
DVT(C4∗ ) = conv{± 21 ei ± 12 ej : 1 ≤ i < j ≤ 4}
= {x ∈ Ê4 : |xi | ≤ 1 for 1 ≤ i ≤ 4, |x1 | + |x2 | + |x3| + |x4 | ≤ 1}.
• Deep vertex truncation of the regular 600-cell (which has 600 regular tetra-
hedra as facets) yields a 2s2s 4-polytope with 720 vertices, whose vertex fig-
ures are prisms over regular pentagons; its facets are 600 octahedra, and 120
regular icosahedra. It seems that this remarkable polytope, with f -vector
(720, 3600, 3600, 720), first occured in the literature in 1994, as the dual of
the “dipyramidal 720-cell” constructed by Gévay [36]. See also Exercise 3.2.
Theorem 3.9 (Paffenholz & Ziegler [57]). Any combinatorial type of a stacked
d-polytope Stack(n, d) can be realized so that it admits a deep vertex truncation.
Proof. We proceed by induction on n, starting at n = 0, with a d-simplex, and a
deep vertex truncation that takes the convex hull of the edge midpoints.
Ê
Assume now that Stack(n, d) has been realized as P ⊂ d such that DVT(P )
can be obtained by a suitable choice of points pe on the edges e ⊂ P . Assume
that Stack(n + 1, d) arises by stacking onto a facet of Stack(n, d) that is realized
by the facet F ⊂ P with vertex set {v1 , . . . , vd }. The “new” vertex w is now
chosen “beyond” the facet DVT(F ) of DVT(P ), and “beneath” all other facets
of DVT(P ). That is, addition of w to DVT(P ) would mean stacking a pyramid
onto the facet DVT(F ) of DVT(P ). In particular, w lies “beyond” the facet F of P ,
and “beneath” all other facets of P , so P := conv({w} ∪ P ) is a stacked polytope
realizing Stack(n + 1, d), as required.
The facet hyperplanes Hvi of DVT(P ) cut the edges [vi , w] of P in points pi :
This is since w is beneath Hvi , while vi is cut off by Hvi . Thus we obtain points pi
on the new edges of P , and the hyperplane Hw := aff{p1 , . . . , pd } may be taken to
cut off the new vertex w of P . This new truncation plane is determined uniquely
by the d intersection points, because the new vertex w of P is simple.
This theorem is valid for all d ≥ 3; in particular, 3D-pictures work. (Figure 3.8
is a feeble attempt.) However, the construction produces by far the most interesting
results for d = 4.
w
pi
pj
vi
vj
Figure 3.8. The induction step in Theorem 3.9, for d = 3. DVT(F ) is drawn shaded.
Corollary 3.10 ([57]). For each n ≥ 0, and for every type of stacked 4-polytope
Stack(n, 4) with f -vector (5 + n, 10 + 4n, 10 + 6n, 5 + 3n), there is a corresponding
2-simple 2-simplicial 4-polytope DVT(Stack(n, 4)), with f -vector
f (DVT(Stack(n, 4))) = (10 + 4n, 30 + 18n, 30 + 18n, 10 + 4n).
In particular, this yields infinitely many combinatorial types of 2-simple 2-
simplicial 4-polytopes. Moreover, with a bit of care the proof of Theorem 3.9 yields
these polytopes with rational vertex coordinates. See [54] for explicit examples of
such coordinates.
Corollary 3.11 ([57]). The number of combinatorial types of 2-simple 2-simplicial
4-polytopes with 10 + 4n vertices grows exponentially in n.
LECTURE 3. 2-SIMPLE 2-SIMPLICIAL 4-POLYTOPES 663
See Paffenholz & Werner [56] for further constructions of 2-simple 2-simplicial
4-polytopes with interesting f -vectors. In particular, they describe the “smallest”
example of such a polytope (other than the simplex), which has only 9 vertices.
Exercises
3.1. Show that any simple or simplicial d-polytope with f0 = fd−1 must be a
simplex, or 2-dimensional.
3.2. Compute the full f -vectors, as well as the number f03 of vertex-facet incidences,
for the following 4-polytopes, based only on the information given here:
(a) The 24-cell: a 2s2s polytope whose facets are 24 octahedra;
(b) The 600-cell: a simple polytope whose facets are 120 dodecahedra;
(c) The 720-cell: a 2s2s 4-polytope whose facets are 720 bipyramids over pen-
tagons;
(d) A neighborly cubical polytope NCPn4 , a cubical polytope with the graph
of the n-cube (n ≥ 4).
3.3. Compute the full f -vectors of the stacked d-polytopes Stack(n, d).
3.4. Show that if a 4-polytope P is not simplicial, then DVT(P ) cannot be 2-
simplicial.
3.5. Find coordinates for DVT(Stack(1, 4)). Check them with polymake.
(This is Braden’s “glued hypersimplex” [18].)
3.6. Show that there are exponentially many distinct combinatorial types of stacked
d-polytopes with d + 1 + n vertices, for any d ≥ 3. Derive that there are
exponentially many types of 2-simple 2-simplicial 4-polytopes with the same
f -vector.
3.7. Show that f13 = f03 + 2f2 − 2f3 , and dually f02 = f03 + 2f1 − 2f0 , holds for
the flag vector of each 4-polytope.
(Hint: Sum the Euler equations for the facets, which are 3-polytopes.)
Derive from this that the inequality 2f03 ≥ (f1 + f2 ) + 2(f0 + f3 ) is valid
for all 4-polytopes, and that it is tight exactly for the 2-simple 2-simplicial
4-polytopes.
3.8. Show that there is no f -vector inequality (not involving f03 ) that characterizes
the 2s2s 4-polytopes.
3.9. If P is a d-dimensional simplicial polytope, and if DVT(P ) exists, is DVT(P )
then 2-simple? 2-simplicial?
LECTURE 4
f -Vectors of 4-Polytopes
for some large M . But let’s not get too ambitious too fast.
665
666 GÜNTER M. ZIEGLER, CONVEX POLYTOPES
f0 (f0 − 3)
f3 ≤
2
(equality:
f3 neighborly
15 polytopes)
14
13
12
11
10
9
8
7
6 f3 (f3 − 3)
5 f0 ≤
2
(equality:
dual-to-neighborly polytopes)
5 6 7 8 9 10 1112131415
f0
Ê
Equivalently, cone(F4 ) ⊂ 4 is the solution set to all the linear inequalities that
are valid for all f -vectors for 4-polytopes, and that are tight at the f -vector of the
simplex.
The equivalence between the two versions of the definition rests on basic facts
about closed convex sets, which you should put together yourself (Exercise 4.1).
You are also asked to verify that the cone generated by the f -vectors is not closed,
so we do have to take the topological closure (Exercise 4.2.)
The closed convex cone we are looking at is 3-dimensional, so we may view it
as the cone over a 2-dimensional convex figure, which might be just a pentagon or
hexagon. Instead of looking at a 2-dimensional section (say intersecting by f1 +f2 =
100), we may equivalently introduce homogeneous (“projective”) coordinates, which
are rational linear functions, normalized to yield “ 00 ” at the f -vector of a simplex
(compare Lecture 1). There is no unique best way to do this; we choose
f0 − 5 f3 − 5
ϕ0 := and ϕ3 :=
f1 + f2 − 20 f1 + f2 − 20
as our homogeneous coordinates. (Figure 4.2 illustrates the geometry of such a
rational function on a cone.) So we are trying to describe proj(F4 ) ⊂ 2 , the Ê
closure of
Ê
conv{(ϕ0 (P ), ϕ3 (P )) ∈ 2 : P a convex 4-polytope}.
LECTURE 4. F -VECTORS OF 4-POLYTOPES 667
ϕ0 = 1 ϕ0 = 2
ϕ0 = 0 ϕ0 = 3
Figure 4.2. The function ϕ0 is constant on certain planes that contain the
apex of the cone. It is not defined on the line where all those planes intersect.
(In terms of (f0 , f1 , f2 )-coordinates, is defined by f0 = 5 and f1 + f2 = 20.)
Any 4-polytope yields a (rational) point in the (ϕ0 , ϕ3 )-plane. Any valid linear
inequality, tight at the 4-simplex, translates into a linear inequality in ϕ0 and ϕ3 .
So let’s look at some families of polytopes and of linear inequalities that we know,
and let’s see what they buy us.
Some 4-polytopes we know:
Stacked: (5 + n, 10 + 4n, 10 + 6n, 5 + 3n) −→ 1
( 10 3
, 10 )
Truncated: (5 + 3n, 10 + 6n, 10 + 4n, 5 + n) −→ 3
( 10 , 101
)
n→∞
Cyclic: (n, n(n−1)
2 , n(n − 3), n(n−3)
2 ) −→ 1
(0, 3 )
n→∞
Dual-to-cyclic: ( 2 , n(n − 3), n(n−1)
n(n−3)
2 , n) −→ ( 13 , 0).
The truncated polytopes are the duals of the stacked polytopes, so they are simple.
Similarly, the duals of cyclic polytopes are simple. Thus we find the four points
1 3 3 1
( 10 , 10 ), ( 10 , 10 ), (0, 13 ), and ( 13 , 0), which span a quadrilateral subset of proj(F4 ).
This quadrilateral also represents the f -vectors of simple and of simplicial polytopes
and “everything in between.” (Note that duality interchanges the coordinates ϕ0
and ϕ3 , and thus proj(F4 ) is symmetric with respect to the main diagonal.)
Five linear constraints we know:
“Few Vertices”: f0 ≥ 5 ⇐⇒ ϕ0 ≥ 0,
“Few Facets”: f3 ≥ 5 ⇐⇒ ϕ3 ≥ 0,
“Simple”: f1 ≥ 2f0 ⇐⇒ 3ϕ0 + ϕ3 ≤ 1,
“Simplicial”: f2 ≥ 2f3 ⇐⇒ ϕ0 + 3ϕ3 ≤ 1,
“Lower bound”: 2f1 + 2f2 ≥ 5f0 + 5f3 − 10 ⇐⇒ ϕ0 + ϕ3 ≤ 25 .
668 GÜNTER M. ZIEGLER, CONVEX POLYTOPES
The first four inequalities are quite trivial, and we have named them by the poly-
topes that satisfy them with equality, at least asymptotically. The translation
into (ϕ0 , ϕ3 )-inequalities, using the Euler-Poincaré relation, poses no problem.
There is no polytope with ϕ0 = 0, but the condition is satisfied asymptotically
by any family of 4-polytopes with far more vertices than facets. For example, the
products of n-gons, with f (Pn × Pn ) = (n2 , 2n2 , n2 + 2n, 2n), yield (ϕ0 , ϕ3 ) =
2
−5
( 3n2n+2n−20 +2n−20 ) ∈ proj(F4 ), which in the limit n → ∞ yields ( 3 , 0).
, 3n22n−5 1
The one non-trivial inequality in our table above is the last one, a “Lower
Bound Theorem.” It may be derived quite easily [11] from the inequality f03 ≥
3f0 + 3f3 − 10, which was first established by Stanley [70] in terms of the so-called
toric g-vector (it is the inequality “g2tor (P ) ≥ 0”); a proof via rigidity theory was
later given by Kalai [44].
Figure 4.3 summarizes our discussion up to this point: We are interested
in proj(F4 ), the closure of the set
conv{(ϕ0 (P ), ϕ3 (P )) : P is a 4-polytope, not a simplex } ⊂ Ê2.
This set is contained in the pentagon cut out by the five linear inequalities discussed
above, and it contains the shaded trapezoid, which represents “everything between
simple and simplicial polytopes.” Indeed, simple and simplicial polytopes satisfy
the additional linear inequality ϕ0 + ϕ3 ≥ 13 .
Thus we are left with the following “upper bound problem”:
“Upper Bound Problem”. Are there 4-polytopes with ϕ0 + ϕ3 → 0 ?
The inequality ϕ0 + ϕ3 ≥ 13 is certainly not valid for all (possibly non-simple
non-simplicial) 4-polytopes: Already for the hypersimplex we get (ϕ0 , ϕ3 ) = ( 18 , 18 ).
cyclic polytopes
ϕ3 ϕ0 + 3ϕ3 ≤ 1
simplicial polytopes
2
5
stacked polytopes
1
3
3
10
ϕ0 + ϕ3 ≤ 2
5
1
5
truncation polytopes
3ϕ0 + ϕ3 ≤ 1
1 simple polytopes
10
dual-to-cyclic polytopes
1 1 3 1 2
10 5 10 3 5 ϕ0
However, currently it is not clear how small ϕ0 + ϕ3 can be for convex polytopes.
Thus the Upper Bound Problem is the key remaining problem in the description of
the f -cone for 4-polytopes.
(!) If the answer is YES to the problem as posed above, then the five inequalities
above constitute a complete linear description of cone(F4 ).
(!) If the answer is NO, then this is also exciting, since it means that the answers
for cellular spheres and for convex polytopes are distinct! Indeed, cellular
3-spheres with arbitrarily small ϕ0 + ϕ3 have been constructed by Eppstein,
Kuperberg & Ziegler [26]; see our discussion in Section 4.3.
few facets
many ridges
many edges
few vertices
f0 f3
f1 f2
Figure 4.4. Fatness for a 4-polytope face lattice, and for an f -vector
cyclic polytopes
ϕ3
simplicial polytopes
2
5
stacked polytopes
1
3
3
10
neighborly
cubical F = 2.5
polytopes [43] 1
5
F =3 truncation polytopes
projected
products 1 F =5 simple polytopes
of polygons 10
F =9
dual-to-cyclic polytopes
1 1 3 1 2
10 5 10 3 5 ϕ0
What do polytopes “of very high fatness” look like? You can verify (via Exer-
cise 4.6) that they have two properties:
(1) The facets have many vertices (on average).
(2) The vertices are in many facets (on average).
Either of these properties are easy to satisfy — just look at the products Pn × Pn
for the first property, and at their duals, the free sums Pn ⊕ Pn , for the second one.
The key question is whether they can simultaneously be satisfied.
Finally, here is a problem on 3-dimensional polytopal tilings that is “essentially”
Ê
equivalent to the fatness problem: Consider face-to-face tilings of 3 (cf. [65]) that
satisfy some regularity properties, e.g. one of the following (each implies the next):
LECTURE 4. F -VECTORS OF 4-POLYTOPES 671
– the tiling is triply periodic (that is, there are three linearly independent trans-
lational symmetries),
– there are only finitely many distinct congruence classes of tiles,
– in- and circumradius of the tiles are uniformly bounded.
For such tilings, we may define notions of “average” vertex degrees, face numbers,
etc. The question is whether there is such a tiling where the tiles have lots of
vertices on average, and the vertices are in many tiles on average. Again, either
property is easy to achieve (look at tilings by Schlegel diagrams), but can they be
simultaneously satisfied?
Exercises
4.1. Show that the two definitions of the f -vector cone given in Definition 4.1 are
indeed equivalent.
Hint: You need a separation lemma; see for example Matoušek [48, p. 6].
Ê
4.2. Show that the union of the line segments [f (Δ4 ), f (C4 (n))] ⊂ 4 has the whole
ray {(5, 10 + t, 10 + 2t, 5 + t) : t ≥ 0} in its closure. Note that f0 ≥ 5 is a valid
linear inequality, which is tight at f (Δ4 ), but for no other f -vector.
Conclude that the cone with apex f (Δ4 ) spanned by the f -vectors of 4-
polytopes is not closed.
4.3. Compute the fatness and the (ϕ0 , ϕ3 )-pair for the hypersimplex, the 24-cell,
and for DVT(600-cell).
4.4. Compute the fatness of the 2s2s polytopes DVT(Stack(n, 4)), and show that
it lies in the interval [4, 4.5).
Show that for any simplicial 4-polytope P , the fatness of DVT(P ) is smaller
than 6.
Where would the f -vectors of the polytopes DVT(P ) lie in proj(F4 ), as graphed
in Figure 4.5?
672 GÜNTER M. ZIEGLER, CONVEX POLYTOPES
4.5. If C4n is a cubical 4-polytope with the graph of an n-cube (see Exercise 3.2),
compute the fatness and the pair (ϕ0 , ϕ3 ).
4.6. Define the complexity of a 4-polytope to be the quotient
f03 − 20
C(P ) := .
f0 + f3 − 10
(a) Show that F (P ) ≤ 2C(P ) − 2, with equality if and only if P is 2-simple
and 2-simplicial.
(b) Show that C(P ) ≤ 2F (P ) − 2, with equality if and only if if all facets of P
are simple, or equivalently, if all vertex figures are simple.
(c) Derive from this that fatness is high if and only if both the average number
of vertices per facet, f03 /f3 , and the average number of facets per vertex,
f03 /f0 , are large.
LECTURE 5
Projected Products of Polygons
algorithm; cf. [45] [3]) that some projections of “deformed products” have very
interesting extremal properties. For a very simple example, look at a 3-cube (which
is a product of three 1-polytopes). Any orthogonal cube projected to the plane will
produce a hexagon (at best), while a deformed cube can be projected to the plane
to yield an octagon: All the vertices “survive the projection” (see Figure 5.1).
And indeed, it is not so hard to show that one can realize an n-cube in such a
way that all its vertices “survive the projection” to the plane — this was first proved
by Murty [51] and Goldfarb [37]; in the context of linear programming it yields
exponential examples for the “shadow vertex” pivot rule for linear programming.
Ê
Similarly, if you project an orthogonal product of polygons to 4 , you cannot
expect that all the edges survive the projection, but with deformed products, this
is possible (although hard to visualize — proofs are mostly based on linear algebra
criteria rather than on geometric intuition).
Here are linear algebra descriptions of the polytopes we’ll be looking at:
Polygons: If V is any (n × 2)-matrix whose rows are non-zero, (w.l.o.g.) ordered
Ê
in cyclic order, and positively span 2 , then for a suitable positive right-hand side
Ê
vector b ∈ n the system V x ≤ b describes a convex n-gon Pn ⊂ 2 . Ê
Product of polygons: Given such a V , we immediately get the system
⎛ ⎞ ⎛ ⎞
⎜ ⎟ ⎜ ⎟
⎜ V ⎟ ⎜b ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜
⎜
⎜ V
0 ⎟
⎟
⎟
⎜ ⎟
⎜ ⎟
⎜b ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ V ⎟x ≤ ⎜ b ⎟,
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜. ⎟
⎜
⎜
⎜
0 ..
.
⎟
⎟
⎟
⎜ .. ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎝ V ⎠ ⎝b ⎠
676 GÜNTER M. ZIEGLER, CONVEX POLYTOPES
with a block-diagonal matrix of size rn×2r, which describes the orthogonal product
Ê
of polygons (Pn )r ⊂ n .
The combinatorics of the product of polygons is reflected in the facet-vertex
incidences, as follows: Each inequality defines a facet; each of the nr vertices is the
unique solution of a linear system of equations that is obtained by requiring that
from each block, two cyclically adjacent inequalities are tight.
Ê
If the diagonal V ε -blocks V ε ∈ n×2 satisfy the conditions above, then the blocks
Ê
U, W ∈ n×2 below the diagonal blocks can be arbitrary — the right-hand sides can
be adjusted in such a way that we get a deformed product, which is combinatorially
equivalent to a product (Pn )r .
For this, we only have to verify that we get the correct combinatorics: If for each
of the r blocks we choose two cyclically adjacent inequalities and require them to
be tight, then this should yield a linear system of equations with a unique solution,
for which all other inequalities are satisfied, but not tight. It is now easy to prove
(by induction on r) that our system satisfies this — if we just choose the right-
Ê
hand sides suitably; in particular, bi := M i−1 b ∈ n works, for large enough M , if
V ε x ≤ b (as above) defines an n-gon.
LECTURE 5. PROJECTED PRODUCTS OF POLYGONS 677
Example 5.3. To illustrate this in a simple case, let’s consider deformed products
in the low-dimensional case of I × I, a square, where I denotes an interval (a
1-dimensional polytope) such as I = [0, 1].
In this case I can be written as
0
I = x∈ : Ê −1
1
x ≤
1
.
leaves the first two inequalities (and thus the first I factor) intact, but it changes
the slopes of the other two inequalities — and if you are unlucky (that is, for
a + b > 1) the resulting polytope will not be equivalent to I × I any more. This
situation is depicted in the middle part of Figure 5.2. However, it can be remedied
by increasing right-hand sides: For any given a and b, a suitably large M , namely
M > a + b, in
⎛ ⎞ ⎛ ⎞
−1 0 0
⎜ 1 0 ⎟ ⎜ 1 ⎟
⎜ ⎟ ⎜ ⎟
⎝ a −1 ⎠ x ≤ ⎝ 0 ⎠
b 1 M
will result in a product again (as in the right part of Figure 5.2).
x2 = M − bx1
x2 = 1 M
x2 = 1 − bx1 adjusting
1 right hand
deformation sides
I ×I
1 1 1
x2 = ax1
x2 = 0
nG
n2 n2
F2 G F1
For Theorem 5.1 we have to specify a deformed product realization for (Pn )r of the
type given in Ansatz (5.1), such that all n-gon 2-faces are strictly preserved by the
projection. That is,
• if we choose two cyclically adjacent rows from each block except for one, and
• truncate these rows to the first 2r − 4 coordinates,
then the resulting 2r − 2 vectors must be positively dependent and span.
5.5. Construction
Now we want to specify the lower-triangular block matrix in our Ansatz (5.1) so
that it satisfies the following two main properties:
(1) the diagonal blocks have “rows in cyclic order,” and
(2) any “choice of two cyclically adjacent rows” from all but one of the blocks,
truncated to the first 2r − 4 components, yields a positively-spanning set of
vectors.
Five observations (you may call them “tricks”) help us to achieve this:
(i) Condition (2) is stable under perturbation. So, we first construct a matrix
that satisfies (2), then perturb it in order to achieve (1).
(The diagonal blocks of the matrix that we construct to satisfy (2) are de-
noted V ; after perturbation, they will be V ε .)
LECTURE 5. PROJECTED PRODUCTS OF POLYGONS 679
(ii) The submatrices V , W , and U of size n×2 are constructed to have alternating
rows: So if you choose two cyclically adjacent rows from a block, you know
what you get!
Specifically, we will let matrix V have rows that alternate between (1, 0)
and (0, 0), matrix W gets rows (0, 1) and (a, b), and matrix U gets rows (c, d)
and (e, f ), with the six parameters a, b, c, d, e, f to be determined.
(iii) To make sure that you get the positive linear dependence for (2), we specify a
positive coefficient sequence and compute the matrix entries to satisfy them.
(iv) Rather than admitting that from one of the blocks no row is chosen, we will
prescribe coefficient sequences that could have zeroes on any one of the blocks,
which yields linear dependencies for which the vectors from one block are “not
used.”
Specifically, we take coefficient sequences of the form αk := (2k−t − 1)2
and βk := (2k−t − 1)(2k−t − 32 ) for the odd resp. even-index rows. These
coefficients are clearly positive for integral k, except they vanish at k = t.
Moreover, they are linear combinations of the three exponential functions
2k−t , 4k−t , and 1. If we write out the condition that “the rows chosen should
be dependent, with coefficients αk (for the even-index row chosen from the
k-th block) and βk (for the odd-index row chosen from the k-th block), then
this leads to a system of six linear equations, in six unknowns a, b, c, d, e, f —
solve it!
(v) The properties “alternating rows” and “rows in cyclic order” are, of course,
incompatible — but a matrix with rows in cyclic order can be obtained as a
perturbation V ε of the matrix V that has rows (1, 0) and (0, 0) in alternation.
Figure 5.4 suggests a way to do this.
This completes our sketch of “how to do it” — it should be sufficient to let you
construct the polytopes Pnr and thus prove Theorem 5.1 (but [81] provides these
details, too.)
If you think about this construction, can you perhaps manage to simplify some
of the details, or even to improve the construction? After all, on the way to the
characterization of the f -vector cone for 4-polytopes, this construction of polytopes
of fatness up to 9 should be taken just as an intermediate step. Can you get further?
There is a long way to go from 9 to infinity . . .
680 GÜNTER M. ZIEGLER, CONVEX POLYTOPES
Exercises
5.1. Show that every polytope arises as a projection of a simplex.
Ê Ê
5.2. Show that if π : n → d , P → π(P ) is a polytope projection, for an n-
polytope P , and n > d, then the strictly preserved faces have dimension at
most d − 1. In particular, no facet of P is strictly preserved by the projection.
5.3. Give examples of polytope projections where no faces are strictly preserved.
5.4. Realize the prism over an n-gon (n ≥ 3) in such a way that the projection
Ê3
Ê
→ 2 strictly preserves all 2n vertices.
5.5. Show that the product Δ2 × Δ2 of two triangles cannot be realized in such a
Ê
way that all 9 vertices are strictly preserved in a projection to 2 .
5.6. How large do we have to choose r and n in order to obtain 4-polytopes of
fatness F (π(Pnr )) > 8?
5.7. Any projection of a (non-deformed) product of centrally symmetric polygons
is a zonotope. For these, one knows that f1 < 3f0 (for such inequalities, for
the dual polytope, see [16, pp. 198/199]). Deduce from this information that
the fatness is smaller than 5.
APPENDIX
A Short Introduction to polymake
by Thilo Schröder and Nikolaus Witte
The software project polymake [31] has been developed since 1997 in the Discrete
Geometry group at TU Berlin by Ewgenij Gawrilow and Michael Joswig, with
contributions by several others. It was initially designed to work with convex poly-
topes. Due to its open design the polymake framework can also be used on other
types of objects; the current release includes a second application, topaz, which
treats simplicial complexes.
polymake is designed to run on any Linux or Unix system, including Mac OS X.
It runs in a shell using command line input. This introduction is for polymake
versions 2.0 and 2.1. polymake is free software and you can redistribute it and/or
modify it under the terms of the GNU General Public License as published by the
Free Software Foundation.
This introduction aims at getting you to work on the computer rather than
explaining the details about the machinery of polymake. Therefore, there will be
only a short description of the software design, in Section A.2. For further reading
we refer to Gawrilow & Joswig [32] [33]. On the polymake website
http://www.math.tu-berlin.de/polymake
you will find extensive online documentation as well as an introductory tutorial.
This section explains the command line syntax of polymake for constructing,
analyzing and visualizing polytopes. The polymake file format is briefly described
at the end of this section.
1 − x0 − x1 ≥ 0
x1 ≥ 0
1
x0 ≥ 0
Example. If you want to construct the triangle Δ (cf. Figure A.1) that is given as
Δ := conv{(0, 0), (0, 1), (1, 0)}
your input file should look as follows (homogeneous coordinates):
POINTS
1 0 0
1 0 1
1 1 0
If you prefer to enter the same example by the inequality description
Δ := {x ∈ R2 : 1 − x0 − x1 ≥ 0, x0 ≥ 0, x1 ≥ 0},
this should be your polymake file:
INEQUALITIES
1 -1 -1
0 1 0
0 0 1
The difference between POINTS and VERTICES is that the former may contain
redundancies. So you should enter POINTS, but ask for VERTICES. Similarly, one
should enter INEQUALITIES and ask for FACETS.
684 GÜNTER M. ZIEGLER, CONVEX POLYTOPES
external
software
scratch object
The rule base is a collection of rules, each rule containing a set of input and output
properties and an algorithm (that is a client or external software) which computes
the output properties directly from the input properties. If you request a property of
an object, the server has to determine how to compute the requested property from
the ones which are already known. There might not be an algorithm computing the
requested property directly, so other properties might have to be computed first.
Therefore, the server has to compose a sequence of rules (from the rule base) to be
executed in order to compute the requested property.
appendix: a short introduction to polymake 685
A.8. Use the client rand sphere to create a random polytope by uniformly dis-
tributing 1000 points on the unit 2-sphere, visualize it and compare it to Fig-
ure 1.4. Take a look at its VERTEX DEGREES and cut off a vertex of maximal
degree.
A.9. Take a 3-polytope and truncate all its vertices. Is the resulting polytope
always simple?
A.10. Truncate the vertices of the 4-dimensional cross polytope, and let polymake
compute the f -vector, and whether the resulting polytope is simple.
Can you justify the results theoretically?
A.11. For a given planar 3-connected graph G containing a triangular face, produce
a 3-polytope with G as its graph (see Section 1.2). Use the client tutte
lifting.
A.12. Visualize the Schlegel diagrams of the dwarfed cube that you get by using
different projection facets.
A.13. Visualize the effect of standard constructions (such as truncation, stellar
subdivision) on the Schlegel diagrams of 4-polytopes.
BIBLIOGRAPHY