K-homology The assembly map Coarse homotopy A generalized conjecture

Coarse Index Theory Lecture 4

May 2006

John Roe

Coarse Index Theory Lecture 4

K-homology The assembly map Coarse homotopy A generalized conjecture

Outline

K-homology The assembly map Coarse homotopy A generalized conjecture

John Roe

Coarse Index Theory Lecture 4

K-homology The assembly map Coarse homotopy A generalized conjecture

K -homology

At the end of the last lecture we saw that coarse K-theory for open cones is a generalized homology theory. In fact it turns out to be K-homology the homology theory dual to K-theory. Well begin this lecture by reviewing the denition and properties of K -homology.

John Roe

Coarse Index Theory Lecture 4

K-homology The assembly map Coarse homotopy A generalized conjecture

Let X be a locally compact space and let H be an X -module. Denition Let 1 (X ; H) denote the C -algebra of locally compact operators on the X -module H. Reminder: an operator T is locally compact if T (f ) and (f )T are compact, for every f C0 (X ). Here is the representation dening the X -module structure of H.

John Roe

Coarse Index Theory Lecture 4

K-homology The assembly map Coarse homotopy A generalized conjecture

Pseudolocal operators

Denition We will say that a bounded operator T on H is pseudolocal if T (f ) (f )T is compact for all f C0 (X ). We denote the C -algebra of pseudolocal operators by 0 (X ; H). It is easy to see that 1 is an ideal in 0 . Lemma (Kasparov) An operator T is pseudolocal if and only if (f )T (g) is compact whenever f and g have disjoint supports.

John Roe

Coarse Index Theory Lecture 4

K-homology The assembly map Coarse homotopy A generalized conjecture

The algebras depend on the choice of X -module, just as C (X ) does. To dene their K -theory we shall use the same colimit construction as we do in the case of C (X ). Denition The K -homology groups of X are dened by Ki (X ) := Ki+1 (0 (X )/1 (X )). (This is sometimes referred to as the Paschke duality theorem. However, with our colimit interpretation of K -theory, it becomes almost a tautology.) Kasparov (following a suggestion of Atiyah) proved that the K -homology groups do constitute a generalized homology theory they are homotopy invariant and have exact sequences.
John Roe Coarse Index Theory Lecture 4

K-homology The assembly map Coarse homotopy A generalized conjecture

Example Let X be a point. Then 0 (X ) = B(H) and 1 (X ) = K(H). Thus the quotient is the Calkin algebra, and Ki (X ) = Ki+1 (B/K) = Z (i = 0) . 0 (i = 1)

This explains the dimension shift in the denition of K -homology.

John Roe

Coarse Index Theory Lecture 4

K-homology The assembly map Coarse homotopy A generalized conjecture

Suppose that X is a manifold. Then every elliptic operator D on X gives rise to a K -homology class. If D is essentially self-adjoint we can construct this as [(D)], just as in our discussion of the coarse index but neglecting the coarse structure. Remark: This version of K -homology is a locally nite homology theory it pairs with K -cohomology with compact supports.

John Roe

Coarse Index Theory Lecture 4

K-homology The assembly map Coarse homotopy A generalized conjecture

Notice that the algebras make no use of any coarse structure. Denition Let X be a proper coarse space and H an X -module. Dene D (X ; H) to be the C -algebra generated by the controlled, pseudolocal operators on H. Lemma Forgetting the coarse structure induces an isomorphism of C -algebras D (X )/C (X ) 0 (X )/1 (X ). = We will outline the proof.

John Roe

Coarse Index Theory Lecture 4

K-homology The assembly map Coarse homotopy A generalized conjecture

Choose a locally nite partition of unity i subordinate to a uniformly bounded open cover of X . (This uses properness.) The operation (T ) =
i

(i

1/2

)T (i

1/2

denes a positive linear map B(H) B(H), and its range consists of controlled operators. Observe that if T is pseudolocal, then (T ) T =
i

[(i

1/2

), T ](i

1/2

is locally compact.
John Roe Coarse Index Theory Lecture 4

K-homology The assembly map Coarse homotopy A generalized conjecture

It now sufces to show

0 = D + 1 C = D 1

Item (1) follows directly from the last observation on the previous slide. Indeed, if T is pseudolocal then (T ) T is locally compact (belongs to 1 ) and (T ) is pseudolocal and of nite propagation, so belongs to D (X ).

John Roe

Coarse Index Theory Lecture 4

K-homology The assembly map Coarse homotopy A generalized conjecture

As for item (2), let T D 1 . Then T is locally compact and there is a sequence Tn of nite propagation, pseudolocal operators tending to T . The operators Tn (Tn ) are locally compact and of nite propagation, so their limit T (T ) belongs to C (X ). But (T ) is of nite propagation, and it is locally compact because T is, so belongs to C (X ). Thus T = (T (T )) + (T ) belongs to C (X ) also.

John Roe

Coarse Index Theory Lecture 4

K-homology The assembly map Coarse homotopy A generalized conjecture

Consider the short exact sequence 0 C (X ) D (X ) D (X )/C (X ) 0 of C -algebras. The associated K -theory long exact sequence contains Ki+1 (D (X )/C (X )) Ki (C (X )). But Ki+1 (D (X )/C (X )) = Ki+1 (0 (X )/1 (X )) = Ki (X ) by the lemma we just proved. Thus we get a map A : Ki (X ) Ki (C (X )) from K -homology to the K -theory of the coarse C -algebra. Denition A is called the (coarse) assembly map.
John Roe Coarse Index Theory Lecture 4

K-homology The assembly map Coarse homotopy A generalized conjecture

The coarse assembly map takes the K -homology class of an elliptic operator to its coarse index. Thus, the basic problem of coarse index theory can be reformulated as: Compute the coarse assembly map. Theorem If X = O(Z ) is the open cone on a nite simplicial complex Z , then the coarse assembly map for X is an isomorphism. We have already proved this, essentially. The proof is an inductive argument simplex-by-simplex on Z , using the Mayer-Vietoris sequences in homology and coarse K -theory, and the ve lemma.

John Roe

Coarse Index Theory Lecture 4

K-homology The assembly map Coarse homotopy A generalized conjecture

Could the assembly map always be an isomorphism? Clearly not, because the left hand side depends on the small scale (topological) structure of X only and the right hand side depends on the large scale (coarse) structure only. For cones, the small and large scale structures exactly match that is why the assembly map is an isomorphism. In the next denition, let E be a controlled set in a coarse space X . A map Y X will be called an E-map if its range lies within some ball Ex . Denition Let X be a proper coarse space. Say that X is uniformly contractible if for every n and every controlled E, there is a controlled F such that every continuous E-map S n1 X extends to a continuous F -map D n X .
John Roe Coarse Index Theory Lecture 4

K-homology The assembly map Coarse homotopy A generalized conjecture

The coarse BaumConnes conjecture

John Roe

Coarse Index Theory Lecture 4

K-homology The assembly map Coarse homotopy A generalized conjecture

A natural test case for the coarse BaumConnes conjecture: consider complete simply-connected Riemannian manifolds of non-positive curvature. The properties of geodesics (i.e., the Cartan-Hadamard theorem) tell us that such a manifold is uniformly contractible. In fact, each metric ball is contractible. The logarithm map M Tx0 M is a coarse map to Euclidean space, but it is not a coarse equivalence. So it does not immediately gives rise to a computation of K (C (M)). We need a more exible notion of equivalence.

John Roe

Coarse Index Theory Lecture 4

K-homology The assembly map Coarse homotopy A generalized conjecture

Coarse homotopy
Let X and Y be proper coarse spaces. Denition A proper and continuous map H : X [0, 1] Y is a coarse homotopy if the associated family of maps ht : X Y is equi-coarse: that is, for every controlled set E X X , the union t (ht ht )(E) Y Y is controlled. This is strictly weaker than the assertion that H is a coarse map (when X [0, 1] is given a product coarse structure). (It is equivalent to the assertion that H is coarse from another, weaker coarse structure.) The tracks t ht (x) can be arbitrarily long.
John Roe Coarse Index Theory Lecture 4

K-homology The assembly map Coarse homotopy A generalized conjecture

Key example
Consider X = Rn and let (r , ) be polar coordinates ( S n1 ). Let be any Lipschitz function R+ R+ which is increasing and tends to (e.g. (r ) = log+ log+ log+ r ). Then f : (r , ) ((r ), ) denes a coarse map. Moreover, F : (r , , t) ((1 t)(r ) + tr , ) gives a coarse homotopy between f and the identity. Let Y be a complete, simply-connected, nonpositively curved Riemannian manifold. Let : Y X be the logarithm map and let g : X Y be exp f where f is as above. By suitable choice of we can make g coarse. Then g and are inverses up to coarse homotopy. Thus Y is coarsely homotopy equivalent to Euclidean space.
John Roe Coarse Index Theory Lecture 4

K-homology The assembly map Coarse homotopy A generalized conjecture

Theorem Coarsely homotopy equivalent coarse maps X Y induce the same homomorphism K (C (X )) K (C (Y )). This theorem was proved by HigsonR. However, recently VietTrung Luu has found a more elegant proof. It involves replacing the Hilbert spaces in the denition of C (X ) by Hilbert A-modules. A Hilbert A-module is just a Hilbert space in which the scalars are taken from the C -algebra A. Using this notion one can dene C (X ; A), the coarse C -algebra with coefcients in A.

John Roe

Coarse Index Theory Lecture 4

K-homology The assembly map Coarse homotopy A generalized conjecture

Luu shows that a coarse homotopy H denes a map H : K (C (X )) K (C (Y ; C[0, 1])). Then he proves Proposition There is a natural product K (C (Y ; C(Z ))) K (Z ) K (C (Y )). The group K (Z ) appearing here is the K -homology of Z . In the case Z = [0, 1], let i0 , i1 be the two natural inclusions of a point into Z .Then one can see that the composite K (C (X ))
H

/ K (C (Y ; C[0, 1]))

it (1)

/ K (C (Y ))

is equal to H(, t) . But i0 (1) = i1 (1) by the homotopy invariance of K -homology. It follows that H(, 0) = H(, 1)
John Roe Coarse Index Theory Lecture 4

K-homology The assembly map Coarse homotopy A generalized conjecture

Using coarse homotopy one can show that the coarse BaumConnes conjecture is true for Hadamard manifolds M. Further progress has come from a different approach. First, let us reformulate the conjecture in a way that does not use uniform contractibility. In fact it makes most sense to formulate this new version for discrete bounded geometry spaces. Denition Let X be a discrete bg coarse space and let E be a controlled set. The Rips complex PE (X ) is the simplicial complex whose vertices are the points of X , and such that x0 , . . . , xp span a p-simplex iff {x0 , . . . , xp } is contained in some E-ball.

John Roe

Coarse Index Theory Lecture 4

K-homology The assembly map Coarse homotopy A generalized conjecture

If E E then PE (X ) is a subcomplex of PE (X ). Therefore, the homology groups K (PE (X )) form a direct system. Denition The coarse K -homology of X , denoted KX (X ), is the direct limit lim K (PE (X )).
E

It can be shown that, if there exists a uniformly contractible space EX coarsely equivalent to X , then KX (X ) = K (EX ).

John Roe

Coarse Index Theory Lecture 4

K-homology The assembly map Coarse homotopy A generalized conjecture

Passing to the limit we obtain the coarse assembly map KX (X ) K (C (X )) for any (discrete, but this condition can be removed) coarse space X . It is natural to conjecture Conjecture The coarse assembly map is always an isomorphism (?) The work of Guoliang Yu has shown that this conjecture is true in a very many cases (e.g., whenever X can be coarsely embedded in a Hilbert space). However, it is false in general! (Higson 2000)
John Roe Coarse Index Theory Lecture 4

K-homology The assembly map Coarse homotopy A generalized conjecture

Box spaces
Let G be a nitely generated discrete group. Denition G is residually nite if there exists a sequence Gn of nite index normal subgroups of G with Gn = {1}. If G is residually nite, we dene the box space G to be the coarse disjoint union G/Gn . Notice that G acts on each G/Gn and thus on G; we dene the coarse structure on G by requiring that E is controlled if and only if there is a nite subset F of G such that, for each (x, y ) E, there is g F such that x = gy .

John Roe

Coarse Index Theory Lecture 4

K-homology The assembly map Coarse homotopy A generalized conjecture

Denition A translation of a coarse space X is a bijection X X whose graph is controlled. Proposition Let G be a (discrete) group that acts on a (discrete) coarse space X by translations. There is an induced -homomorphism
Cmax (G) C (X ).

It is important that we use the maximal C -algebra here. In particular, we get a homomorphism Cmax (G) C ( G).

John Roe

Coarse Index Theory Lecture 4

K-homology The assembly map Coarse homotopy A generalized conjecture

Property T

Denition Let G be a discrete group. G has property T if there exists a projection p Cmax (G) that has the following property: for every representation : G U(H), the image (p) B(H) is the orthogonal projection onto the subspace of G-invariant vectors. The denition is due to Kazhdan, and p is called the Kazhdan projection. Obviously every nite group has property T, but there exist innite examples also (this is rather surprising).

John Roe

Coarse Index Theory Lecture 4

K-homology The assembly map Coarse homotopy A generalized conjecture

Let G be an innite, residually nite, property T group (e.g. G = SL(3, Z)) and let X = G. Let q C (X ) be the image of the Kazhdan projection p under the canonical homomorphism Cmax (G) C (X ). We know that q is the orthogonal projection onto the G-invariant functions, i.e. those functions constant on each coarse component of X . In particular, q is a projection with innite-dimensional range. Remark q is not a controlled operator. Property T tells us that it is a limit of controlled operators, but this is mysterious!

John Roe

Coarse Index Theory Lecture 4

K-homology The assembly map Coarse homotopy A generalized conjecture

Higson showed that (under appropriate hypotheses, fullled e.g. in the case of SL(3, Z)) the K -theory class [q] K0 (C (X )) does not belong to the image of the coarse assembly map. Well prove a related but weaker result, namely that q is not a limit of nite propagation idempotents. (The relationship is that any index can, as we know, be represented by an idempotent of nite propagation.) Denition (Yu) Let X be a discrete coarse space. An operator T C (X ) is a ghost if its matrix entries tend to 0. The ghosts form an ideal in C (X ) which is (possibly) not geometric, i.e. not of the form IY .

John Roe

Coarse Index Theory Lecture 4

K-homology The assembly map Coarse homotopy A generalized conjecture

Lemma Every compact operator is a ghost, and every nite propagation ghost is compact. The element q C ( G) coming from the Kazhdan projection is a ghost but not compact (it is a projection with innite-dimensional range). Suppose that there is a sequence qn of nite propagation idempotents tending in norm to q. Then some qn is similar to q; hence it is a ghost (because the ghosts form an ideal); hence it is compact (because nite propagation ghosts are compact); hence q is compact also. This is a contradiction.

John Roe

Coarse Index Theory Lecture 4