Loop groups and twisted K-theory I

2011, Journal of Topology

Journal of Topology 4 (2011) 737–798 ❡ 2011 London Mathematical Society C doi:10.1112/jtopol/jtr019 Loop groups and twisted K-theory I Daniel S. Freed, Michael J. Hopkins and Constantin Teleman Abstract This is the first in a series of papers investigating the relationship between the twisted equivariant K-theory of a compact Lie group G and the “Verlinde ring” of its loop group. In this paper we set up the foundations of twisted equivariant K-groups, and more generally twisted K-theory of groupoids. We establish enough basic properties to make eﬀective computations. We determine the twisted equivariant K-groups of a compact connected Lie group G with torsion free fundamental group. We relate this computation to the representation theory of the loop group at a level related to the twisting. Contents 1. Twisted K-theory by example 2. Twistings of K-theory . 3. Twisted K-groups . . τ (G) 4. Computation of KG . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741 747 758 768 797 Introduction Equivariant K-theory focuses a remarkable range of perspectives on the study of compact Lie groups. One finds tools from topology, analysis, and representation theory brought together in describing the equivariant K-groups of spaces and the maps between them. In the process all three points of view are illuminated. Our aim in this series of papers [18, 19] is to begin the development of similar relationships when a compact Lie group G is replaced by LG, the infinite-dimensional group of smooth maps from the circle to G. There are several features special to the representation theory of loop groups. First of all, we focus only on the representations of LG that have ‘positive energy.’ This means that the representation space V admits an action of the rotation group of the circle that is (projectively) compatible with the action of LG, and for which there are no vectors v on which rotation by θ acts by multiplication by einθ with n < 0. It turns out that most positive energy representations are projective, and so V must be regarded as a representation of a central extension LGτ of LG by U (1). The topological class of this central extension is known as the level. One thing a topological companion to the representation theory of loop groups must take into account is the level. Next there is the fusion product. Write Rτ (LG) for the group completion of the monoid of positive energy representations of LG at level τ . In [34], Verlinde introduced a multiplication on Rτ (LG) ⊗ C making it into a commutative ring (in fact a Frobenius algebra). This Received 25 April 2010; revised 13 July 2011. 2010 Mathematics Subject Classification 19L50 (primary), 22E67. During the course of this work the first author was supported by the National Science Foundation under the grants DMS-0072675 and DMS- 0305505, the second by grants DMS-9803428 and DMS-0306519, and the third by DMS-0072675. Collaboration between the authors was greatly facilitated by the KITP of Santa Barbara (NSF Grant PHY99-07949) and the Aspen Center for Physics. 738 D. S. FREED, M. J. HOPKINS AND C. TELEMAN multiplication is called fusion, and Rτ (LG) ⊗ C equipped with the fusion product is known as the Verlinde algebra. The fusion product also makes Rτ (LG) into a ring, which we call the Verlinde ring. A good topological description of Rτ (LG) should account for the fusion product in a natural way. The positive energy representations of loop groups turn out to be completely reducible, and, somewhat surprisingly, there are only finitely many irreducible positive energy representations at a fixed level. Moreover, an irreducible positive energy representation is determined by its lowest non-trivial energy eigenspace, V (n0 ), which is an irreducible (projective) representation of G. Thus, the positive energy representations of LG correspond to a subset of the representations of the compact group G. This suggests that G-equivariant K-theory might somehow play a role in describing the representations of LG. In fact this is the case. The following is our main theorem. Theorem 1. Let G be a connected compact Lie group and τ be a level for the loop group. ζ(τ ) The Grothendieck group Rτ (LG) at level τ is isomorphic to a twisted form KG (G), of the equivariant K-theory of G acting on itself by conjugation. Under this isomorphism the fusion product, when it is defined, corresponds to the Pontryagin product. The twisting ζ(τ ) is given in terms of the level ζ(τ ) = g + ȟ + τ, where ȟ is the ‘dual Coxeter’ twisting. Several aspects of this theorem require clarification. The main new element is the ‘twisted form’ of K-theory. Twisted K-theory was introduced by Donovan and Karoubi [14] in connection with the Thom isomorphism, and generalized and further developed by Rosenberg [29]. Interest in twisted K-theory was rekindled by its appearance in the late 1990s (see [26, 35]) in string theory. Our results came about in the wake of this revival when we realized that the work of the first author [17] on Chern–Simons theory for finite groups could be interpreted in terms of twisted K-theory. The twisted forms of G-equivariant K-theory are classified by the nerve of the category of invertible modules over the equivariant K-theory spectrum KG . What comes up in geometry though, is only a small subspace, and throughout this paper the term ‘twisting’ refers to twistings in this restricted, more geometric class. These geometric twistings of KG -theory on a G-space X are classified up to isomorphism by the set 0 1 3 (X; Z/2) × HG (X; Z/2) × HG (X; Z). HG (2) The component in H 0 corresponds to the ‘degree’ of a K-class, and the fact that the coeﬃcients are the integers modulo 2 is a reflection of Bott periodicity. In this sense ‘twistings’ refine the notion of ‘degree’, although, when considering twistings, it is important to remember more than just the isomorphism class. The tensor product of KG -modules makes these spaces of twistings into infinite loop spaces and provides a commutative group structure on the sets of isomorphism classes. The group 0 1 3 structure in (2) is the product of HG (X; Z/2) with the extension of HG (X; Z/2) by HG (X; Z) with cocycle β(x ∪ y), where β is the Bockstein homomorphism. A twisting is a form of equivariant K-theory on a space. A level for the loop group, on the other hand, corresponds to a central extension of LG. One way of relating these two structures 3 (G), is via the classification (2). A central extension of LG has a topological invariant HG and so give rises to a twisting, up to isomorphism. When the group G is simple and simply 1 (G) connected, this invariant determines the central extension up to isomorphism, the group HG 3 vanishes, and there is a canonical isomorphism HG (G) ≈ Z. In this case, an integer can be used LOOP GROUPS AND TWISTED K-THEORY I 739 to specify both a level and a twisting. There is a more refined version of this correspondence directly relating twistings to the central extension, and the approach to twistings we take in this paper is designed to make this relationship as transparent as possible. There is a map from vector bundles to twistings, which associates to a vector bundle V over X the family of K-modules K V̄x , where V̄x is the one point compactification of the fiber of V over x ∈ X, and for a space S, K S is the K-module with π0 K S = K 0 (S). We denote the twisting associated to V by τV , although when no confusion is likely to arise, we just use the symbol V . The invariants of τV in (2) are dim V , w1 (V ), and βw2 (V ). These twistings are described by Donovan and Karoubi [14] from the point of view of Cliﬀord algebras. −1 (X) to the group of twistings of equivariant There is also a homomorphism from KOG K-theory on X. In topological terms it corresponds to the map from the stable orthogonal group O to its third Postnikov section O0, . . . , 3. It sends an element of KO−1 to the twisting whose components are (σw1 , σw2 , x), where σ : H ∗ (BO) → H ∗−1 (O) is the cohomology suspension, and x ∈ H 3 (O; Z) is the unique element, twice of which is the cohomology suspension of p1 . In terms of operator algebras this homomorphism sends a skew-adjoint Fredholm operator to its graded Pfaﬃan gerbe. We call this map pfaff . We now describe two natural twistings on G, which are equivariant for the adjoint action. The first comes from the adjoint representation of G regarded as an equivariant vector bundle over a point, and pulled back to G. We write this twisting as g. For the other, first note that the ∗ (G) is just H ∗ (LBG). The vector bundle associated to the equivariant cohomology group HG adjoint representation gives a class ad ∈ KO0 (BG), which we can transgress to KO−1 (LBG), and then map to twistings by the map pfaff . We call this twisting ȟ. When G is simple and simply connected, the integer corresponding to ȟ is the dual Coxeter number, and g is just a degree shift. With these definitions, the formula ζ(τ ) = g + ȟ + τ in the statement of Theorem 1 should be clear. The twistings g and ȟ are those just described, and τ is the twisting corresponding to the level. Theorem 1 provides a topological description of the group Rτ (LG) and its fusion product ζ(τ ) when it exists. But it also gives more. The twisted K-group KG (G) is defined for any compact Lie group G, and it makes sense for any level τ . This points the way to a formulation of an analog of the group Rτ (LG) for any compact Lie group G (even one that is finite). In Parts II and III, we take up these generalizations and show that the assertions of Theorem 1 remain true. One thing that emerges from our topological considerations is the need to consider Z/2graded central extensions of loop groups. Such extensions are necessary when working with a group like SO(3), whose adjoint representation is not Spin. Another interesting case is that of O(2). When the adjoint representation is not orientable, the dual Coxeter twist makes a 0 (G; Z/2). One sees an extra change in the degree in which the non-trivial contribution to HG interesting K-group occurs. In the case of O(2) these degree shifts are diﬀerent on the two connected components, again emphasizing the point that twistings should be regarded as a generalization of degree. The ‘Verlinde ring’ in this case is composed of an even K-group on one component and an odd K-group on another. Such inhomogeneous compositions are not typically considered when discussing ordinary K-groups. The fusion product on Rτ (LG) has been defined for simple and simply connected G, and ζ(τ ) in a few further special cases. The Pontryagin product on KG (G) is defined exactly when τ is primitive in the sense that its pullback along the multiplication map of G is isomorphic to the sum of its pullbacks along the two projections. This explains, for example, why a fusion product on Rτ (LSO(n)) exists only at half of the levels. Using the Pontryagin product, we are able to define a fusion product on Rτ (LG) for any G at any primitive level τ . We do not, however, give a construction of this product in terms of representation theory. 740 D. S. FREED, M. J. HOPKINS AND C. TELEMAN When the fusion product is defined on Rτ (LG), it is part of a much more elaborate structure. For one thing, there is a trace map Rτ (LG) → Z making Rτ (LG) into a Frobenius algebra. Using twisted K-theory, we construct this trace map for general compact Lie groups at primitive levels τ , which are non-degenerate in the sense that the image of τ in HT3 (T ; R) ≈ H 1 (T ; R) ⊗ H 1 (T ; R) is a non-degenerate bilinear form. Again, in the cases when the fusion product has been defined, there are operations on Rτ (LG) coming from the moduli spaces of Riemann surfaces with boundary, making Rτ (LG) part of what is often called a topological conformal field theory. Using topological methods, we are able to construct a topological conformal field theory for any compact Lie group G, at levels τ that are transgressed from (generalized) cohomology classes on BG and are non-degenerate. Some of this work appears in [20, 21]. Another impact of Theorem 1 is that it brings the computational techniques of algebraic topology to bear on the representations of loop groups. One very interesting approach, for connected G, is to use the Rothenberg–Steenrod spectral sequence relating the equivariant K-theory of ΩG to that of G. In this case, one gets a spectral sequence G TorK∗ (ΩG) (R(G), R(G)) =⇒ KτG+∗ (G), (3) relating the untwisted equivariant K-homology of ΩG, and the representation ring of G to the Verlinde algebra. The ring K∗G (ΩG) can be computed using the techniques of Bott [5] and Bott–Samelson [6] and has also been described by Bezrukavnikov, Finkelberg, and Mirković [4]. The K-groups in the E 2 -term are untwisted. The twisting appears in the way that the representation ring R(G) is made into an algebra over K G (ΩG). The equivariant geometry of ΩG has been extensively studied in connection with the representation theory of LG, and the spectral sequence (3) seems to express yet another relationship. We do not know of a representation-theoretic construction of (3). An analog of the spectral sequence (3) has been used by Douglas [15] to compute the (non-equivariant) twisted K-groups K τ (G) for all simple, simply connected G. Using the Lefschetz fixed point formula, one can easily conclude for connected G that ∆−1 K∗G (ΩG) = ∆−1 Z[Λ × Π], where ∆ is the square of the Weyl denominator, Π = π1 T is the coweight lattice, and Λ is the weight lattice. When the level τ is non-degenerate there are no higher Tor groups, and the spectral sequence degenerates to an isomorphism ∆−1 KτG+∗ (G) ≈ ∆−1 R(G)/I τ , where I τ is the ideal of representations whose characters vanish on certain conjugacy classes. The main computation of this paper asserts that such an isomorphism holds without inverting ∆ when G is connected, and π1 G is torsion-free. The distinguished conjugacy classes are known ζ(τ ) as Verlinde conjugacy classes, and the ideal I τ as the Verlinde ideal. In [21], the ring KG (G) ⊗ C is computed using a fixed point formula, and shown to be isomorphic to the Verlinde algebra. The plan of this series of papers is as follows. In Part I, we define twisted K-groups, and ζ (G) for connected G with torsion-free fundamental group, at noncompute the groups KG degenerate levels ζ. Our main result is Theorem 4.27. In Part II, we introduce a certain family of Dirac operators and our generalization of Rτ (LG) to arbitrary compact Lie groups. We ζ(τ ) construct a map from Rτ (LG) to KG (G) and show that it is an isomorphism when G is connected with torsion-free fundamental group. In Part III, we show that our map is an isomorphism for general compact Lie groups G, and develop some applications. The bulk of this paper is concerned with setting up twisted equivariant K-theory. There are two things that make this a little complicated. First, when working with twistings it is important to remember the morphisms between them, and not just the isomorphism classes. LOOP GROUPS AND TWISTED K-THEORY I 741 The twistings on a space form a category and spelling out the behavior of this category as the space varies becomes a little elaborate. The other thing has to do with the kind of G-spaces we use. We need to define twistings on G-equivariant K-theory in such a way as to make clear what happens as the group G changes, and for the constructions in Part II we need to make the relationship between twistings and (graded) central extensions as transparent as possible. We work in this paper with groupoids and define twisted equivariant K-theory for groupoids. Weakly equivalent groupoids (see Appendix A) have equivalent categories of twistings and isomorphic twisted K-groups. A group G acting on a space X forms a special kind of groupoid X//G called a ‘global quotient groupoid’. A central extension of G by U (1) defines a twisting of K-theory for X//G. If G is a compact connected Lie group acting on itself by conjugation, and P G denotes the space of paths in G starting at the identity, acted upon by LG by conjugation, then P G//LG −→ G//G is a local equivalence, and a (graded) central extension of LG defines a twisting of P G//LG and hence of G//G. In general, we define a twisting of a groupoid X to consist of a local equivalence P → X and a graded central extension P̃ of P . Although most of the results we prove reduce, ultimately, to ordinary results about compact Lie groups acting on spaces, not all do. In Part III, it becomes necessary to work with groupoids that are not equivalent to a compact Lie group acting on a space. Kitchloo [24] has pointed out that the space P G is the universal LG space for proper actions. Using this, he has described a generalization of our computation to other Kac–Moody groups. At the time we began this work, the paper of Atiyah and Segal [1] was in preparation, and we benefited a great deal from early drafts. Since that time several other approaches to twisted K-theory have appeared. In addition to [1], we refer the reader to [10, 33]. We have chosen to use ‘graded central extensions’ because of the close connection with loop groups and the constructions we wish to make in Part II. Of course, our results can be presented from any of the points of view mentioned above, and the choice of which is a matter of personal preference. We have attempted to organize this paper so that the issues of implementation are independent of the issues of computation. Section 1 is a kind of field guide to twisted Ktheory. We describe a series of examples intended to give the reader a working knowledge of twisted K-theory suﬃcient to follow the main computation in § 4. Section 2 contains our formal discussion of twistings of K-theory for groupoids, and our definition of twisted K-groups appears in § 3. We have attempted to axiomatize the theory of twisted K-groups in order to facilitate comparison with other models. Our main computation appears in § 4. We assume throughout this paper that all spaces are locally contractible, paracompact, and completely regular. These assumptions imply the existence of partitions of unity [13] and locally contractible slices through actions of compact Lie groups [27, 28]. 1. Twisted K-theory by example The K-theory of a space is assembled from data that are local. To give a vector bundle V on X is equivalent to giving vector bundles Vi on the open sets Ui of a covering, and isomorphisms λij : Vi −→ Vj on Ui ∩ Uj satisfying a compatibility (cocycle) condition on the triple intersections. In terms of K-theory this is expressed by the Mayer–Vietoris (spectral) sequence relating K(X) and the K-groups of the intersections of the Ui . In forming twisted K-theory, we modify this descent or gluing datum, by introducing a line bundle Lij on Ui ∩ Uj , and asking for an isomorphism λij : Lij ⊗ Vi −→ Vj 742 D. S. FREED, M. J. HOPKINS AND C. TELEMAN satisfying a certain cocycle condition. In terms of K-theory, this modifies the restriction maps in the Mayer–Vietoris sequence. In order to formulate the cocycle condition, the Lij must come equipped with an isomorphism Ljk ⊗ Lij −→ Lik on the triple intersections, satisfying an evident compatibility relation on the quadruple intersections. In other words, the {Lij } must form a 1-cocycle with values in the groupoid of line bundles. Cocycles diﬀering by a 1-cochain give isomorphic twisted K-groups, and so, up to isomorphism, we can associate a twisted notion of K(X) to an element τ ∈ H 1 (X; {Line Bundles}). On good spaces there are isomorphisms H 1 (X; {Line Bundles}) ≈ H 2 (X; U (1)) ≈ H 3 (X; Z), and correspondingly, twisted notions of K-theory associated to an integer-valued 3-cocycle. In this paper, we find we need to allow the Lij to be ± line bundles, and so, in fact, we consider a twisted notion of K(X) classified by elements† τ ∈ H 1 (X; {±Line Bundles}) ≈ H 3 (X; Z) × H 1 (X; Z/2). We write K τ +n for the version of K n , twisted by τ . In practice, to compute twisted K(X), one represents the twisting τ as a Cech 1-cocycle on an explicit covering of X. The twisted K-group is then assembled from the Mayer–Vietoris sequence of this covering, involving the same (untwisted) K-groups one would encounter in computing K(X). The presence of the 1-cocycle is manifest in the restriction maps between the K-groups of the open sets. They are modified on the two-fold intersections by tensoring with the (±) line bundle given by the 1-cocycle. This ‘operational definition’ suﬃces to make most computations. See § 3 for a more careful discussion. Here are a few examples. Example 1.4. Suppose that X = S 3 , and that the isomorphism class of τ is n ∈ H (X; Z) ≈ Z. Let U+ = X \ (0, 0, 0, −1) and U− = X \ (0, 0, 0, 1). Then U+ ∩ U− ∼ S 2 , and τ is represented by the 1-cocycle whose value on U+ ∩ U− is Ln , with L the tautological line bundle. The Mayer–Vietoris sequence for K τ (X) takes the form 3 · · · −→ K τ +0 (X) −→ K τ +0 (U+ ) ⊕ K τ +0 (U− ) −→ K τ +0 (U+ ∩ U− ) −→ K τ +1 (X) −→ K τ +1 (U+ ) ⊕ K τ +1 (U− ) −→ K τ +1 (U+ ∩ U− ) −→ · · · . As the restriction of τ to U± is isomorphic to zero, we have K τ +0 (U± ) ≈ K 0 (U± ) ≈ Z, K τ +1 (U± ) ≈ K 1 (U± ) ≈ 0, K τ +1 (U+ ∩ U− ) ≈ K 1 (S 2 ) ≈ 0, and K τ +0 (U+ ∩ U− ) ≈ K 0 (S 2 ) ≈ Z ⊕ Z, with basis the trivial bundle 1, and the tautological line bundle L. The Mayer–Vietoris sequence reduces to the exact sequence 0 −→ K τ +0 (X) −→ Z ⊕ Z −→ Z ⊕ Z −→ K τ +1 (X) −→ 0. † The set of isomorphism classes of twistings has a group structure induced from the tensor product of graded line bundles. Although there is, as indicated, a set-theoretic factorization of the isomorphism classes of twistings, the group structure is not, in general, the product. 743 LOOP GROUPS AND TWISTED K-THEORY I In ordinary (untwisted) K-theory, the middle map is 1 −1 : Z ⊕ Z −→ Z ⊕ Z. 0 0 In twisted K-theory, with suitable conventions, the middle map becomes 1 n−1 : Z ⊕ Z −→ Z ⊕ Z, 0 −n and so K τ +n 0 (S ) = Z/n 3 (1.5) n = 0, n = 1. In the language of twistings, the map (1.5) is accounted for as follows. To identify the twisted K-groups with ordinary twisted K-groups, we have to choose isomorphisms t± : τ |U± −→ 0. If we use the t+ to trivialize τ on U+ ∩ U− , then the following diagram commutes: K τ +∗ (U+ ) t+ / K 0+∗ (U+ ) restr. restr. K τ +∗ (U+ ∩ U− ) t+ / K 0+∗ (U+ ∩ U− ) and we can identify the restriction map in twisted K-theory from U+ to U+ ∩ U− with the restriction map in untwisted K-theory. On U+ ∩ U− we have t− = (t− t−1 + ) ◦ t+ , so the restriction map in twisted K-theory is identified with the restriction map in untwisted Ktheory, followed by the map (t− t−1 + ). By definition of τ , this map is given by multiplication by Ln . This accounts for the second column of the matrix in (1.5). Example 1.6. Now consider the twisted K-theory of U (1) acting trivially on itself. In this case, the twistings are classified by HU3 (1) (U (1); Z) × HU1 (1) (U (1); Z/2) ≈ Z ⊕ Z/2. We consider twisted K-theory, twisted by τ = (n, ǫ). Regard U (1) as the unit circle in the complex plane, and set U+ = U (1) \ {−1} U− = U (1) \ {+1}. The twisting τ restricts to zero on both U+ and U− . Write KU0 (1) = R(1 ) = Z[L, L−1 ]. Then the Mayer–Vietoris sequence becomes ±1 0 −→ KUτ +0 ] ⊕ Z[L±1 ] −→ Z[L±1 ] ⊕ Z[L±1 ] −→ KUτ +1 (1) (U (1)) −→ 0. (1) (U (1)) −→ Z[L The 1-cocycle representing τ can be taken to be the equivariant vector bundle whose fiber over −i is the trivial representation of U (1) and whose fiber over +i is (−1)ǫ Ln . With suitable conventions, the middle map becomes 1 −(−1)ǫ Ln : Z[L±1 ]2 → Z[L±1 ]2 . 1 −1 It follows that KUτ +k (1) (U (1)) 0 = Z[L±1 ]/((−1)ǫ Ln − 1) k = 0, k = 1. 744 D. S. FREED, M. J. HOPKINS AND C. TELEMAN When ǫ = 0, this coincides with the Grothendieck group of representations of the Heisenberg extension of Z × U (1) of level n, and in turn with the Grothendieck group of positive energy representations the loop group of U (1) at level n. Example 1.7. Consider the twisted K-theory of SU(2) acting on itself by conjugation. 1 The group HSU(2) (SU(2); Z/2) vanishes, whereas 3 (SU(2); Z) = Z, HSU(2) so a twisting τ in this case is given by an integer 3 (SU(2); Z) = Z. n ∈ HSU(2) Set U+ = SU(2) \ {−1} U− = SU(2) \ {+1}. The spaces U+ and U1 are equivariantly contractible, whereas U+ ∩ U− is equivariantly homotopy equivalent to S 2 = SU(2)/T , where T = U (1) is a maximal torus. The restrictions of τ to U+ and U− are isomorphic to zero. We have 0 0 (U± ) ≈ KSU(2) (pt) = R(SU(2)) = Z[L, L−1 ]W KSU(2) with the Weyl group W ≈ Z/2 acting by exchanging L and L−1 , and 0 0 (U+ ∩ U− ) ≈ KSU(2) (SU(2)/T ) ≈ KT0 (pt) ≈ Z[L, L−1 ]. KSU(2) The ring R(SU(2)) has an additive basis consisting of the irreducible representations, ρk = Lk + Lk−2 + . . . + L−k , k 0, which multiply according to the Clebsch–Gordon rule ρl ρk = ρk+l + ρk+l−2 + . . . + ρk−l , k l. As in our other example, the Mayer–Vietoris sequence is short exact: τ +1 τ +0 (SU(2)) −→ 0. (SU(2)) −→ R(SU(2)) ⊕ R(SU(2)) −→ Z[L±1 ] −→ KSU(2) 0 −→ KSU(2) The 1-cocycle representing the diﬀerence between the two trivializations of the restriction of τ to U+ ∩ U− can be taken to be the element Ln ∈ KSU(2) (SU(2)/T ) ≈ R(T ). The sequence is a sequence of R(SU(2))-modules. With suitable conventions, the middle map is 1 −Ln : R(SU(2))2 → R(T ). (1.8) To calculate the kernel and cokernel, note that R(T ) is a free module of rank 2 over R(SU(2)). We give R(SU(2)) ⊕ R(SU(2)) the obvious basis, and R(T ) the basis {1, L}. It follows from the identity Ln = Lρn−1 − ρn−2 that (1.8) is represented by the matrix and that τ +k (SU(2)) KSU(2) 1 0 ρn−2 , −ρn−1 0 k = 0, = R(SU(2))/(ρn−1 ) k = 1. This coincides with the Grothendieck group of positive energy representations the loop group of SU(2) at level (n − 2). LOOP GROUPS AND TWISTED K-THEORY I 745 Examples 1.6 and 1.7 illustrate the relationship between twisted K-theory and the representations of loop groups. In both cases the Grothendieck group of positive energy representations of the loop group of a compact Lie group G is described by the twisted equivariant K-group of G acting on itself by conjugation. Two minor discrepancies appear in this relationship. On one hand, the interesting K-group is in degree k = 1. As explained in the Introduction, the representations of the loop group at level τ correspond to twisted K-theory at the twisting ζ(τ ) = g + ȟ + τ . The shift in K-group to k = 1 corresponds to the term g. In both examples the adjoint representation is Spinc and so contributes only its dimension to ζ(τ ). This term could be disposed of by working with twisted equivariant K-homology rather than K-cohomology. We have chosen to work with K-cohomology in order to make better contact with our geometric constructions in Parts II and III. The other discrepancy is the shift in level in Example 1.7: twisted equivariant K-theory at level n corresponds to the representations of the loop group at level (n − 2). The shift of 2 here corresponds to the term ȟ in our formula for ζ(τ ). We now give a series of examples describing other ways in which twistings of K-theory arise. Example 1.9. Let V be a vector bundle of dimension n over a space X, and write X V for the Thom complex of V . Then K̃ n+k (X V ) is a twisted form of K k (X). To identify the twisting, choose local Spinc structures µi on the restrictions Vi = V |Ui of V to the sets in an open cover of X. The K-theory Thom classes associated to the µi allow one to identify K̃ n+k (UiVi ) ≈ K k (Ui ). The diﬀerence between the two identifications on Ui ∩ Uj is given by multiplication by the graded line bundle representing the diﬀerence between µi and µj . Figuratively, the cocycle 1 representing the twisting is µj µ−1 i and the cohomology class is (w1 (V ), W3 (V )) ∈ H (X; Z/2) × H 3 (X; Z), where W3 = βw2 . This is one of the original examples of twisted K-theory, described by Donovan and Karoubi [14] from the point of view of Cliﬀord algebras. We review their description in § 3.6. Example 1.10. Let G be a compact Lie group. The central extensions τ T −→ G̃ − →G 3 of G by T = U (1) are classified by HG ({pt}; Z) = H 3 (BG; Z). The Grothendieck group Rτ (G) of representations of G̃ on which T acts according to its defining representation, can be thought of as a twisted form of R(G). In this case, our definition of equivariant twisted K-theory gives Rτ (G) k = 0, τ +k KG (pt) = 0 k = 1. τ +k 3 3 More generally, if S is a G-space, and τ ∈ HG (S) is pulled back from HG (pt), then KG (S) k is the summand of KG̃ (S) corresponding to G̃-equivariant vector bundles on which T acts according to its defining character. Example 1.11. Now suppose that H −→ G −→ Q is an extension of groups, and V is an irreducible representation of H that is stable, up to isomorphism, under conjugation by elements of G. Then the Grothendieck group of representations of G, whose restriction to H is V -isotypical, forms a twisted version of the Grothendieck group of representations of Q. When H is central, equal to T, and V is the 746 D. S. FREED, M. J. HOPKINS AND C. TELEMAN defining representation, this is the situation in Example 1.10. We now describe how to reduce to this case. Fix an H-invariant Hermitian metric on V , and write V ∗ = hom(V, C) for the representation dual to V . Let G̃ denote the group of pairs (g, f ) ∈ G × hom(V ∗ , V ∗ ), for which f is unitary and satisfies f (hv) = ghg −1 f (v), h ∈ H. As V is irreducible, and (ad g)∗ V ≈ V , the same is true of V ∗ , and the map G̃ −→ G (g, f ) −→ g is surjective, with kernel T. The inclusion H ⊂ G̃ h −→ (h, action of h) is normal, and lifts the inclusion of H into G. We define Q̃ = G̃/H. The group Q̃ is a central extension of Q by T, which we define by τ Q̃ − → Q. We now describe an equivalence of categories between V -isotypical G-representations, and τ projective representations of Q (representations of Q̃ on which T acts according to its defining character). By definition, the representations V and V ∗ of H come equipped with extensions to unitary representation of G̃. Given a V -isotypical representation W of G, we let M denote the Hinvariant part of V ∗ ⊗ W : M = (V ∗ ⊗ W )H . The action of G̃ on V ∗ ⊗ W factors through an action of Q̃ on M . This defines a functor from V -isotypical representations of G to τ -projective representations of Q. Conversely, suppose that M is a τ -projective representation of Q̃. Let G̃ act on M through the projection G̃ −→ Q̃, and form W = V ⊗ M. The central T of G̃ acts trivially on W , giving W a G-action. This defines a functor from the category of τ -projective representations of Q to the category of V -isotypical representations of G. One easily checks these two functors to form an equivalence of categories. Example 1.12. Continuing with the situation of Example 1.11, consider an extension H −→ G −→ Q, and an irreducible representation V of H, which this time is not assumed to be stable under conjugation by G. Write G0 = {g ∈ G | (ad g)∗ V ≈ V }, and Q0 = G0 /H. Let S be the set of isomorphism classes of irreducible representations of H of the form (ad g)∗ V . The conjugation action of G on S factors through Q, and we have an identification S = Q/Q0 . Let us call a representation of G S-typical if its restriction to LOOP GROUPS AND TWISTED K-THEORY I 747 H involves only the irreducible representations in S. One easily checks that ‘induction’ and ‘passage to the V -isotypical part of the restriction’ give an equivalence of categories {S, typical representations of G} ↔ {V, isotypical representations of G0 }, and therefore an isomorphism of the Grothendieck group RS (G) of S-typical representations of G with τ (pt). Rτ (Q0 ) ≈ KQ 0 We can formulate this isomorphism a little more cleanly in the language of groupoids. For each α ∈ S choose an irreducible H-representation Vα representing α. Consider the groupoid S//Q, with set of objects S, and in which a morphism α −→ β is an element g ∈ Q for which (ad g)∗ α = β. We define a new groupoid P with objects S, and with P (α, β) the set of equivalence classes of pairs (g, φ) ∈ G × hom(Vα∗∗ ,Vβ ), with φ unitary, and satisfying φ(hv) = ghg −1 φ(v) (so that, among other things, (ad g)∗ α = β). The equivalence relation is generated by (g, φ) ∼ (hg, hφ), h ∈ H. There is an evident functor τ : P → S//Q, representing P as a central extension of S//Q by T. The automorphism group of V in P is the central extension Q̃0 of Q0 . An easy generalization of the construction of Example 1.11 gives an equivalence of categories {τ, projective representations of P } ↔ {S, typical representations of G}. Central extensions of S//Q are classified by 3 3 H 3 (S//Q; Z) = HQ (S; Z) ≈ HQ (pt; Z), 0 and so represent twistings of K-theory. Our definition of twisted K-theory of groupoids will identify the τ -twisted K-groups of (S//Q) with the summand of the K-theory of P on which the central T acts according to its defining representation. We therefore have an isomorphism τ +0 RS (G) ≈ K τ +0 (S//Q) = KQ (S). Example 1.13. Now let S denote the set of isomorphism classes of all irreducible representations of H. Decomposing S into orbit types, and using the construction of Example 1.12 gives a central extension τ : P → (S//Q), and an isomorphism τ +0 R(G) ≈ K τ +0 (S//Q) = KQ (S). More generally, if X is a space with a Q-action, there is an isomorphism τ +k k KG (X) ≈ KQ (X × S), in which τ is the Q-equivariant twisting of X × S pulled back from the Q-equivariant twisting τ of S, given by P . 2. Twistings of K-theory We now turn to a more careful discussion of twistings of K-theory. Our terminology derives from the situation of Example 1.10 in which a central extension of a group gives rise to a twisted notion of equivariant K-theory. By working with graded central extensions of groupoids (rather than groups), we are able to include in a single point of view both the twistings that come from 1-cocycles with values in the group of Z/2-graded line bundles and the twistings that come from central extensions. In order to facilitate this, in the rest of this paper we use the 748 D. S. FREED, M. J. HOPKINS AND C. TELEMAN language of T-bundles and T-torsors instead of ‘line bundles’, where T is the group U (1). We begin with a formal discussion of (Z/2-)graded T-bundles. 2.1. Graded T-bundles Let X be a topological space. Definition 2.1. A graded T-bundle over X consists of a principal T-bundle P → X, and a locally constant function ǫ : X → Z/2. We call a graded T-bundle (P, ǫ) even (respectively odd) if ǫ is the constant function 1 (respectively −1). The collection of graded T-bundles forms a symmetric monoidal groupoid. A map of graded T-bundles (P1 , ǫ1 ) → (P1 , ǫ2 ) exists only when ǫ1 = ǫ2 , in which case it is a map of principal bundles P1 → P2 . The tensor structure is given by (P1 , ǫ1 ) ⊗ (P2 , ǫ2 ) = (P1 ⊗ P2 , ǫ1 + ǫ2 ), in which P1 ⊗ P2 is the usual ‘tensor product’ of principal T-bundles: (P1 ⊗ P2 )x = (P1 )x × (P2 )x /(vλ, w) ∼ (v, wλ). It is easiest to describe the symmetry transformation T : (P1 , ǫ1 ) ⊗ (P2 , ǫ2 ) −→ (P2 , ǫ2 ) ⊗ (P1 , ǫ1 ) fiberwise. In the fiber over a point x ∈ X, it is given by (v, w) −→ (w, vǫ1 (x)ǫ2 (x)). We write BT± for the contravariant functor that associates to a space X the category of graded T-bundles over X, and BT for the functor ‘category of T-bundles’. We also write H 1 (X; T± ) for the group of isomorphism classes in BT± (X), and H 0 (X; T± ) for the group of automorphisms of any object. There is an exact sequence BT −→ BT± −→ Z/2 (2.2) in which the rightmost arrow is ‘forget everything but the grading’. In fact, this sequence can be split by associating to a locally constant function ǫ : X −→ Z/2 the ‘trivial’ graded T-bundle 1ǫ := (X × T, ǫ). This splitting is compatible with the monoidal structure, but not with its symmetry. It gives an equivalence BT± ≈ BT × Z/2 (2.3) of monoidal categories (but not of symmetric monoidal categories). It follows that the group H 1 (X; T± ), of isomorphism classes of graded T-bundles over X, is isomorphic to H 0 (X; Z/2) × H 1 (X; T). As X is assumed to be paracompact, this in turn is isomorphic to H 0 (X; Z/2) × H 2 (X; Z). The automorphism group of any graded T-bundle is the group of continuous maps from X to T. 2.2. Graded central extensions Building on the notion of graded T-bundles, we now turn to graded central extensions of groupoids. The reader is referred to the Appendix A for our conventions on groupoids, and a recollection of the fundamental notions. Unless otherwise stated, all groupoids will be assumed to be local quotient groupoids (§ A.2.2), in the sense that they admit a countable open cover by LOOP GROUPS AND TWISTED K-THEORY I 749 subgroupoids, each of which is weakly equivalent to a compact Lie group acting on a Hausdorﬀ space. Let X = (X0 , X1 ) be a groupoid. Write BZ/2 for the groupoid associated to the action of Z/2 on a point. Definition 2.4. A graded groupoid is a groupoid X equipped with a functor ǫ : X → BZ/2. The map ǫ is called the grading. The collection of gradings on X forms a groupoid, in which a morphism is a natural transformation. Spelled out, a grading of X is a function ǫ : X1 → Z/2 satisfying ǫ(g ◦ f ) = ǫ(g) + ǫ(f ), and a morphism from ǫ0 to ǫ1 is a continuous function η : X0 → Z/2 satisfying, for each (f : x → y) ∈ X1 , ǫ1 (f ) = ǫ0 (f ) + (η(y) − η(x)). Example 2.5. Suppose that X = S//G, with S a connected topological space. Then a grading of X is just a homomorphism G → Z/2, making G into a graded group. We denote the groupoid of gradings of X Hom(X, BZ/2). Definition 2.6. A graded central extension of X is a graded T-bundle L over X1 , together with an isomorphism of graded T-bundles on X2 λg,f : Lg ⊗ Lf → Lg◦f satisfying the cocycle condition, that the diagram ≈ / Lh (Lg Lf ) (Lh Lg )Lf NNN NNN NNN N& Lh◦g Lf / Lh Lg◦f / Lh◦g◦f of graded T-bundles on X3 commutes. If L → X1 is a graded central extension of X, then the pair X̃ = (X0 , L) is a graded groupoid over X, and the functor X̃ → X represents X̃ as a graded central extension of X by T in the evident sense. Our terminology comes from this point of view. Some constructions are simpler to describe in terms of the graded T-bundles L and others in terms of X̃ → X. The collection of graded central extensions of X forms a symmetric monoidal 2-category which we denote Ext(X, T± ) = ExtX . The category of morphisms in ExtX from L1 → L2 is the groupoid of graded T-torsors (η, ǫ) over X0 , equipped with an isomorphism ηb L1f → L2f ηa making ηc L1g L1f ηc L1g◦f / L2g ηb L1f / L2g L2f ηa / L2g◦f ηa 750 D. S. FREED, M. J. HOPKINS AND C. TELEMAN commute. The tensor product L ⊗ L′ is the graded central extension L ⊗ L′ −→ X1 with structure map λg,f ⊗λ′g,f 1⊗T⊗1 Lg ⊗ L′g ⊗ Lf ⊗ L′f −−−−−→ Lg ⊗ Lf ⊗ L′g ⊗ L′f −−−−−−→ Lg◦f ⊗ L′g◦f . The symmetry isomorphism L ⊗ L′ → L′ ⊗ L is derived from the symmetry of the tensor product of graded T-bundles. For the purposes of twisted K-groups it suﬃces to work with the 1-category quotient of ExtX . Definition 2.7. The category Ext(X; T± ) = ExtX is the category with objects the graded central extensions of X, and with morphisms from L to L′ the set of isomorphism classes in ExtX (L, L′ ). The symmetric monoidal structure on ExtX makes ExtX into a symmetric monoidal groupoid in the evident way. Remark 2.8. A 1-automorphism of L consists of a graded T-torsor η over X0 , together with an isomorphism ηa → ηb over X1 , satisfying the cocycle condition. In this way, the category of automorphisms of any twisting can be identified with the groupoid of graded line bundles over X. A graded line bundle on X defines an element of K 0 (X) (even line bundles go to line bundles, and odd line bundles go to their negatives). The fundamental property relating twistings and twisted K-theory is that the automorphism η acts on twisted K-theory as multiplication by the corresponding element of K 0 (X) (Proposition 3.3). Remark 2.9. (1) The formation of ExtX is functorial in X, in the sense of 2-categories. If F : Y → X is a map of groupoids, and L → X1 is a graded central extension of X, then F ∗ L → Y1 gives a graded central extension, F ∗ L of Y . If η → X0 is a morphism from L1 to L2 , then F ∗ η defines a morphism from F ∗ L1 to F ∗ L2 . (2) If T : Y0 → X1 is a natural transformation from F to G, then the graded line bundle T ∗ L determines a morphism from F ∗ L to G∗ L. This is functorial in the sense that, given η : L1 → L2 , there is a 2-morphism relating the two ways of going around F ∗ L?1 ∗ ??F η ?? ∗ 1 =⇒ F ∗ L2 G L? ?? ? G∗ η ? T ∗ L2 G∗ L2 . T ∗ L1 The 2-morphism is the isomorphism (G∗ η) ◦ (T ∗ L1 ) −→ (T ∗ L2 ) ◦ (F ∗ η) obtained by pulling back map η : L1 → L2 along T . It is given pointwise over y ∈ Y , T y : F y → Gy by (ηGy )(L1T y ) −→ (L2T y )(ηF y ). (3) It follows that the formation of ExtX is functorial in X, making ExtX a (weak) presheaf of groupoids. 751 LOOP GROUPS AND TWISTED K-THEORY I Example 2.10. Suppose that G is a group, and X = pt//G. Then a graded central extension of X is just a graded central extension of G by T. Forgetting the T-bundle, gives a functor from ExtX to the groupoid of gradings of X, and the decomposition (2.3) gives a 2-category equivalence Ext(X, T± ) ≈ Ext(X, T) × Hom(X, BZ/2), (2.11) which is not, in general, compatible with the monoidal structure. Here Ext(X, T) is the 2-category of evenly graded (that is, ordinary) central extensions of X by T. 2.2.1. Classification of graded central extensions. We now turn to the classification of graded central extensions of a groupoid X. In view of (2.11), it suﬃces to separately classify T-central extensions (graded central extensions that are purely even) and gradings. For the T-central extensions, first recall that the category of T-torsors on a space Y is equivalent to the category whose objects are T-valued Cech 1-cocycles, Ž 1 (Y ; T), and in which a morphism from z0 to z1 is a Cech 0-cochain c ∈ Č 0 (Y ; T) with the property that δc = z1 − z0 . Now consider the double complex for computing the Cech hypercohomology groups of the nerve X• , with coeﬃcients in T: (2.12) O O O Č 2 (X0 ; T) O / Č 2 (X1 ; T) O / Č 2 (X2 ; T) O / Č 1 (X0 ; T) O / Č 1 (X1 ; T) O / Č 1 (X2 ; T) O / Č 0 (X0 ; T) / Č 0 (X1 ; T) / Č 0 (X2 ; T) /. In terms of the Cech cocycle model for T-bundles, the 2-category of T-central extension of X is equivalent to the category whose objects are cocycles in (2.12), of total degree 2, whose component in Č 2 (X0 ; T) is zero. The 1-morphisms are given by cochains of total degree 1, whose coboundary has the property that its component in Č 2 (X0 ; T) vanishes. The 2-morphisms are given by cochains of total degree 0. Write Ȟ ∗ (X) and Ȟ ∗ (X0 ), for the Cech hypercohomology of X and the Cech cohomology of X0 , respectively. Then the group of isomorphism classes of even graded T-gerbes is given by the kernel of the map Ȟ 2 (X; T) → Ȟ 2 (X0 ; T), the group of isomorphism classes of 1-automorphisms of any even graded T-gerbe is Ȟ 1 (X; T), and the group of 2-automorphisms of any 1-morphism is Ȟ 0 (X; T). As for the gradings, the group of isomorphism classes of gradings is ker{Ȟ 1 (X; Z/2) −→ Ȟ 1 (X0 ; Z/2)}, and the automorphism group of any grading is Ȟ 0 (X; Z/2). For convenience write t Ȟrel (X; A) = ker{Ȟ t (X; A) −→ Ȟ t (X0 ; A)}. 752 D. S. FREED, M. J. HOPKINS AND C. TELEMAN Proposition 2.13. The group π0 ExtX of isomorphism classes of graded central extension of X is given by the set-theoretically split extension 1 2 Ȟrel (X; Z/2) (X; T) −→ π0 ExtX −→ Ȟrel (2.14) with cocycle c(ǫ, µ) = β(ǫ ∪ µ), where β : Z/2 = {±1} ⊂ T is the inclusion. The group of isomorphism classes of automorphisms of any graded central extension of X is Ȟ 1 (X; T) × Ȟ 0 (X; Z/2), and the group of 2-automorphisms of any morphism of graded central extensions is Ȟ 0 (X; T). Proof. Most of this result was proved in the discussion leading up to its statement. The decomposition (2.11) gives the exact sequence, as well as a set-theoretic splitting 1 s : Ȟrel (X; Z/2) −→ π0 ExtX . It remains to identify the cocycle describing the group structure. Suppose that ǫ, µ : X −→ BZ/2 are two gradings of X. Then, by the discussion leading up to (2.3), s(ǫ)s(µ) is the graded central extension given by (s(ǫ)s(µ))f = 1ǫ(f ) 1µ(f ) ≈ 1ǫ(f )+µ(f ) , and structure map 1ǫ(g) 1µ(g) 1ǫ(f ) 1µ(f ) −→ 1ǫ(g) 1ǫ(f ) 1µ(g) 1µ(f ) −→ 1ǫ(g◦f ) 1µ(g◦f ) . (2.15) Using the canonical identifications 1a 1b = 1a+b , and ǫ(g ◦ f ) = ǫ(g) + ǫ(f ), µ(g ◦ f ) = µ(g) + µ(f ), one checks that (2.15) can be identified with the automorphism (−1)ǫ(f )µ(g) of the trivialized graded line 1ǫ(g)+µ(g)+ǫ(f )+µ(f ) ≈ 1ǫ(g◦f )+µ(g◦f ) . Similarly, the structure map of s(ǫ + µ) can be identified with the identity map of the same trivialized graded line. It follows that s(ǫ)s(µ) = c(ǫ, µ)s(ǫ + µ), where c(ǫ, µ) is the graded central extension with Lf = 1, and λg,f = (−1)ǫ(f )µ(g) . Now the 2-cocycle ǫ(f )µ(g) is precisely the Alexander-Whitney formula for the cup product ǫ ∪ µ ∈ Z 2 (X; Z/2). This completes the proof. One easy, but very useful, consequence of Proposition 2.13 is the stacky nature of the morphism categories in ExtX . LOOP GROUPS AND TWISTED K-THEORY I Corollary 2.16. functor 753 Let P, Q ∈ ExtX , and f : Y → X be a local equivalence. Then the f ∗ : ExtX (P, Q) −→ ExtY (f ∗ P, f ∗ Q) is an equivalence of categories, and so f ∗ : ExtX (P, Q) −→ ExtY (f ∗ P, f ∗ Q) is a bijection of sets. Another useful, although somewhat technical, consequence of the classification is the following corollary. Corollary 2.17. Suppose that X is a groupoid with the property that the maps Ȟ 2 (X; T) −→ Ȟ 2 (X0 ; T), Ȟ 1 (X; Z/2) −→ Ȟ 1 (X0 ; Z/2) are zero. If Y → X is a local equivalence, then the maps Ȟ 2 (Y ; T) −→ Ȟ 2 (Y0 ; T), 1 Ȟ (Y ; Z/2) −→ Ȟ 1 (Y0 ; Z/2) are zero, and ExtX −→ ExtY , and ExtX −→ ExtY are equivalences. Proof. The first assertion is a simple diagram chase. It has the consequence that the maps 2 Ȟrel (X; T) −→ Ȟ 2 (X0 ; T), 1 Ȟrel (X; Z/2) −→ Ȟ 1 (X0 ; Z/2), 2 Ȟrel (Y ; T) −→ Ȟ 2 (Y0 ; T), 1 Ȟrel (Y ; Z/2) −→ Ȟ 1 (Y0 ; Z/2) are isomorphisms, and the second assertion follows. For later reference we note the following additional consequence of Proposition 2.13. Corollary 2.18. Suppose that X is a local quotient groupoid, and that τ is a twisting of X represented by a local equivalence P → X and a graded central extension P̃ → P . Then P̃ is a local quotient groupoid. Proof. As the property of being a local quotient groupoid is an invariant of local equivalence, we know that P is a local quotient groupoid. The question is also local in P , so we may assume that P is of the form S//G for a compact Lie group G. By our assumptions, the action of G on S has locally contractible slices. Working still more locally in S, we may assume that S is contractible. But then Proposition 2.13 implies that τ is given by a (graded) central extension of G̃ → G, and P̃ = S//G̃. 754 D. S. FREED, M. J. HOPKINS AND C. TELEMAN 2.3. Twistings We now describe the category TwistX of K-theory twistings on a local quotient groupoid X (§ A.2.2). The objects of TwistX are pairs a = (P, L) consisting of a local equivalence P → X, and a graded central extension L of P . The set of morphisms from a = (P0 , L0 ) to b = (P1 , L1 ) is defined to be the colimit TwistX (a, b) = ExtP (p∗ π1∗ a, p∗ π2∗ b), lim −→ p:P →P12 where P12 = P1 ×X P2 , the limit is taken over CovP12 , and our notation refers to the diagram P P1 o π1 p P12 π2 / P2 . We leave to the reader to check that ExtP (a, b) does indeed define a functor on the 1-category quotient CovP12 . The colimit appearing in the definition of TwistX (a, b) is present in order that the definition be independent of any extraneous choices. In fact, the colimit is attained at any stage. Lemma 2.19. For any local equivalence p : P −→ P12 (2.20) the map ExtP (p∗ π1∗ a, p∗ π2∗ b) → TwistX (a, b) is an isomorphism. Proof. By Corollary 2.16, for any P ′ −→ P in CovP12 , the map ExtP (a, b) −→ ExtP ′ (a, b) is an isomorphism. The result now follows from Corollary A.12. For the composition law, suppose that we are given three twistings a = (P1 , L1 ), b = (P2 , L2 ), and c = (P3 , L3 ). Find a P123 ∈ CovX and maps P123D DD p zz DD 3 z z p2 DD z z D! z }z P2 P3 P1 p1 (for example, one could take P123 to be the (2-category) fiber product P1 ×X P2 ×X P3 , and pi projection to the ith factor). By Lemma 2.19 the maps ExtP123 (p∗1 a, p∗2 b) −→ TwistX (a, b), ExtP123 (p∗1 a, p∗2 c) −→ TwistX (a, c), ExtP123 (p∗1 b, p∗2 c) −→ TwistX (b, c) are bijections. With these identifications, the composition law in TwistX is formed from that in ExtP123 . We leave it to the reader to check that this is well defined. LOOP GROUPS AND TWISTED K-THEORY I 755 The formation of TwistX is functorial in X. Given f : Y → X, and a = (P, L) ∈ TwistX form π Y ×X P −−−−→ ⏐ ⏐ Y P ⏐ ⏐ −−−−→ X f and set f ∗ a = (f ∗ P, π ∗ L). Proposition 2.21. The association X → TwistX is a weak presheaf of groupoids. If Y → X is a local equivalence, then TwistY → TwistX is an equivalence of categories. There is an evident functor ExtX −→ TwistX . Proposition 2.22. When X satisfies the condition of Corollary 2.17, the functor ExtX −→ TwistX (2.23) is an equivalence of categories. Proof. Lemma 2.19 shows that (2.23) is fully faithful. Essential surjectivity is a consequence of Corollary 2.17. Corollary 2.24. If Y −→ X is a local equivalence and Y satisfies the conditions of Corollary 2.17, then the functors TwistX −→ TwistY ← ExtY are equivalences of groupoids. Combining this with Proposition 2.13 gives the following corollary. Corollary 2.25. The group π0 TwistX of isomorphism classes of twistings on X is the set-theoretically split extension Ȟ 2 (X; T) −→ π0 TwistX −→ Ȟ 1 (X; Z/2) with cocycle c(ǫ, µ) = β(ǫ ∪ µ). The group of automorphisms of any twisting is Ȟ 1 (X; T) × Ȟ 0 (X; Z/2). We now switch to the point of view of ‘fibered categories’ in order to more easily describe the functorial properties of twisted K-groups. Let Ext denote the category whose objects are pairs (X, L) consisting of a groupoid X and a graded central extension L of X. A morphism (X, L) → (Y, M ) is a functor f : X → Y , and 756 D. S. FREED, M. J. HOPKINS AND C. TELEMAN an isomorphism L → f ∗ M in ExtX . We identify morphisms f, g : (X, L) −→ (Y, M ) if there is a natural transformation T : f → g for which the following diagram commutes: f ∗M o7 o o ooo ηT L OOO OOO ' g ∗ M. The functor (X, L) → X from Ext to groupoids represents Ext as a fibered category, fibered over the category of groupoids. Similarly, we define a category Twist with objects (X, a) consisting of a groupoid X, and a K-theory twisting a ∈ TwistX , and morphisms (X, a) → (Y, b) to be equivalence classes of pairs consisting of a functor f : X → Y and an isomorphism a → f ∗ b in TwistX . There is an inclusion Ext → Twist corresponding to the inclusion ExtX → TwistX . Corollary 2.24 immediately implies the following lemma. Lemma 2.26. Suppose that F : Ext → C is a functor sending every morphism (f, t) : (X, L) → (Y, M ) in which f is a local equivalence to an isomorphism. Then there is a factorization F Ext x Twist . x x x / x< C F′ Moreover, any two such factorizations are naturally isomorphic by a unique natural isomorphism. 2.4. Examples of twistings Example 2.27. Suppose that X is a space, P → X is a principal G bundle, and G̃ −→ G ǫ G− → Z/2 is a graded central extension of G. Then P//G → X is a local equivalence, P//G̃ is a graded central extension of P//G, and (P//G̃, P//G) represents a twisting of X. Example 2.28. As a special case, we note that any double cover P → X defines a twisting. In this case, G = Z/2, G̃ → G is the trivial bundle and ǫ : G → Z/2 is the identity map. Any cohomology theory can be twisted by a double cover, and in fact these are the only twistings of ordinary cohomology with integer coeﬃcients. Example 2.29. Suppose that X = pt//G, with G a compact Lie group. In this case, every local equivalence X̃ → X admits a section. Moreover, the inclusion {IdX } −→ CovX LOOP GROUPS AND TWISTED K-THEORY I 757 of the trivial category consisting of the identity map of X into CovX is an equivalence. It follows that Ext(pt//G, T± ) −→ TwistX is an equivalence of categories, and so twistings of X in this case are just graded central extensions of G. Using Corollary 2.25, one can draw the same conclusion for S//G when S is contractible. We now describe the main example of twistings used in this paper. Example 2.30. Suppose that G is a connected compact Lie group, and consider the pathloop fibration ΩG −→ P G −→ G. (2.31) We regard P G as a principal bundle over G with structure group ΩG. The group G acts on everything by conjugation. Write LG for the group Map(S 1 , G) of smooth maps from S 1 to G. The homomorphism ‘evaluation at 1’ : LG → G is split by the inclusion of the constant loops. This exhibits LG as a semidirect product LG ≈ ΩG ⋊ G. The group LG acts on the fibration (2.31) by conjugation. The action of LG on G factors through the action of G on itself by conjugation, through the map LG → G. This defines a map of groupoids P G//LG −→ G//G, which is easily checked to be a local equivalence. A graded central extension L̃G → LG then defines a twisting of G//G. We write τ for a typical twisting of X, and write the monoidal structure additively: τ1 + τ2 . We use the notations (P̃ τ , P τ ) and (Lτ , P τ ) for typically representing graded central extensions. This is consistent with writing the monoidal structure additively: Lτ1 +τ2 = Lτ1 ⊗ Lτ2 . Example 2.32. Suppose that Y = S//G and H ⊂ G is a normal subgroup. Write X = S//H, and f : X → Y for the natural map. If τ is a twisting of Y , then f ∗ τ has a natural action of G/H (in the 2-category sense), and the map X → Y is invariant under this action (again, in the 2-category sense). To see this, it is easiest to replace X by the weakly equivalent groupoid X ′ = (S × G/H)//G, and factor X → Y by i f′ X− → X ′ −→ Y. As i∗ : TwistX ′ → TwistX is an equivalence of 2-categories, it suﬃces to exhibit an action of ∗ G/H on f ′ τ . The obvious left action of G/H on S × G/H commutes with the right action of ∗ G, giving an action of G/H on X ′ commuting with f ′ . The action of G/H on f ′ τ is then a consequence of naturality. 758 D. S. FREED, M. J. HOPKINS AND C. TELEMAN Example 2.33. By way of illustration, consider the situation of Example 2.32 in which H is commutative, S = {pt}, and τ is given by a central extension T −→ G̃ −→ G. Write T −→ H̃ −→ H, for the restriction of τ to H and assume, in addition, that H̃ is commutative. Then the action of G/H on f ∗ τ constructed in Example 3.23 works out to be the natural action of G/H on H̃ given by conjugation. 3. Twisted K-groups 3.1. Axioms Before turning to the definition of twisted K-theory, we list some general properties describing it as a cohomology theory on the category Twist of local quotient groupoids equipped with a twisting. These properties almost uniquely determine twisted K-theory, and suﬃce to make our main computation in § 4. Twisted K-theory is going to be homotopy invariant, so we need to define the notion of homotopy. Definition 3.1. A homotopy between two maps f, g : (X, τX ) −→ (Y, τY ) is a map (X × [0, 1], π ∗ τX ) −→ (Y, τY ) (π : X × [0, 1] → X is the projection) whose restriction to X × {0} is f , and to X × {1} is g. Twisted K-theory is also a cohomology theory. To state this properly involves defining the relative twisted K theory of a triple (X, A, τ ) consisting of a local quotient groupoid X, a subgroupoid A, and a twisting of X. We form a category of the triples (X, A, τ ) in the same way we formed Twist. We call this the category of pairs in Twist. We now turn to the axiomatic properties of twisted K-theory. Proposition 3.2. The association (X, A, τ ) → K τ +n (X, A) to be constructed in § 3.4 is a contravariant homotopy functor on the category of pairs (X, A, τ ) in Twist, taking local equivalences to isomorphisms. Proposition 3.3. The functors K τ +n form a cohomology theory. (i) There is a natural long exact sequence · · · −→ K τ +n (X, A) −→ K τ +n (X) −→ K τ +n (A) −→ K τ +n+1 (X, A) −→ K τ +n+1 (X) −→ K τ +n+1 (A) −→ · · · . (ii) If Z ⊂ A is a (full) subgroupoid whose closure is contained in the interior of A, then the restriction (excision) map K τ +n (X, A) −→ K τ +n (X \ Z, A \ Z) LOOP GROUPS AND TWISTED K-THEORY I is an isomorphism. (iii) If (X, A, τ ) = α (Xα , Aα , τα ), 759 then the natural map K τ +n (X, A) −→ K τα +n (Xα , Aα ) α is an isomorphism. The combination of excision and the long exact sequence of a pair gives the Mayer–Vietoris sequence · · · −→ K τ +n (X) −→ K τ +n (U ) ⊕ K τ +n (V ) −→ K τ +n (U ∩ V ) −→ K τ +n+1 (X) −→ · · · when X is written as the union of two subgroupoids whose interiors form a covering. Proposition 3.4. (i) There is a bilinear pairing K τ +n (X) ⊗ K µ+m (X) −→ K τ +µ+n+m (X), which is associative and (graded) commutative up to the natural isomorphisms of twistings coming from Proposition 3.2. (ii) Suppose that η : τ → τ is a 1-morphism, corresponding to a graded line bundle L on X. Then η∗ = multiplication by L : K τ +n (X) −→ K τ +n (X), where L is regarded as an element of K 0 (X) and the multiplication is that of (i). Twisted K-theory also reduces to equivariant K-theory in special cases. Proposition 3.5. Let X = S//G be a global quotient groupoid, with G a compact Lie group, and τ be a twisting given by a graded central extension T −→ Gτ −→ G ǫ : G −→ Z/2. (i) If ǫ = 0, then K τ +n (X) is the summand n n KG τ (S)(1) ⊂ KGτ (S) on which T acts via its standard (defining) representation. This isomorphism is compatible with the product structure. (ii) For general ǫ, K τ +n (X) is isomorphic to n+1 KG τ (S × (R(ǫ), R(ǫ) \ {0}))(1), in which the symbol R(ǫ) denotes the 1-dimensional representation (−1)ǫ of Gτ . In (ii), when ǫ = 0, then R(ǫ) is the trivial representation, and the isomorphism can be composed with the suspension isomorphism to give the isomorphism of (i). 760 D. S. FREED, M. J. HOPKINS AND C. TELEMAN When τ = 0, so that Gτ ≈ G × T, Proposition 3.5 reduces to an isomorphism n K τ +n (X) ≈ KG (S). In view of this, we often write τ +n KG (S) = K τ +n (X) in case X = S//G and ǫ = 0. Of course, there is also a relative version of Proposition 3.5. The reader is referred to [21, Section 4] for a more in-depth discussion of the twistings of equivariant K-theory, and interpretation of the ‘ǫ’ part of the twisting in terms of graded representations. Using the Mayer–Vietoris sequence, one can easily check that the result of (i) of Proposition 3.5 holds for any local quotient groupoid X. If the twisting τ is represented by a central extension P → X, then the restriction mapping is an isomorphism K τ +n (X) ≈ K n (P )(1). In this way, once K-theory is defined for groupoids, twisted K-theory is also defined. 3.2. Twisted Hilbert spaces Our definition of twisted K-theory will be in terms of Fredholm operators on a twisted bundle of Hilbert spaces. In this section, we describe how one associates to a graded central extension of a groupoid, a twisted notion of Hilbert bundle. We refer the reader to the Appendix A, § A.4 for our notation and conventions on bundles over groupoids, and to § A.4 for a discussion of Hilbert space bundles. Let X be a groupoid, and τ : X̃ → X be a graded central extension, whose associated graded T-bundle we define by Lτ → X1 . As in Appendix A.3, we use f a− →b f g and a − →b− →c to refer to generic points of X1 and X2 , respectively, and so, for example, in a context describing bundles over X2 , the symbol Hb will refer to the pullback of X along the map X2 −→ X0 (a −→ b −→ c) −→ b. Definition 3.6. A τ -twisted Hilbert bundle on X consists of a Z/2-graded Hilbert bundle H on X0 , together with an isomorphism (on X1 ) Lτf ⊗ Ha −→ Hb satisfying the cocycle condition that Lτg (Lτf Ha ) o Lτg Hb / (Lτg Lτf )Ha / Hc o Lτg◦f Ha commutes on X2 . Remark 3.7. Phrased diﬀerently, a twisted Hilbert bundle is just a graded Hilbert bundle f over X̃, with the property that the map Ha → Hb induced by (a − → b) ∈ X̃1 has degree ǫ(f ), and for which the central T acts according to its defining character. LOOP GROUPS AND TWISTED K-THEORY I 761 Example 3.8. Suppose that X is of the form P//G, and that our twisting corresponds to a central extension of Gτ → G. Then a projective unitary representation of G (meaning a representation of Gτ on which the central T acts according to its defining character) defines a twisted Hilbert bundle over X. Suppose that τ and µ are graded central extensions of X, with associated graded T-torsors Lτ and Lµ . If H is a τ -twisted Hilbert bundle over X and W is a µ-twisted Hilbert bundle, then the graded tensor product H ⊗ W is a (τ + µ)-twisted Hilbert bundle, with structure map Lfτ +µ ⊗ Ha ⊗ Wa = Lτf ⊗ Lµf ⊗ Ha ⊗ Wa −→ Lτf ⊗ Ha ⊗ Lµf ⊗ Wa −→ Hb ⊗ Wb . Now suppose that H 1 and H 2 are τ -twisted, graded Hilbert bundles over X. Definition 3.9. A linear transformation T : H 1 → H 2 consists of a linear transformation of Hilbert bundles T : H 1 → H 2 on X0 for which the following diagram commutes on X1 : Lf Ha1 −−−−→ ⏐ ⏐ 1⊗T Hb1 ⏐ ⏐ T Lf Ha2 −−−−→ Hb2 . τ If Lτ is a graded central extension of X, then we write UX (or just U τ if X is understood) for the category in which the objects are τ -twisted Z/2-graded Hilbert bundles, and with morphisms the linear isometric embeddings. If f : Y → X is a map, then there is an evident functor ∗ τ f ∗ : UX −→ UYf τ . A natural transformation T : f → g of functors X → Y gives a natural transformation T ∗ : f ∗ → g ∗ . Using Remark 3.7 and descent, one easily checks that f ∗ is an equivalence of categories when f is a local equivalence. τ The category UX is also functorial in τ . Indeed, suppose that η : τ → σ is a morphism, given by a graded T-bundle η, and an isomorphism ηb ⊗ τf −→ σf ⊗ ηa . If H is a τ -twisted Hilbert bundle, then H ⊗ η is a σ-twisted Hilbert bundle. One easily checks τ σ → UX , with inverse H → H ⊗ η −1 . The that H → H ⊗ η gives an equivalence of categories UX 2-morphisms η1 → η2 give natural isomorphisms of functors. The tensor product of Hilbert spaces gives a natural tensor product µ τ +µ τ × UX −→ UX . UX 3.3. Universal twisted Hilbert bundles We now turn to the existence of special kinds of τ -twisted Hilbert bundles, following the discussion of § A.4. We keep the notation of § 3.2. Definition 3.10. A τ -twisted Hilbert bundle H on X is (i) universal if, for every τ -twisted Hilbert space bundle V , there is a unitary embedding V → H; (ii) locally universal if H|U is universal for every open subgroupoid U ⊂ X; 762 D. S. FREED, M. J. HOPKINS AND C. TELEMAN (iii) absorbing if, for every τ -twisted Hilbert space bundle V , there is an isomorphism H ⊕ V ≈ H; (iv) locally absorbing if H|U is absorbing for every open subgroupoid U ⊂ X. As in Appendix A.4, if H is (locally) universal, then H is automatically absorbing. Lemma 3.11. Suppose that X̃ → X is a graded central extension and H is a graded Hilbert bundle on X̃. Let H(1) ⊂ H be the eigenbundle on which the central T acts according to its defining representation. Then H(1) is a τ -twisted Hilbert bundle on X, which is (locally) universal if H is. Lemma 3.12. If X is a local quotient groupoid and τ : X̃ → X is a graded central extension, then there exists a locally universal τ -twisted Hilbert bundle H on X. The bundle H is unique up to unitary equivalence. Proof. By Corollary 2.18, X̃ is a local quotient groupoid, which, by Corollary A.33, admits a locally universal Hilbert bundle. The result now follows from Lemma 3.11. 3.4. Definition of twisted K-groups Our task is to define twisted K-groups for pairs (X, A, τ ) in Twist. In view of Lemma 2.26, it suﬃces to define functors K τ +∗ (X, A) for (X, A, τ ) in Ext, and show that they take local equivalences to isomorphisms. We do this by using spaces of Fredholm operators to construct a spectrum K τ (X, A) and defining K τ +n (X, A) = π−n K τ (X, A). The reader is referred to § A.5 for some background discussion on spaces of Fredholm operators. Suppose then that (X, τ ) is an object of Ext and H is a locally universal, τ -twisted Hilbert bundle over X. With the notation of § 3.2, H is given by a Hilbert bundle H over X0 , equipped with an isomorphism Lτf ⊗ Ha −→ Hb (3.13) over f : a → b ∈ X1 satisfying the cocycle condition. The map T −→ Id ⊗ T is a homeomorphism between the spaces of Fredholm operators (see § A.5) Fred(n) (Ha ) and Fred(n) (Lτf ⊗ Ha ) compatible with the structure maps (3.13). The spaces Fred(n) (Ha ) therefore form a fiber bundle over Fred(n) (H) over X. We define spaces K τ (X)n by Γ(X; Fred(0) (H)) n even, τ K (X)n = Γ(X; Fred(1) (H)) n odd. By an obvious modification of the arguments of Atiyah and Singer [3], the results described in § A.5 hold for the bundle Fred(n) (H) over X. In particular, the maps (A.44) and the homeomorphism (A.45) give weak homotopy equivalences K τ (X)n −→ ΩK τ (X)n+1 , making the collection of spaces K τ (X) = {K τ (X)n } into a spectrum. LOOP GROUPS AND TWISTED K-THEORY I 763 Definition 3.14. Suppose that (X, τ ) is a local quotient groupoid equipped with a graded central extension τ , and H is a locally universal, τ -twisted Hilbert bundle over X. The twisted K-theory spectrum of X is the spectrum K τ (X) defined above. To keep things simple, we do not indicate the choice of Hilbert bundle H in the notation K τ (X). The value of the twisted K-group is, in the end, independent of this choice; see Remark 3.17. We now turn to the functorial properties of X → K τ (X). Suppose that f : Y → X is a map of local quotient groupoids, and τ is a twisting of X. Let HX be a τ -twisted, locally universal Hilbert bundle over X, and HY be an f ∗ τ -twisted, locally universal Hilbert space bundle over Y . As HY is universal, there is a unitary embedding f ∗ HX ⊂ HY . Pick one. There is then an induced map f ∗ Fred(n) (HX ) −→ Fred(n) (HY ) T −→ T ⊕ ǫ, (ǫ is the basepoint) and so a map of spectra f ∗ : K τ (X) −→ K τ (Y ). Suppose that η : σ → τ is a morphism of central extensions of X, given by a graded T -bundle η over X0 , and isomorphism ηb ⊗ σf −→ τf ⊗ ηa . If H is a locally universal σ-twisted Hilbert bundle, then H ⊗ η is a locally universal τ -twisted Hilbert bundle. The map T −→ T ⊗ Idη then gives a homeomorphism Fred(n) (H) → Fred(n) (H ⊗ η), and so an isomorphism of spectra η∗ : K σ (X) −→ K τ (X). As automorphisms of η commute with the identity map, 2-morphisms of twistings have no eﬀect on η∗ . In this way, the association τ → K τ (X) can be made into a functor on ExtX . Now we come to an important point. Suppose Y → X is the inclusion of a (full) subgroupoid of a local quotient groupoid, and HX is locally universal. By Corollary A.34 we may then take HY to be f ∗ HX . The bundle of spectra K τ (Y ) is then just the restriction of K τ (X). This would not be true for general groupoids and is the reason for our restriction to local quotient groupoids. We use this restriction property in the definition of the twisted K-theory of a pair. Although this could be avoided, the restriction property plays a key role in the proof of excision, and does not appear to be easily avoided there. Definition 3.15. Suppose that A ⊂ X is a subgroupoid of a local quotient groupoid, and that τ is a graded central extension of X. The twisted K-theory spectrum of (X, A, τ ) is the homotopy fiber K τ (X, A) of the restriction map K τ (X) → K τ (A). If we write Γ(X, A; Fred(n) (H)) ⊂ Γ(X; Fred(n) (H)), for the subspace of sections whose restriction to A is the basepoint ǫ, then K τ (X, A)n = Γ(N, A; Fred(n) (H)), 764 D. S. FREED, M. J. HOPKINS AND C. TELEMAN where N is the mapping cylinder of A ⊂ X, given by N = X ∐ A × [0, 1]/ ∼ . Definition 3.16. The twisted K-group K τ +n (X, A) is the group π−n K τ (X, A) = π0 K τ (X, A)n . Remark 3.17. There are several unspecified choices that go into the definition of the spectra K τ (X, A), and the induced maps between them as X, A, and τ vary. It follows from Propositions A.31 and A.32 that these choices are parameterized by (weakly) contractible spaces, and so have no eﬀect on the homotopy invariants (such as twisted K-groups, and maps of twisted K-groups) derived from them. 3.5. Verification of the axioms 3.5.1. Proof of Proposition 3.2 : functoriality. Most of this result was proved in the process of defining the groups K τ +n (X, A). Functoriality in Ext follows from the discussion of § 3.4 and Remark 3.17. For homotopy invariance, note that if H is a locally universal τ -twisted Hilbert bundle over (X, A), then π ∗ H is a locally universal Hilbert space bundle over (X, A) × I, and so K τn ((X, A) × I) = K τn (X, A)I , and the two restriction maps to K τn (X) correspond to evaluation of paths at the two endpoints. The two restriction maps are thus homotopic, and homotopy invariance follows easily. The assertion about local equivalences is an immediate consequence of descent. As remarked at the beginning of § 3.4, this, in turn, gives functoriality on the category of pairs in Twist. 3.5.2. Proof of Proposition 3.3 : cohomological properties. The long exact sequence of a pair (assertion i) is just the long exact sequence in homotopy groups associated to the fibration of spectra K τ (X, A) −→ K τ (X) −→ K τ (A). The ‘wedge axiom’ (iii) is immediate from the definition. More significant is excision (ii). In describing the proof, we freely use, in the context of groupoids, the basic constructions of homotopy theory as described in § A.2.1. Write U = X \ Z and let N be the double mapping cylinder of U ←− U ∩ A −→ A. Then the map N → X is a homotopy equivalence of groupoids, and so, by the diagram of fibrations K τ (X, A) −−−−→ K τ (X) −−−−→ K τ (A) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ K τ (N, A) −−−−→ K τ (N ) −−−−→ K τ (A), the map K τ (X, A) −→ K τ (N, A) is a weak equivalence. Similarly, if N ′ denotes the mapping cylinder of U ∩ A → U , then K τ (X \ Z, A \ Z) = K τ (U, U ∩ A) −→ K τ (N ′ , U ∩ A) is a weak equivalence. We therefore need to show that, for each n, the map K τ (N, A)n −→ K τ (N ′ , U ∩ A)n LOOP GROUPS AND TWISTED K-THEORY I 765 is a weak equivalence. Let H be a locally universal τ -twisted Hilbert bundle over X. Then the pullback of H to each of the (local quotient) groupoids N , N ′ A, U , U ′ , U ∩ A is also locally universal. It follows that the twisted K-theory spectra of each of these groupoids is defined in terms of sections of the bundle pulled back from Fred(n) (H ⊗ C1 ). To simplify the notation a little, let us denote all of these pulled back bundles by Fred(n) . Now consider the diagram Γ(N, A; Fred(n) ) −−−−→ Γ(N ′ , U ⏐ ⏐ K τ (N, A)n −−−−→ ∩ A; Fred(n) ) ⏐ ⏐ K τ (N ′ , U ∩ A)n . We are to show that the bottom row is a weak equivalence. But the top row is a homeomorphism, and the vertical arrows are weak equivalences as the maps (N ∪ cyl(A), A) −→ (N, A) (N ′ ∪ cyl(U ∩ A), U ∩ A) −→ (N ′ , U ∩ A) are relative homotopy equivalences. 3.5.3. Proof of Proposition 3.4 : multiplication. pairing The multiplication is derived from the Fred(n) (H1 ) × Fred(m) (H2 ) −→ Fred(n+m) (H1 ⊗ H2 ) (S, T ) −→ S ∗ T = S ⊗ Id + Id ⊗ T, the tensor structure on the category of twisted Hilbert space bundles described in § 3.2, and the natural identification of the Z/2-graded tensor product Cℓ(Rn ) ⊗ Cℓ(Rm ) ≈ Cℓ(Rn+m ). Verification of (i) is left to the reader. Even if one of S, T is not acting on a locally universal Hilbert bundle, the product S ∗ T will. This is particularly useful when describing the product of an element of untwisted K-theory, with one of twisted K-theory. For example, if V is a vector bundle over X, then we can choose a Hermitian metric on V , regard V as a bundle of finite-dimensional graded Hilbert spaces, with odd component 0, and take S = 0. Then S ∗ T is just the identity map of V tensored with T . More generally, a virtual diﬀerence V − W of K 0 (X) can be represented by the odd, skew-adjoint Fredholm operator S = 0 on the graded Hilbert space whose even part is V and whose odd part is W , and S ∗ T represents the product of V − W with the class represented by T . The assertion of (ii) is the special case in which V is a graded line bundle. 3.5.4. Proof of Proposition 3.5 : equivariant K-theory. Let X = S//G be a global quotient, and τ be a twisting given by a graded central extension Gτ of G, and a homomorphism ǫ : G → Z/2. Replacing X with X × (R(ǫ), R(ǫ) \ {0}) and using (3.19), if necessary, we may reduce to the case ǫ = 0. Write V (1) for the summand of V = C1 ⊗ L2 (Gτ ) ⊗ ℓ2 on which the central T acts according to its defining character. Then H = S × V (1) is a locally universal Hilbert bundle. Our definition of K τ (X) becomes K τ −n (X) = [S, Fred(n) (Cn ⊗ V (1))]G , which is the summand of [S, Fred(n) (Cn ⊗ V )]G τ corresponding to the defining representation of T. So the result follows from the fact that −n Fred(n) (Cn ⊗ V ) is a classifying space for KG τ . Although this is certainly well known, we were unable to find an explicit statement in the literature. It follows easily from the case in which 766 D. S. FREED, M. J. HOPKINS AND C. TELEMAN G is trivial. Indeed, the universal index bundle is classified by a map to any classifying space for equivariant K-theory, and it suﬃces to show that this map is a weak equivalence on the fixed point spaces for the closed subgroups H of G. The assertion for the fixed point spaces easily reduces to the main result of Atiyah and Singer [3]. 3.6. The Thom isomorphism, pushforward, and the Pontryagin product t n ΩEn+1 }∞ We begin with a general discussion. Let E = {En −→ n=0 be a spectrum. For a real vector space V , equipped with a positive definite metric, let ΩV (En ) denote the space of maps from the unit ball B(V ) to En , sending the unit sphere S(V ) to the basepoint. The collection of spaces ΩV En forms a spectrum ΩV E. An isomorphism V ≈ Rk gives an identification of ΩV En with En−k , and of ΩV E with the spectrum derived from E by simply shifting the indices. Such a spectrum is called a ‘shift desuspension’ of E (see [25]). Some careful organization is required to avoid encountering signs by moving loop coordinates past each other. The reader is referred to [25] for more details. Of course, for a space X one has (ΩV E)n (X) ≈ E n (X × (V, V \ {0})) ≈ E −k+n (X). Now suppose that V is a vector bundle of dimension k over a space X. The construction described above can be formed fiberwise to form a bundle ΩV E = {ΩV En } of spectra over X. The group of vertical homotopy classes of sections π0 Γ(X, ΩV En ) can then be thought of as a twisted form of [X, E−k+n ] = E (generalized) cohomology group E −τV +n (X). (3.18) −k+n (X). We denote this twisted Now the group (3.18) is the group of pointed homotopy classes of maps [X V , En ] from the Thom complex of V to En . This gives a tautological Thom isomorphism Ẽ n (X V ) = E n (B(V ), S(V )) ≈ Ẽ −τV +n (X). The more usual Thom isomorphisms arise when a geometric construction is used to trivialize the bundle ΩV E. Such a trivialization is usually called an ‘E-orientation of V ’. We now return to the case E = K, with the aim of identifying the twisting τV with type defined in § 2. The main point is that the action of the orthogonal group O(k) on Ωk Fred(n) lifts through the Atiyah–Singer map Fred(k+n) → Ωk Fred(n) . Our discussion of this matter is inspired by the Stoltz–Teichner [31] description of Spin structures, and, of course, Donovan–Karoubi [14]. Let X be a local quotient groupoid, V be a real vector bundle over X of dimension k, and Cℓ(V ) be the associated bundle of Cliﬀord algebras. The bundle Cℓ(V ) ⊗ H is a locally universal Cℓ(V )-module. The Atiyah–Singer construction [3] gives a map FredCℓ(V ) (Cℓ(V ) ⊗ H) −→ ΩV Fred(0) (Cℓ(V ) ⊗ H), which is a weak equivalence on global sections. We can therefore trivialize the bundle of spectra ΩV K by trivializing the bundle of Cliﬀord algebras Cℓ(V ). Of course something weaker will also trivialize ΩV K. We do not really need a bundle isomorphism Cℓ(V ) ≈ X × Ck . We just need a way of going back and forth between Cℓ(V )modules and Ck modules. It is enough to have a bundle of irreducible Cℓ(V )−Ck bimodules giving a Morita equivalence. Let M = Ck , regarded as a Cℓ(Rk )−Ck -bimodule. We equip M with the Hermitian metric in which the monomials in the ǫi are orthonormal. Consider the group Pinc (k) of pairs (t, f ) LOOP GROUPS AND TWISTED K-THEORY I 767 in which t : Rk → Rk is an orthogonal map, and f : t∗ M −→ M is a unitary bimodule isomorphism. The group Pinc (k) is a graded central extension of O(k), graded by the sign of the determinant. We now identify the twisting τV in the terms of § 2.3. Let E → X be the bundle of orthonormal frames in V . Thus, E → X is a principal bundle with structure group O(k). Write P = E//O(k), P̃ = E//Pinc (k). Then P −→ X is a local equivalence and P̃ → P is a graded central extension, defining a twisting τ . Over P̃ we can form the bundle of bimodules M̃ = (E × M )//Pinc (k), giving a Morita equivalence between bundles of Cℓ(V )-modules and bundles of Ck -modules. In particular, H ′ = homCℓ(V ) (M̃ , Cℓ(V ) ⊗ H) is a locally universal τ -twisted Ck -module, and the map Γ(FredCℓ(V ) (Cℓ(V ) ⊗ H)) −→ Γ(Fred(k) (H ′ )) T −→ T ◦ ( − ) is a homeomorphism. Thus, the group K −τV +n (X) is isomorphic to the twisted K-group K −τ +n (X), and, as in Donovan and Karoubi [14], we have a tautological Thom isomorphism K n (X V ) ≈ K −τ +n (X). More generally, the same construction leads to a tautological Thom isomorphism K σ+n (B(V ), S(V )) ≈ K −τ +σ+n (X), (3.19) when V is a vector bundle over a groupoid X. With the Thom isomorphism in hand, one can define the pushforward, or umkehr map in the usual way. Let f : X → Y be a map of smooth manifolds, or a map of groupoids forming a bundle of smooth manifolds, T = TX/Y the corresponding relative (stable) tangent bundle, and τ0 the twisting on X corresponding to T . Given a twisting τ on Y , and an isomorphism f ∗ τ ≈ τ0 , one can combine the Pontryagin–Thom collapse with the Thom isomorphism to form a pushforward map f! : K f ∗ σ+n (X) −→ K −τ +σ+n (Y ), where σ is any twisting on Y . We leave the details to the reader. We apply this to the situation in which X = (G × G)//G, Y = G//G (both with the adjoint action) and X → Y is the multiplication map µ. In this case, the twisting τ0 can be taken to be the twisting we denoted by g in the introduction. As g is pulled back from pt//G, there are canonical isomorphisms µ∗ g ≈ p∗1 g ≈ p∗2 g. We just write g for any of these twistings. Suppose that σ is any twisting of G//G that is ‘primitive’ in the sense that it comes equipped with an associative isomorphism µ∗ σ ≈ σ+g (G) acquires a Pontryagin product p∗1 σ + p∗2 σ. Then the group KG ∗ µ σ+g σ+g KG (G) ⊗ KG (G) −→ KG 0 making it into an algebra over KG (pt) = R(G). σ+2g µ! σ+g (G × G) −→ KG (G), 768 D. S. FREED, M. J. HOPKINS AND C. TELEMAN 3.7. The fundamental spectral sequence Our basic technique of computation will be based on a variation of the Atiyah–Hirzebruch spectral sequence, which is constructed using the technique of Segal [30]. The identification of the E 2 -term depends only on the properties listed in § 3.1. Suppose that X is a local quotient groupoid, and write Ǩτ +t for the presheaf on [X] given by Ǩτ +t (U ) = K τ +t (XU ). Write Kτ +t = sh Ǩτ +t , for the associated sheaf. The limit of the Mayer–Vietoris spectral sequences associated to the (hyper-)covers of [X] is a spectral sequence H s ([X]; Kτ +t ) =⇒ K τ +s+t (X). As X admits locally contractible slices, the stalk of Kτ +t at a point c ∈ [X] is 0 t odd, τ +t K (Xc ) ≈ τ R (Gx ) t even, where x ∈ X0 is a representative of c, and Gx = X(x, x). τ +t There is also a relative version. Suppose that A ⊂ X is a pair of groupoids, and write Krel for the sheaf on [X] associated to the presheaf U −→ K τ +t (XU , X[A]∩U ). Then the limit of the Mayer–Vietoris spectral sequences associated to the hyper-covers of [X] gives τ +t H s ([X]; Krel ) =⇒ K τ +s+t (X, A). This spectral sequence is most useful when A ⊂ X is closed, and has the property that, for all suﬃciently small U ⊂ [X], the map K τ +t (XU ) → K τ +t (X[A]∩U ) is surjective. In that case there is (for suﬃciently small U ) a short exact sequence K τ +t (XU , X[A]∩U ) −→ K τ +t (XU ) −→ K τ +t (X[A]∩U ), τ +t and the sheaf Krel can be identified with the extension of i∗ Kτ +t by zero τ +t Krel = i! (Kτ +t ), where i : V ⊂ [X] is the inclusion of the complement of A. We make use of this situation in the proof of Proposition 4.41. τ (G) 4. Computation of KG τ +∗ The aim of this section is to compute the groups KG (G) for non-degenerate τ . We begin by considering general twistings, and adopt the non-degeneracy hypothesis as necessary. Our main results are Theorem 4.27 and Corollaries 4.38 and 4.39. 4.1. Notation and assumptions We first fix some notation. Let (i) G be a compact connected Lie group; (ii) g be the Lie algebra of G; (iii) T be a fixed maximal torus of G; (iv) t be the Lie algebra of T ; LOOP GROUPS AND TWISTED K-THEORY I 769 (v) N be the normalizer of T ; (vi) W = N/T the Weyl group; (vii) Π = ker exp : t → T ; (viii) Λ = hom(Π, Z), the character group of T ; e = Π ⋊ NT; (ix) Naﬀ e e = Π ⋊ W = Naﬀ /T , the extended aﬃne Weyl group. (x) Waﬀ e The group Waﬀ can be identified as the group of symmetries of t generated by translations in Π and the reflections in W . When G is connected, the exponential map, from the orbit space e to the space of conjugacy classes in G, is a homeomorphism. t/ Waﬀ We make our computation for groups satisfying the equivalent conditions of the following lemma. Lemma 4.1. For a Lie group G the following are equivalent: (i) for each g ∈ G the centralizer Z(g) is connected; (ii) G is connected and π1 G is torsion-free; (iii) G is connected and any central extension T −→ Gτ −→ G splits. Proof. The equivalence of (ii) and (iii) is elementary: as G is connected, its classifying space BG is simply connected, and from the Hurewicz theorem and the universal coeﬃcient theorem the torsion subgroup of π1 G is isomorphic to the torsion subgroup of H2 (BG), and so too the torsion subgroup of H 3 (BG; Z). But for any compact Lie group the odd-dimensional cohomology of the classifying space is torsion, the real cohomology of the classifying space is in even degrees (and is given by invariant polynomials on the Lie algebra). The implication (ii) =⇒ (i) is [7, (3.5)]. For the converse (i) =⇒ (ii), we note first that G = Z(e) is connected by hypothesis. Let G′ ⊂ G denote the derived subgroup of G, the connected Lie subgroup generated by commutators in G, and Z1 ⊂ G the connected component of the center of G. Set A = Z(G′ ) ∩ Z1 . Then from the principal fiber bundle G′ → G → Z1 /A we deduce that the torsion subgroup of π1 G is π1 G′ . We must show that the latter vanishes. Now the inclusion π1 Z(g) → π1 G is surjective for any g ∈ G, as any centralizer contains a maximal torus T of G and the inclusion π1 T → π1 G is surjective, the flag manifold G/T being simply connected. It follows that Z(g) is connected if and only if the conjugacy class G/Z(g) is simply connected. Furthermore, the conjugacy class in G of an element of G′ equals its conjugacy class in G′ , from which we deduce that all conjugacy classes in G′ are connected and simply connected. Let G′ denote the simply connected (finite) cover of G′ . Then the set of conjugacy classes in G′ may be identified as a bounded convex polytope in the Lie algebra of a maximal torus, and furthermore π1 G′ acts on it by aﬃne transformations with quotient G′ /G′ ; see [16, § 3.9]. The center of mass of the vertices of the polytope is a fixed point of the action. The finite group π1 G′ acts freely on the corresponding conjugacy class of G′ with quotient a conjugacy class in G′ . As the former is connected and the latter simply connected, it follows that π1 G′ is trivial, as desired. 4.2. The main computation Let X = G//G be the groupoid formed from G acting on itself by conjugation. We compute τ +∗ K τ +∗ (X) = KG (G) using the spectral sequence described in § 3.7. In this case, it takes the form τ +s+t (G). H s (G/G; Kτ +t ) =⇒ KG (4.2) 770 D. S. FREED, M. J. HOPKINS AND C. TELEMAN The orbit space G/G is the space of conjugacy classes in G, which is homeomorphic via the e e exponential map to t/ Waﬀ . Our first task is to identify the sheaf Kτ +t on G/G ≈ t/ Waﬀ . τ +t As G//G admits locally contractible slices, the stalk of K at a conjugacy class c ∈ G/G is the twisted equivariant K-group τ +t KG (c). A choice of point g ∈ c gives an identification c = G/Z(g), and an isomorphism Rτg (Z(g)) t even, τg +t τ +t KG (c) ≈ KZ(g) ({g}) ≈ 0 t odd. (4.3) We have denoted by τg the restriction of τ to {g}//Z(g), in order to emphasize the dependence on the choice of g. Among other things, this proves that Kτ +odd = 0. The twisting τg corresponds to a graded central extension T −→ Z̃(g) −→ Z(g). (4.4) The group Z(g) has T for a maximal torus, and is connected when G satisfies the equivalent conditions of Lemma 5.1. Denote T −→ T̃ −→ T (4.5) the restriction of (4.4) to T . Then T̃ is a maximal torus in Z̃(g). The map from the Weyl group of Z̃(g) to the Weyl group Wg of Z(g) is an isomorphism, and Rτg (Z(g)) −→ Rτg (T )Wg is an isomorphism. We can therefore rewrite (4.3) as Rτg (T )Wg τ +t KG (c) ≈ 0 t even, t odd. (4.6) We now reformulate these remarks in order to eliminate the explicit choice of g ∈ c. We can cut down the size of c by requiring that g lie in T . That helps, but it does not eliminate the dependence of τg on g. We can get rid of the reference to g by choosing a geodesic in T from each g to the identity element, and using it to identify the twisting τg with τ0 . This amounts to considering the set of elements of t, which exponentiate into c. This set admits a e , and the stabilizer of an element v is canonically isomorphic to Wg transitive action of Waﬀ where g = exp(v). e We are thus led to look at the groupoid t//T , and the action of Waﬀ . Writing it this e way, however, does not conveniently display the action of Waﬀ on the twisting τ . Following e e ×t)//Naﬀ . Example 3.23, we work instead with the weakly equivalent groupoid (Waﬀ Consider the map τ +t τ +t e (G) −→ KN KG e (Waﬀ ×t) aff induced by projection exp e e e ×t)//Naﬀ −−−−−−→ t//Naﬀ −−→ G//G. (Waﬀ e e e e As the right action of Waﬀ = Naﬀ /T commutes with the diagonal left action of Naﬀ on Waﬀ ×t, e e e the group Waﬀ acts on the groupoid Waﬀ ×t//Naﬀ . The twisting τ is fixed by this action as it e e e on (Waﬀ ×t)//Naﬀ therefore induces a right is pulled back from G//G. The left action of Waﬀ τ +t τ +t e action of Waﬀ on KN e (t), and the image of KG (G) is invariant: aff e τ +t τ +t e Waff KG . (G) −→ KN e (Waﬀ ×t) aff (4.7) 771 LOOP GROUPS AND TWISTED K-THEORY I e e As Waﬀ = Naﬀ /T , the map e e t//T − → (Waﬀ ×t)//Naﬀ is a local equivalence, and so gives an isomorphism τ +t τ +t e KN (t). e (Waﬀ ×t) ≈ KT aff There is therefore an action of e Waﬀ on KTτ +t (t), and we may rewrite (4.7) as e τ +t (G) −→ KTτ +t (t)Waff . KG e For c ∈ G/G ≈ t/ Waﬀ , let Sc = {s ∈ t | exp(s) ∈ c} be the corresponding e Waﬀ -orbit in t. A similar discussion gives a map e e τ +t τ +t e (Waﬀ ×Sc )Waff ≈ KTτ +t (Sc )Waff . (c) −→ KN KG (4.8) Proposition 4.9. If G satisfies the conditions of Lemma 5.1, then the map e τ +t (c) −→ KTτ +t (Sc )Waff KG constructed above is an isomorphism. e be the stabilizer of v. We then have an Proof. Choose v ∈ Sc , and let Wv ⊂ Waﬀ e identification Sc ≈ Waﬀ /Wv , and so an isomorphism e KTτ +t (Sc )Waff ≈ KTτ +t ({v})Wv . e → G identifies Wv with the Weyl group of Z(g), Write g = exp(v). The restriction of Naﬀ {v}//T with {g}//T , and the restriction of τ to {v}//T with τg . By Example 2.33 the action of Wv on KTτ +t ({v}) coincides with the action Wv by conjugation. The result then follows from (4.6). We now identify the sheaf Kτ +t . As {0} → t is an equivariant homotopy equivalence, the restriction map KTτ +t (Sc × t) −→ KTτ +t (Sc × {0}) is an isomorphism. Next note that the aggregate of the restriction maps to the points of Sc gives a map from e KTτ +t (Sc × t)Waff e to the set of Waﬀ -equivariant maps Sc −→ KTτ +t (t), e acts transitively on Sc , is easily checked to be an isomorphism. which, using the fact that Waﬀ e e , set Write p : t → t/ Waﬀ for the projection, and, for an open U ⊂ G/G = t/ Waﬀ SU = p−1 (U ). Let F τ +t be the presheaf that associates to U ⊂ G/G the set of locally constant e Waﬀ -equivariant maps SU −→ KTτ +t (t). There is then a map of presheaves Ǩτ +t −→ F τ +t , 772 D. S. FREED, M. J. HOPKINS AND C. TELEMAN hence a map of sheaves Kτ +t −→ F τ +t . Corollary 4.11. (4.10) The map (4.10) is an isomorphism. Proof. Proposition 4.9 implies that (4.10) is an isomorphism of stalks, hence an isomorphism. We now re-interpret the sheaf F in a form more suitable to describing its cohomology. As Twistt//T −→ Twist{0}//T is an equivalence of categories, the restriction of τ to t//T corresponds to a graded central extension T −→ T τ −→ T (4.12) e e equipped with an action of Waﬀ . The Waﬀ -action fixes T and acts on T through its quotient τ W , the Weyl group. Write Λ for the set of splittings of (4.12). Note that Λτ is a torsor for Λ e and inherits a compatible Waﬀ action from (4.12). By Proposition 3.5 the group KTτ +0 (t) ≈ KTτ +0 ({0}) may be identified with the set of compactly supported functions on Λτ with values in Z. We e is the combination of its natural action on Λτ and an will see shortly that the action of Waﬀ e action on Z given by a homomorphism ǫ : Waﬀ → Z/2. Writing Z(ǫ) for the sign representation e associated to ǫ, we then have an isomorphism of Waﬀ -modules KTτ +0 ({0}) ≈ Homc (Λτ , Z(ǫ)). (4.13) To verify the claim about the action, first note that the automorphism group of the restriction of τ to t//T is H 2 (BT ; Z) × H 0 (BT ; Z/2) ≈ Λ × Z/2 ≈ R(T )× ≈ KT0 (pt)× . By part (ii) of Proposition 3.4, the factor Λ acts on KTτ +0 ({0}) through its natural action on Λτ , whereas the Z/2 acts by its sign representation. e e As Waﬀ = Π ⋊ W , and the action of Waﬀ on KTτ +0 ({0}) is determined by its restriction to Π and W . The group Π acts trivially on T and so it acts on KTτ +0 ({0}) through a homomorphism (b,ǫΠ ) Π −−−−→ Λ × Z/2. The group W does act on T , and so on H 2 (BT ; Z) × H 0 (BT ; Z/2) ≈ Λ × Z/2, by the product of the natural (reflection) action on Λ and the trivial action on Z/2. The e restriction of the action of Waﬀ to W is therefore determined by a crossed homomorphism W −→ Λ compatible with b, and an ordinary homomorphism ǫW : W → Z/2. The maps ǫΠ and ǫW e combine to give the desired map ǫ : Waﬀ → Z/2, whereas the map b : Π → Λ and the crossed e homomorphism W → Λ correspond to the natural action of Waﬀ on Λτ . This verifies the e isomorphism (4.13) of Waﬀ -modules. We can now give a useful description of F. First recall a construction. Suppose that X is a space equipped with an action of a group Γ, and that G is an equivariant sheaf on X. Write LOOP GROUPS AND TWISTED K-THEORY I 773 p : X → X/Γ for the projection to the orbit space. There is then a sheaf, (p∗ G)Γ on X/Γ whose value on an open set V is the set of Γ-invariant elements of G(p−1 V ). A very simple situation is when G is the constant sheaf Z. In that case (p∗ G)Γ is again the constant sheaf Z. This will be useful in the proof of Proposition 4.18. Corollary 4.14. Write τ e Λ t̃ = t ×Waff and let p : t × Λτ −→ t̃ and e f : t̃ −→ t/ Waﬀ denote the projections. There is a canonical isomorphism e F τ +0 ≈ f∗c (p∗ Z(ǫ)Waff ), where f∗c denotes a pushforward with proper supports. To go further, we need to make an assumption. Assumption 4.15. The twisting τ is non-degenerate in the sense that b is a monomorphism. In terms of the classification of twistings, this is equivalent to requiring that the image of the isomorphism class of τ in HT3 (T ; R) ≈ H 1 (T ; R) ⊗ H 1 (T ; R) is a non-degenerate bilinear form. Next note the following lemma. Lemma 4.16. The map ǫW : W → Z/2 is trivial. 1 Proof. The homomorphism in question corresponds to the element in HW (pt) = H 1 (BW ; Z/2) given by restricting the isomorphism class of the twisting along 1 3 1 1 HG (G; Z/2) × HG (G; Z) −→ HG (G; Z/2) −→ HG ({e}; Z/2) 1 1 −→ HN ({e}; Z/2) ≈ HW ({e}; Z/2). 1 As G is assumed to be connected, HG (pt) = 0 and the result follows. Corollary 4.17. There is an isomorphism Z ≈ Z(ǫ) of equivariant sheaves on t × Λτ . Proof. The sheaf Z(ǫ) is classified by the element 1 τ ǫ̃ ∈ HW e (t × Λ ; Z/2) aff 1 pulled back from the ǫ ∈ HW e (pt; Z/2). By Lemma 4.16 the restriction of ǫ̃ to W is trivial. By aff 1 τ Assumption 4.15, the group Π acts freely on Λτ , and so the restriction of ǫ̃ to HW e (t × Λ ; Z/2) aff is also trivial. This proves that ǫ̃ = 0. 774 D. S. FREED, M. J. HOPKINS AND C. TELEMAN Proposition 4.18. There is an isomorphism F τ +0 ≈ f∗c (Z), τ e Λ where f : t̃ = t ×Waff → t/W is the projection and f∗c denotes a pushforward with proper supports. Proof. By Corollaries 4.17 and 4.14 there is an isomorphism e F τ +0 ≈ f∗c (p∗ ZWaff ), e so the result follows from the fact that p∗ ZWaff ≈ Z. Finally, using the existence of contractible local slices, we can describe the cohomology of Kτ +0 ≈ F τ +0 . Lemma 4.19. The edge homomorphism of the Leray spectral sequence for f is an isomorphism e H ∗ (t/ Waﬀ ; Kτ ) ≈ Hc∗ (t̃; Z), where Hc∗ denotes cohomology with compact supports. To calculate Hc∗ (t̃; Z), we need to understand the structure of t̃. This amounts to describing e on Λτ . more carefully the action of Waﬀ Lemma 4.20. There exists an element λ0 ∈ Λτ fixed by W . Proof. The inclusion {0}//T ⊂ t//T is equivariant for the action of W . We can therefore study the action of W on the central extension of T defined by the restriction of τ to {0}//T . Now τ started out as a twisting of G//G, so our twisting of {0}//T is the restriction of a twisting τG of {e}//G. Moreover, the action of W is derived from the action of inner automorphisms of G on τG . Now the twisting τG corresponds to a central extension T −→ Gτ −→ G. By our assumptions on G (Lemma 5.1), this central extension splits. Choose a splitting Gτ −→ T, (4.21) and let λ0 ∈ Λτ be the composition T̃ −→ Gτ −→ T. As T is abelian, the splitting (4.21) is preserved by inner automorphisms of G. It follows that splitting e is fixed by the inner automorphisms of G̃, which normalize T̃ . The claim follows. Remark 4.22. Any two choices of λ0 diﬀer by a character of G, so the element λ0 is unique if and only if the character group of G is trivial. As we have assumed that G is connected and π1 G is torsion-free, this is in turn equivalent to requiring that G be simply connected. LOOP GROUPS AND TWISTED K-THEORY I 775 Remark 4.23. A primitive twisting τ comes equipped with a trivialization of its restriction to {e}//G, or in other words a splitting of the graded central extension Gτ → G. A primitive twisting therefore comes equipped with a canonical choice of λ0 . Using a fixed choice of λ0 , we can identify Λτ with Λ as a W -space. To sum up, we can make an identification Λτ ≈ Λ, the action of Π is given by a W -equivariant homomorphism Π → Λ, and the W -action is the natural one on Λ. e Lemma 4.24. When τ is non-degenerate, the Waﬀ -set Λτ admits an (equivariant) embede τ e ding in t. There are finitely many Waﬀ -orbit in Λ , and each orbit is of the form Waﬀ /Wc , with (Wc , t) a finite (aﬃne) reflection group. Proof. As b is a monomorphism, the map t = Π ⊗ R → Λ ⊗ R is an isomorphism. The first assertion now follows from our identification of Λτ with Λ. As for the finiteness of the number of orbits, as b is a monomorphism, the group Λ/Π is finite, and there are already only finitely many Π-orbits in Λτ . The remaining assertions follow from standard facts about the action of e Waﬀ on t (Propositions 4.46 and 4.47). Corollary 4.25. When τ is non-degenerate, there is a homeomorphism t̃ ≡ t/Ws s∈S with S finite, and Ws a finite reflection group of isometries of t. Moreover, t/Ws ≡ Rn1 × [0, ∞)n2 with n2 = 0 if and only if Ws is trivial. Proof. This is immediate from Lemma 4.24 and Proposition 4.47. As Hc∗ ([0, ∞); Z) = 0 and Hc∗ (R; Z) Z ∗ = 1, = 0 otherwise, the Kunneth formula gives Hc∗ (Rn1 Z n2 = 0 and ∗ = n1 , × [0, ∞) ; Z) = 0 otherwise. n2 In summary, we have the following proposition. Proposition 4.26. The cohomology group Hc∗ (t̃; Z) is zero unless ∗ = n, and Hcn (t̃; Z) is e isomorphic to the free abelian group on the set of free Waﬀ -orbits in Λτ . More functorially, τ n e (Λ , H (t) ⊗ Z(ǫ)). Hcn (t̃; Z) ≈ HomWaff c Proposition 4.26 implies that the spectral sequence (4.2) collapses, giving the following theorem. 776 D. S. FREED, M. J. HOPKINS AND C. TELEMAN Theorem 4.27. Suppose that G is a Lie group of rank n satisfying the conditions of Lemma 4.1, and that τ is a non-degenerate twisting of G//G, classified by 3 1 [τ ] ∈ HG (G; Z) × HG (G; Z/2). The restriction of τ to pt//T determines a central extension T −→ T τ −→ T (4.28) e e with an action of Waﬀ → Z/2 for the map . Write Λτ for the set of splittings of (4.28), ǫ : Waﬀ corresponding to the restriction of [τ ] to 1 e HN (T ; Z/2) ≈ H 1 (Waﬀ ; Z/2), τ +n+1 and Z(ǫ) for the associated sign representation. Then KG (G) = 0, and the twisted K-group τ +n KG (G) is given by τ +n τ n e (Λ , H (t) ⊗ Z(ǫ)), KG (G) ≈ HomWaff c e which can be identified with the free abelian group on the set of free Waﬀ -orbits in Λτ , after choosing a point in each free orbit. This isomorphism is natural in the sense that if i : H ⊂ G is a subgroup of rank n also satisfying the conditions of Lemma 4.1, then the restriction map τ +n τ +n KG (G) −→ KH (H) is given by the inclusion τ n τ n e (G) (Λ , H (t) ⊗ Z(ǫ)) ⊂ HomW e (H) (Λ , H (t) ⊗ Z(ǫ)). HomWaff c c aff e Remark 4.29. In Theorem 4.27, the group Waﬀ acts on Hcn (t) through the action of W on t. The reflections thus act by (−1) and a choice of orientation on t identifies Hcn (t) with the usual sign representation of W on Z. τ +n The group KG (G) is a module over R(G). Our next goal is to identify this module structure. Because G is connected, we can identify R(G) with the ring of W -invariant elements of Z[Λ] or with the convolution algebra of compactly supported functions Homc (Λ, Z). The algebra Homc (Λ, Z) acts on Hom(Λτ , Hcn (t) ⊗ Z(ǫ)) by convolution, and one easily checks that e the W -invariant elements preserve the Waﬀ -equivariant functions. Proposition 4.30. Under the identification τ +n τ n e (Λ , H (t) ⊗ Z(ǫ)), KG (G) ≈ HomWaff c the action of R(G) ≈ Homc (Λ, Z)W corresponds to convolution of functions. Proof. This is straightforward to check in case G is a torus. The case of general G is reduced to this case by looking at the restriction map to a maximal torus and using Theorem 4.27. τ +n Proposition 4.30 leads to a very useful description of KG (G). Choose an orientation of t n and hence an identification Hc (t) ≈ Z of abelian groups. By definition, the elements of Λτ are characters of T τ , all of which restrict to the defining character of T. To a function f ∈ HomΠ (Λτ , Hcn (t) ⊗ Z(ǫ)) ≈ HomΠ (Λτ , Z(ǫ)), we associate the series f (λ)λ−1 , δf = λ∈Λτ (4.31) LOOP GROUPS AND TWISTED K-THEORY I 777 which is the Fourier expansion of the distribution on T τ satisfying δf (λ) = f (λ). Our next aim is to work out more explicitly which distribution it is, especially when f comes from an element of τ n τ e (Λ , H (t) ⊗ Z(ǫ)) ≈ HomW e (Λ , Z(ǫ)). HomWaff c aff As all of the characters λ restrict to the defining character of the central T, we think of the distribution δf as acting on the space of functions g : T τ → C satisfying g(ζv) = ζg(v) for ζ ∈ T. This space is the space of sections of a suitable complex line bundle Lτ over T . The character χ of a representation of G is a function on T , and the action of χ on δf is given by χ · δf (g) = δf (g · χ). As Π and Λ are duals, we have hom(Π, Z/2) = Λ ⊗ Z/2, and we may regard ǫΠ as an element of Λ/2Λ. This determines an element λǫ = 21 ǫΠ ∈ 21 Λ/Λ ⊂ Λ ⊗ R/Z. Thinking of Π as the character group of Λ ⊗ R/Z, the function ǫΠ : Π → Z/2 is given by evaluation of characters on λǫ : ǫΠ (π) = π(λǫ ). As ǫΠ is W -invariant, so is λǫ . From the embedding b : Π ⊂ Λ we get a map b : T = Π ⊗ R/Z −→ Λ ⊗ R/Z. We write F = Λ/Π for the kernel of this map, and Fǫ for the inverse image of λǫ . Set F τ = Λτ /Π. The elements of F τ can be interpreted as sections of the restriction of Lτ to F . Finally, let τ Fǫ,reg ⊂ Fǫ and Freg ⊂ F τ be the subsets consisting of elements on which the Weyl group W acts freely. Proposition 4.32. For f ∈ HomΠ (Λτ , Hcn (t) ⊗ Z(ǫ)) ≈ HomΠ (Λτ , Z(ǫ)), the value of the distribution δf on a section g of Lτ is given by 1 f (λ)λ−1 (x)g(x). δf (g) = |F | τ (λ,x)∈F ×Fǫ e When f is Waﬀ -invariant, then δf (g) = 1 |F | f (λ)λ−1 (x)g(x). τ ×F (λ,x)∈Freg ǫ,reg Proof. Let us first check that (4.33) is well defined. Under λ −→ λπ, the term f (λ)λ −1 π ∈ Π, (x)g(x) gets sent to f (λπ)(λπ)−1 (x)g(x) = ǫ(π)f (λ)λ−1 (x)π −1 (x)g(x) = ǫ(π)f (λ)λ−1 (x)ǫ(π −1 )g(x) = f (λ)λ−1 (x)g(x), so f (λ)λ−1 (x)g(x) does indeed depend only on the Π-coset of λ. (4.33) 778 D. S. FREED, M. J. HOPKINS AND C. TELEMAN To establish (4.33), it suﬃces by linearity to consider the case in which δf is of the form δf = λ−1 · δΠ , with λ ∈ Λτ , ǫ(π)π −1 , δΠ = π∈Π τ and g is an element of Λ . In this case, f vanishes oﬀ of the Π-orbit through λ, and f (λ) = 1. The sum (4.33) is then 1 λ−1 (x)g(x). (4.34) |F | x∈Fǫ By definition, for η ∈ Π, 1 |F | δΠ (η) = η(λǫ ) = η(x). x∈Fǫ For η ∈ Λ \ Π there is an a ∈ F with η(a) = 1. In that case 1 |F | η(x) = x∈Fǫ 1 |F | η(ax) = η(a) x∈Fǫ 1 |F | η(x), x∈Fǫ so 1 |F | η(x) = 0. x∈Fǫ It follows that, for every η ∈ Λ, δΠ (η) = 1 |F | η(x). x∈Fǫ Now suppose g ∈ Λτ . Then δf (g) = δΠ (λ−1 g) = 1 |F | λ−1 (x)g(x), x∈Fǫ which is (4.34). This proves the first assertion of Proposition 4.32. τ For the second assertion, note that if λ ∈ Freg is fixed by an element of W , it is fixed by an element w ∈ W , which is a reflection. By Weyl equivariance, we have f (λ) = f (wλ) = w · f (λ) = −f (λ), and so f (λ) = 0. This gives δf (g) = 1 |F | f (λ)λ−1 (x)g(x). τ ×F (λ,x)∈Freg If x ∈ F is an element fixed by a reflection w ∈ W , then f (λ)λ−1 (w · x)g(x) f (λ)λ−1 (x)g(x) = τ λ∈Freg τ λ∈Freg f (λ)(λw )−1 (x)g(x) = τ λ∈Freg f (λw )λ−1 (x)g(x) = τ λ∈Freg f (λ)λ−1 (x)g(x), =− τ λ∈Freg LOOP GROUPS AND TWISTED K-THEORY I so the terms involving such an x sum to zero, and 1 δf (g) = |F | τ 779 f (λ)λ−1 (x)g(x). (λ,x)∈Freg ×Fǫ,reg Let I τ ⊂ R(G) be the ideal consisting of virtual representations whose character vanishes on the elements of Fǫ,reg . Corollary 4.35. τ +∗ The ideal I τ annihilates KG (G). τ n e (Λ , H (t) ⊗ Z(ǫ)) Proof. Write χ for the character of an element of I τ . For f ∈ HomWaff c we have, by Proposition 4.32, f (λ)λ−1 (x)g(x)χ(x) = 0. χδf (g) = δf (g · χ) = τ ×F (λ,x)∈Freg reg Remark 4.36. The conjugacy classes in G of the elements in Fǫ,reg are known as the Verlinde conjugacy classes, and the ideal I τ as the Verlinde ideal. τ +n Proposition 4.37. The R(G)-module KG (G) is cyclic. e Proof. Using Lemma 4.20, choose a Waﬀ -equivariant isomorphism Λτ ≈ Λ, and an τ +n n (G) with orientation of t giving an isomorphism Hc (t) ≈ Z. We can then identify KG e (Λ, Z(ǫ)), HomWaff e although we remind the reader that Waﬀ acts on Z through its sign representation. We continue the convention of writing elements e (Λ, Z) f ∈ HomWaff as Fourier series f (λ)λ−1 . Set ǫ(π)π −1 , δΠ = π∈Π and, for λ ∈ Λ, write (−1)w w · λ. a(λ) = w∈W Then the elements a(λ) ∗ δΠ e (Λ, Z(ǫ)). As π1 G is torsion-free, there is an exact sequence span HomWaff G′ −→ G −→ J, where J is a torus and G′ is simply connected. The character group of J is the subgroup ΛW of Weyl-invariant elements of Λ, and the weight lattice for G′ is the quotient Λ/ΛW . Choose 780 D. S. FREED, M. J. HOPKINS AND C. TELEMAN a Weyl chamber for G and let ρ ∈ Λ ⊗ Q be 1/2 the sum of the positive roots of G (which we write as a product of square roots of elements in our Fourier series notation). As J is a torus, the image ρ′ of ρ in Λ/ΛW ⊗ Q is 1/2 the sum of the positive roots of G′ , which, as G′ is simply connected, lies in Λ/ΛW . Let ρ̃ ∈ Λ be any element congruent to ρ′ modulo ΛW . Claim: for any λ ∈ Λ, the ratio a(λ) a(ρ̃) is the character of a (virtual) representation. The claim shows that the class corresponding to τ +n (G). For the claim, first note that the element a(ρ̃) · δΠ is an R(G)-module generator of KG µ = ρ/ρ̃ is W -invariant (and is in fact the square root of a character of J). It follows from the Weyl character formula that a((λρ̃−1 ) · ρ) a(ρ) is, up to sign, the character of an irreducible representation. But then a(λρ̃−1 ρ) a(λµ) a(λ) = =µ a(ρ) a(ρ) a(ρ) a(λ) a(λ) = . = a(µ−1 ρ) a(ρ̃) τ +n (G) be the class corresponding to a(ρ̃) · δΠ . The map Corollary 4.38. Let U ∈ KG ‘multiplication by U ’ is an isomorphism τ +n R(G)/I τ −→ KG (G) of R(G)-modules. Proof. That the map factors through the quotient by I τ is Corollary 4.35, and that it is surjective is Proposition 4.37. The result now follows from the fact that both sides are free of rank equal to the number of free W -orbits in Aǫ,reg (that is, the number of Verlinde conjugacy classes). τ +n As described at the end of § 3.6, when τ is primitive the R(G)-module KG (G) acquires the structure of an R(G)-algebra. Corollary 4.39. When τ is primitive, there is a canonical algebra isomorphism τ +n KG (G) ≈ R(G)/I τ . τ +n τ Remark 4.40. When τ is primitive, the pushforward map KG (e) → KG is a ring homomorphism. Being primitive, the restriction of τ to {e}//G comes equipped with a τ (e) ≈ R(G). trivialization and so KG The isomorphisms of Corollaries 4.38 and 4.39 are proved after tensoring with the complex numbers in [21], where the distributions δf are related to the Kac numerator at q = 1. We refer the reader to [21, §§ 6 and 7] for further discussion. LOOP GROUPS AND TWISTED K-THEORY I 781 We conclude with a further computation, which will be used in Part II. We consider the situation of this section in which G = T is a torus of dimension n. The group KTτ ({e}) is the free abelian group on Λτ , and the pushforward map KTτ ({e}) → KTτ +n (T ) is defined. Proposition 4.41. The pushforward map i! : KTτ ({e}) −→ KTτ +n (T ) sends the class corresponding to λ ∈ Λτ to the class in KTτ +n (T ) ≈ HomΠ (Λτ , Z(ǫ)) corresponding to the distribution with Fourier expansion λ−1 ǫ(π)π −1 . π∈Π Proof. The pushforward map is the composition of the Thom isomorphism KTτ +0 ({e}) −→ KTτ +n (t, t \ {0}) ≈ KTτ +n (T, T \ {e}) with the restriction map KTτ +n (T, T \ {e}) −→ KTτ +n (T ). (4.42) We wish to compute these maps using the spectral sequence for relative twisted K-theory described at the end of § 3.7. In order to do so, however, we need to replace T \ {e} and t \ {0} by the smaller T \ Be and t \ {B0 }, where Be and B0 are small open balls containing e and τ +t 0, respectively. This puts us in the situation described at the end § 3.7, where the sheaf Krel works out to be the extension by zero of the restriction of Kτ +t to Be . Applying the spectral sequence argument of this section to the pairs (t, t \ B0 ) and (T, T \ Be ) gives isomorphisms KTτ +n (T, T \ Be ) ≈ homΠ (Λτ , Hcn (t, t \ BΠ ) ⊗ Z(ǫ)), KTτ +n (t, t \ B0 ) ≈ homc (Λτ , Hcn (t, t \ B0 ) ⊗ Z(ǫ)), (4.43) where BΠ ⊂ t is the inverse image of Be under the exponential map. The same argument identifies the restriction mapping (4.42) with the map induced by Hcn (t, t \ BΠ ) −→ Hcn (t), and the isomorphism KTτ +n (T, T \ Be ) ≈ KTτ +n (t, t \ B0 ) with the map HomΠ (Λτ , Hcn (t, t \ BΠ ) ⊗ Z(ǫ)) −→ Homc (Λτ , Hcn (t, t \ B0 ) ⊗ Z(ǫ)), which first forgets the Π-action and then uses Hcn (t, t \ BΠ ) −→ Hcn (B̄0 , B0 ) ≈ Hcn (t, t \ B0 ). Finally, the isomorphism KTτ +n (t, t \ B0 ) ≈ KTτ (pt) ⊗ K n (t, t \ B0 ) shows that the Thom isomorphism is simply the tensor product of the identity map with suspension isomorphism K 0 (pt) → K n (t, \B0 ) (which uses the orientation of t). In terms of (4.43), this means that the Thom isomorphism KTτ (pt) ≈ Homc (Λτ , Z(ǫ)) −→ KTτ (t, t \ B0 ) ≈ homc (Λτ , Hcn (t, t \ B0 ) ⊗ Z(ǫ)), 782 D. S. FREED, M. J. HOPKINS AND C. TELEMAN is simply the map derived from the suspension isomorphism H 0 (pt) ≈ Hcn (t, t \ B0 ). The result follows easily from this. e 4.3. The action of Waﬀ on t We summarize here some standard facts about aﬃne Weyl groups and conjugacy classes in G. Our basic references are [7, 22]. Recall that we have fixed a maximal torus T of G. We write Λ for the character group of T and R for the set of roots. Following Bourbaki, write N (T, R) for the subgroup of t consisting of elements on which the roots vanish modulo 2πZ. There is a short exact sequence N (T, R) ֌ Π ։ π1 G. Let H be the set of hyperplanes forming the diagram of G. Thus, H = {Hk,α | k ∈ Z, α ∈ R}, where Hk,α = {x ∈ t | α(x) = 2πk}. The collection H is locally finite in the sense that each s ∈ t has a neighborhood meeting only finitely many hyperplanes in H. The aﬃne Weyl group is the group Waﬀ generated by reflections in the hyperplanes Hk,α ∈ H. It has the structure N (T, R) ⋊ W. Proposition 4.44. Let x ∈ t. The stabilizer of x in Waﬀ is the finite reflection group generated by reflections through the hyperplanes Hk,α containing x. e Write Waﬀ = L ⋊ W . There is a short exact sequence e Waﬀ ֌ Waﬀ ։ π1 G. (4.45) e Proposition 4.46. Let x ∈ t. If π1 G is torsion-free, then the stabilizer of x in Waﬀ coincides with the stabilizer of x in Waﬀ . It is therefore the finite reflection group generated by reflections through the hyperplanes Hk,α containing x. e Proof. Write Wx for the stabilizer of x in Waﬀ . The image of Wx in t ⋊ W is conjugate to a subgroup of W , and so Wx is finite. By assumption π1 G has no non-trivial finite subgroups. The exact sequence (4.45) then shows that Wx ⊂ Waﬀ . The result then follows from Proposition 4.44. Write R0 = [0, ∞). Proposition 4.47. Suppose that (W, V ) is a finite reflection group. The orbit space V /W 2 . The group W is generated by n2 reflections. In particular, if is homeomorphic to Rn1 × Rn0 W is non-trivial, then n2 = 0. Proof. This follows immediately from the theorems of Brown [11, pp. 20 and 24]. LOOP GROUPS AND TWISTED K-THEORY I 783 Appendix A. Groupoids We remind the reader that we are assuming throughout this paper that, unless otherwise specified, all spaces are locally contractible, paracompact, and completely regular. These assumptions imply the existence of partitions of unity [13] and locally contractible slices through actions of compact Lie groups [27, 28]. A.1. Definition and first properties A groupoid is a category in which all morphisms are isomorphisms. We consider groupoids in the category of topological spaces. Thus, a groupoid X = (X0 , X1 ) consists of a space X0 of objects, and a space X1 of morphisms, and map ‘identity map’ X0 → X1 , a pair of maps ‘domain’ and ‘range’ X1 → X0 , an associative composition law X1 ×X0 X1 → X1 , and an ‘inverse’ map X1 → X1 . Write Xn = X1 ×X0 . . . ×X0 X1 for the space of n-tuples of composable maps. Then the collection {Xn } is a simplicial space. The ith face map di : Xn −→ Xn−1 is given by ⎧ ⎪ ⎨(f2 , . . . , fn ) di (f1 , . . . , fn ) = (f1 , . . . , fi ◦ fi+1 , . . . , fn ) ⎪ ⎩ (f1 , . . . , fn−1 ) i = 0, 0 < i < n, i = n. Even though a groupoid is a special kind of simplicial space, we refer to the simplicial space as the nerve of X = (X0 , X1 ) and write X• . Finally, we let Xn × ∆n / ∼ |X| = n denote the geometric realization of X• . Example A.1 (cf. Segal [30]). Suppose that G is a topological group acting on a space X. Then the pair (G, X) forms a groupoid with space of objects X and in which a morphism from x to y is an element of g for which g · x = y. In this case, X0 = X and X1 = G × X. The composition law is given by the multiplication in G. We write X//G for this groupoid. Example A.2 (Segal [30]). Suppose that X is a space and U = {Ui } is a covering of X. The nerve of the covering U is the nerve of a groupoid. Indeed, let NU be the category whose objects are pairs (Ui , x) with Ui ∈ U and x ∈ Ui and in which a morphism from (Ui , x) to (Uj , y) is an element w ∈ Ui ×X Uj whose projection to Ui is x and whose projection to Uj is y. Then NU is a groupoid. If Ui and Uj are open subsets of X, then such a map exists if and only if x = y, in which case it is unique. Writing X0 = Ui , we have Xn = X0 ×X . . . ×X X0 , and the nerve of this groupoid is just the nerve of the covering U. Definition A.3. A map of groupoids F : X → Y is an equivalence if it is fully faithful and essentially surjective, that is, if every object y ∈ Y0 is isomorphic to one of the form F x, and if, for every a, b ∈ X0 , the map F : X(a, b) −→ Y (F a, F b) is a homeomorphism. An equivalence of ordinary categories automatically admits an inverse (up to natural isomorphism). Examples A.6 and A.8 below show that the same is not necessarily true of 784 D. S. FREED, M. J. HOPKINS AND C. TELEMAN an equivalence of topological categories, or groupoids. Requiring the existence of a globally defined inverse is, on the other hand, too restrictive. Definition A.4. A local equivalence X → Y is an equivalence of groupoids with the additional property that each y ∈ Y0 has a neighborhood U admitting a lift in the diagram / X0 X̃ G 0 Y1 range / Y0 domain U / Y0 in which the square is Cartesian. Remark A.5. The term local equivalence is derived from thinking of a groupoid X as defining a sheaf U → X(U ) on the category of topological spaces. An equivalence X → Y has a globally defined inverse if and only if, for every space U , the map X(U ) → Y (U ) is an equivalence. As one easily checks, a map X → Y is a local equivalence if and only if it is an equivalence on stalks. We will say that two groupoids X and Y are weakly equivalent if there is a diagram of local equivalences X ← Z → Y. Example A.6. If U is an open covering of a space X, then the map NU −→ X is a local equivalence. More generally, if U → V is a map of coverings of X, then NU −→ NV is a local equivalence. Example A.7. Given groupoids X and A, write X(A) for the groupoid of maps A → X. Then a map X → Y is a local equivalence if and only if, for spaces S, the map lim X(U ) −→ lim Y (U ) −→ −→ U U is an equivalence of groupoids, where U ranges over all coverings of S. Stated more succinctly, a map of groupoids is a local equivalence if and only if the corresponding map of presheaves of groupoids is a stalkwise equivalence. Example A.8. If P −→ X is a principal G-bundle over X, then P//G −→ X is a local equivalence. Example A.9. If H ⊂ G is a subgroup, then the map of groupoids pt//H −→ (G/H)//G is a local equivalence. LOOP GROUPS AND TWISTED K-THEORY I 785 The fiber product of functors P1 11 1 i 11 X Q j is the groupoid P ×X Q whose objects consist of p ∈ P, q ∈ Q, x ∈ X, and isomorphisms ip −→ x ← jq. The morphisms are the evident commutative diagrams. To give a functor S → P ×X Q is to give functors p S− → P, q S− → Q, x S− → X, and natural isomorphisms i ◦ p −→ x ← j ◦ q. The groupoid P ×X Q is usually called a fiber product of P and Q over X, even though strictly speaking it is a kind of homotopy fiber product and not the categorical fiber product. We also say that the morphism P ×X Q → Q is obtained from P → X by a change of base along j : Q → X. A natural transformation T : j1 → j2 gives a natural isomorphism between the groupoids obtained by the change of base along j1 and j2 . One easily checks that the class of local equivalences is stable under composition and change of base. Consequently, if P → X and Q → X are both local equivalences, so is P ×X Q → X. Using Example A.7, one easily checks that if two of three maps in a composition are local equivalences, so is the third. Definition A.10. The 2-category CovX is the category whose objects are local equivalences p : P → X and in which a 1-morphism from p1 : P1 → X to p2 : P2 → X consists of a functor F : P1 → P2 and a natural transformation T : p1 → p2 ◦ F making F / P2 P14 44 T 4 =⇒ p p1 44 2 X commute. A 2-morphism (F1 , T1 ) → (F2 , T2 ) is a natural transformation η : F1 → F2 for which T 2 = p2 η ◦ T 1 . We denote by CovX the 1-category quotient of CovX . The objects of CovX are those of CovX , and CovX (a, b) is the set of isomorphism classes in CovX (a, b). We see that CovX and CovX are not that diﬀerent from each other. Lemma A.11. For every a, b ∈ CovX , the category CovX (a, b) is a codiscrete groupoid: there is a unique morphism between any two objects. 786 D. S. FREED, M. J. HOPKINS AND C. TELEMAN Proof. Write a = (F1 , T1 ) and b = (F2 , T2 ) F1 ,F2 / P2 P1 6 66 T1 ,T2 6 p1 66=⇒ p2 X. A morphism (natural transformation) η ∈ CovX (a, b) associates to x ∈ P1 a map ηx : F1 x −→ F2 x, whose image p2 ηx : p2 F1 x −→ p2 F2 x is prescribed to fit into the diagram of isomorphisms: T1 x p2 F 1 x p1 x? ?? ??T2 x ?? / p2 F2 x. p2 ηx The map p2 ηx is therefore forced to be (T2 x) ◦ (T1 x)−1 , and so ηx is uniquely determined as P2 → X is an equivalence. Corollary A.12. The 1-category quotient CovX is a (co-)directed class. Proof. Suppose that pi : Pi → X, i = 1, 2, are two objects of CovX . The groupoid P12 = P1 ×X P2 comes equipped with maps P12 → P1 and P12 → P2 . If f, g : P → Q are two morphisms in CovX , there is, by Lemma A.11, a unique 2-morphism relating them, and so in fact f = g in CovX A.2. Further properties of groupoids We now turn to several constructions that are invariants of local equivalences. A groupoid is a presentation of a stack, and the invariants of local equivalences are in fact the invariants of the underlying stack. A.2.1. Point set topology of groupoids. The orbit space or coarse moduli space of a groupoid X is the space of isomorphism classes of objects, topologized as a quotient space of X0 . We denote the coarse moduli space of X by [X]. At this level of generality, the space [X] can be somewhat pathological, and without some further assumptions might not be in our class of locally contractible, paracompact, and completely regular spaces. When X = S//G, then [X] is the orbit space S/G, and in that case we revert to the more standard notation S/G. A local equivalence Y → X gives a homeomorphism [Y ] → [X]. For a subspace S ⊂ [X] we denote by XS the full subgroupoid of X consisting of objects in the isomorphism class of S. There is a one to one correspondence between full subgroupoids A ⊂ X containing every object in their X-isomorphism class and subspaces [A] of [X]. With this we can transport many notions from the point set topology of spaces to the context of groupoids. When S is closed or open, we say that XS is a closed or open subgroupoid of X, respectively. We can speak of the interior and closure of a full subgroupoid. By an open covering of a groupoid, we mean an open covering {Sα } of [X], in which case the collection {XSα } forms a covering of X by open subgroupoids. LOOP GROUPS AND TWISTED K-THEORY I 787 More generally, if f : S → [X] is a map, then we can form a groupoid XS with objects the pairs (s, x) ∈ S × X0 for which x is in the isomorphism class of f (s). A map (s, x) → (t, y) is just a map from x to y in X. Phrased diﬀerently, the groupoid XS is the groupoid whose nerve fits into a pullback square (XS )• −−−−→ X• ⏐ ⏐ ⏐ ⏐ S −−−−→ [X]. We say that XS is defined by pullback from the map S → [X]. With this we can transport many of the maneuvers of homotopy theory to the context of groupoids. For instance, if S = [X] × I, and f is the projection, then XS is the groupoid X × I. One can then form mapping cylinders and other similar constructions. For example, suppose that [X] is paracompact, and written as the union of two sets, S0 and S1 , whose interiors cover. Write Ui = XSi . The Ui are (full) subgroupoids whose interiors cover X. Let N denote the groupoid constructed from the map cyl(S0 ← S0 ∩ S1 → S1 ) −→ [X]. It is the double mapping cylinder of U0 ← U0 ∩ U1 → U1 . Following Segal [30], a partition of unity {φ0 , φ1 } subordinate to the covering {S0 , S1 } defines a map [X] −→ cyl(S0 ← S0 ∩ S1 → S1 ). The composite [X] cyl(S0 ← S0 ∩ S1 → S1 ) −→ [X] is the identity, and so we get functors X −→ N −→ X, whose composite is the identity. On the other hand, the composite cyl(S0 ← S0 ∩ S1 → S1 )[X] −→ cyl(S0 ← S0 ∩ S1 → S1 ) is homotopic to the identity, by a homotopy (the obvious linear homotopy) that preserves the map to [X] (it is a homotopy in the category of spaces over [X]). This homotopy then defines by pullback, a homotopy N × ∆1 −→ N from the composite N → X → N to the identity map of N , fixing the map to X. In this way, X becomes a strong deformation retract of N , and N is decomposed in a way especially well suited to constructing sequences of Mayer–Vietoris type. A groupoid X has proper diagonal if the map (domain,range) X1 −−−−−−−−−→ X0 × X0 (A.13) is proper, and [X] is Hausdorﬀ. If (A.13) is proper and X0 is Hausdorﬀ, then X is proper. If Y → X is a local equivalence, then X has proper diagonal if and only if Y has proper diagonal. A.2.2. Local and global quotients. A groupoid that is related by a chain of local equivalences to one of the form S//G, obtained from a group G acting on a space S, is said to be a global quotient. A local quotient groupoid is a groupoid X admitting a countable open cover {Uα } with the property that each XUα is weakly equivalent to a groupoid of the form 788 D. S. FREED, M. J. HOPKINS AND C. TELEMAN S//G with G a compact Lie group, and S a Hausdorﬀ space. If Y → X is a local equivalence, then Y is a local quotient groupoid if and only if X is, so the property of being a local quotient is intrinsic to the underlying stack. If X is a local quotient groupoid, then [X] is paracompact, locally contractible, and completely regular. If X is a local quotient groupoid with the property that there is at most one map between any two objects (that is, X1 → X0 × X0 is an inclusion), then the map X → [X] is a local equivalence, and so X is just a space. The following lemma is straightforward. Lemma A.14. Any groupoid constructed by pullback from a local quotient groupoid is a local quotient groupoid. In particular, any (full) subgroupoid of a local quotient groupoid is a local quotient groupoid, and the mapping cylinder of a map XS → X constructed by pullback along a map S → [X] to the orbit space of a local quotient groupoid is a local quotient groupoid. A.3. Fiber bundles over groupoids and descent In this section, we define the category of fiber bundles over a groupoid, and show that a local equivalence gives an equivalence of categories of fiber bundles (Proposition A.18). Thus, the category of fiber bundles over a groupoid is intrinsic to the underlying stack. A fiber bundle over a groupoid X = (X0 , X1 ) consists of a fiber bundle P on X0 together with identifications of certain pullbacks to Xn for various n. We introduce some convenient notation for describing these pulled-back bundles. Let us denote a typical point of Xn by fn f1 x0 −→ · · · −→ xn . Given a bundle P → X0 , we write Pxi for the pullback of P along the map Xn −→ X0 (x0 −→ · · · −→ xn ) −→ xi . Similarly, if P → X1 is given, then we write Pfi for the pullback of P along the map Xn −→ X1 fi fn f1 (x0 −→ · · · −→ xn ) −→ (xi−1 −→ xi ), and Pfi ◦fi+1 for the pullback along Xn −→ X1 f1 fi ◦fi+1 fn (x0 −→ · · · −→ xn ) −→ (xi−1 −−−−−→ xi+1 ), etc. For small values of n we use symbols like f (a − → b) ∈ X1 , f g (a − →b− → c) ∈ X2 to denote typical points. Definition A.15. A fiber bundle on X consists of a fiber bundle P → X0 , together with a bundle isomorphism tf : Pa −→ Pb (A.16) LOOP GROUPS AND TWISTED K-THEORY I 789 on X1 , for which tId = Id, and satisfying the cocycle condition that tf Pa A AA AA A tg◦f AA Pc / Pb } }} }}tg } ~} commutes on X2 . This way of describing a fiber bundle is convenient when thinking of X as a category. The association a → Pa is a functor from X to spaces, that is, continuous in an appropriate sense. There is a more succinct way of describing a fiber bundle on a groupoid. Namely, a fiber bundle on a groupoid X = (X0 , X1 ) is a groupoid P = (P0 , P1 ) and a functor P → X making Pi → Xi into fiber bundles, and all of the structure maps into maps of fiber bundles (that is, pullback squares). A functor F : Y → X between groupoids defines, in the evident way, a pullback functor F ∗ from the category of fiber bundles over X to the category of fiber bundles over Y . A natural transformation T : F → G defines a natural transformation T ∗ : F ∗ → G∗ . Example A.17. Let U = {Ui } be a covering of a space X. To give a fiber bundle over NU is to give a fiber bundle Pi on each Ui and the clutching (descent) data needed to assemble the Pi into a fiber bundle over X. Indeed, pullback along the map NU → X gives an equivalence between the category of fiber bundles over X and the category of fiber bundles over NU . The following generalization of Example A.17 will be referred to as descent for fiber bundles over groupoids. Proposition A.18. Suppose that F : X → Y is a local equivalence. Then the pullback functor F ∗ : {Fiber bundles on Y } −→ {Fiber bundles on X} is an equivalence of categories. Proof. Suppose that P is a fiber bundle over Y , which we think of as a functor from Y to the category of topological spaces. As Y → X is an equivalence of categories, the functor F ∗ has a left adjoint F∗ , given by F∗ P (x) = lim P, −→ Y /x where Y /x is the category of objects in y ∈ Y equipped with a morphism F y → x. As Y → X is an equivalence of groupoids, there is a unique map between any two objects of Y /x, and so F∗ P (x) is isomorphic to Py for any y ∈ Y /x. For each x ∈ X choose a neighborhood x ∈ U ⊂ X0 , a map t : U → Y0 , and a family of morphisms U → X1 connecting F ◦ t to the inclusion U → X0 . We topologize F∗ P x x∈X by requiring that the canonical map t∗ P −→ F∗ P |U 790 D. S. FREED, M. J. HOPKINS AND C. TELEMAN be a homeomorphism. This gives F∗ P the structure of a fiber bundle over X0 . Naturality provides F∗ P with the additional structure required to make it into a fiber bundle over X. One easily checks that the pair (F∗ , F ∗ ) is an adjoint equivalence of the category of fiber bundles over X with the category of fiber bundles over Y . For a fiber bundle p : P → X, write Γ(P ) for the space of sections Γ(P ) = Γ(X; P ) = {s : X −→ P | p ◦ s = IdX }, topologized as a subspace of X0P0 × X1P1 . If f : Y → X is a local equivalence, and P → X is a fiber bundle, then the evident map Γ(X; P ) −→ Γ(Y ; f ∗ P ) is a homeomorphism. If P is a pointed fiber bundle, with s : X → P as a basepoint, and A ⊂ X is a (full) subgroupoid, write Γ(X, A; P ) for the space of section x of P for which x|A = s. Now suppose that P → Q is a map of fiber bundles over X, and {Uα } is a covering of X by open subgroupoids. Write Pα → Uα for the restriction of P to Uα , and Pα1 ,...,αn for the restriction of P to Uα1 ∩ . . . ∩ Uαn , and similarly for Q. Proposition A.19. If, for each non-empty finite collection {α1 . . . αn }, the map Γ(Pα1 ,...,αn ) −→ Γ(Qα1 ,...,αn ) is a weak homotopy equivalence, then so is Γ(P ) −→ Γ(Q). Proof. This is a straightforward application of the techniques of Segal [30]. Let us first consider the case in which X is covered by just two open subgroupoids U and V . We form the ‘double mapping cylinder’ C = U ∐ U ∩ V × [0, 1] ∐ V / ∼, and consider the functor g : C → X. A choice of partition unity on [X] subordinate to the covering {[U ], [V ]} gives a section of g making Γ(X; P ) → Γ(X; Q) a retract of Γ(C; g ∗ P ) → Γ(C; g ∗ Q). It therefore suﬃces, in this case, to show that Γ(C; g ∗ P ) → Γ(C; g ∗ Q) is a weak equivalence. But Γ(C; g ∗ P ) fits into a homotopy pullback square Γ(C; g ∗ P ) −−−−→ ⏐ ⏐ Γ(U ; P ) ⏐ ⏐ Γ(V ; g ∗ P ) −−−−→ Γ(U ∩ V ; P ) and similarly for Γ(C; g ∗ Q) (to simplify the diagram, we have not distinguished in notation between P and its restriction to U , V , and U ∩ V ). The result then follows from the long exact (Mayer–Vietoris) sequence of homotopy groups. An easy induction then gives that the map on spaces of sections of P → Q restricted to any finite union Uα1 ∪ . . . ∪ Uαn is a weak equivalence. For the case where the collection {Uα } is countable (to which we are reduced when [X] is second countable), order the Uα and write Vn = U1 ∪ . . . ∪ Un . LOOP GROUPS AND TWISTED K-THEORY I 791 Form the ‘infinite mapping cylinder’ C= Vi × [i, i + 1]/ ∼, and consider g : C → X. As before, a partition of unity on [X] subordinate to the covering [V ]i defines a section of [C] → [X] and hence of C → X, making Γ(X; P ) → Γ(X; Q) a retract of Γ(C; g ∗ P ) −→ Γ(C; g ∗ Q). (A.20) It therefore suﬃces to show that (A.20) is a weak equivalence. But (A.20) is the homotopy inverse limit of the tower Γ(Vn ; P ) −→ Γ(Vn ; Q), (A.21) and so its homotopy groups (or sets, in the case of π0 ) are related to those of (A.21) by a Milnor sequence, and the result follows. Alternatively, following Segal [30], one can avoid the countability hypothesis and the induction by using for C the nerve of the covering {Uα } and the homotopy spectral sequences of Bousfield and Kan [9] and Bousfield [8]. A.4. Hilbert bundles A Hilbert bundle over a groupoid X is a fiber bundle whose fibers have the structure of a separable Z/2-graded Hilbert space. Remark A.22. There is a tricky issue in the point set topology here. In defining Hilbert bundles as special kinds of fiber bundles, we are implicitly using the compact-open topology on U (H) and not the norm topology. This causes trouble when we form the associated bundle of Fredholm operators (§ A.5), as we cannot then use the norm topology on the space of Fredholm operators. This issue is raised and resolved by Atiyah and Segal [1], and we are following their discussion in this paper. Definition A.23. A Hilbert bundle H is universal if, for each Hilbert bundle V , there exists a unitary embedding V ⊂ H. The bundle H is said to have the absorption property if, for any V , there is a unitary equivalence H ⊕ V ≈ H. Lemma A.24. A universal Hilbert bundle has the absorption property. Proof. First note that if H is universal, then H ⊗ ℓ2 ≈ H ⊕ H ⊕ . . . has the absorption property. Indeed, given V , write H = W ⊕ V , and use the ‘Eilenberg swindle’ V ⊕ H ⊕ H ⊕ . . . ≈ V ⊕ (W ⊕ V ) ⊕ (W ⊕ V ) ⊕ . . . ≈ (V ⊕ W ) ⊕ (V ⊕ W ) ⊕ (V ⊕ W ) . . . ≈ H ⊕ H ⊕ H ⊕ . . . . We can then write H ≈ H ⊗ ℓ2 ⊕ V ≈ H ⊗ ℓ2 , to conclude that H is absorbing. Definition A.25. A Hilbert bundle H over X is locally universal if, for every open subgroupoid XU ⊂ X, the restriction of H to XU is universal. 792 D. S. FREED, M. J. HOPKINS AND C. TELEMAN Lemma A.26. If H and H ′ are universal Hilbert bundles on X, then there is a unitary equivalence H ≈ H ′ . Remark A.27. As the category of Hilbert bundles on X depends only on X up to local equivalence, if f : Y → X is a local equivalence and H is a (locally) universal Hilbert bundle on X, then f ∗ H is a (locally) universal Hilbert bundle on Y . Similarly, if H ′ is a (locally) universal Hilbert bundle on Y , there is a (locally) universal Hilbert bundle H on X, and a unitary equivalence f ∗ H ≈ H ′ . We now show that the existence of a locally universal Hilbert bundle is a local issue. Lemma A.28. Suppose that X is a groupoid, and that {Ui | i = 1 . . . ∞} is a covering of X by open subgroupoids. If H is a Hilbert bundle with the property that Hi = H|Ui is universal, then H ⊗ ℓ2 is universal. Proof. Let V be a Hilbert space bundle on X. Choose a partition of unity {λi } on [X] subordinate to the open cover [U ]i . For each i choose an embedding ri : V |Ui ֒→ Hi . The map V −→ H ⊕ H ⊕ . . . = H ⊗ ℓ2 with components λi ri is then an embedding of V in H ⊗ ℓ2 . Corollary A.29. is locally universal. In the situation of Lemma A.28, if H|Ui is locally universal, then H ⊗ ℓ2 Lemma A.30. Suppose that X is a groupoid, and that {Ui | i = 1 . . . ∞} is a covering of X by open subgroupoids. If Hi is a locally universal Hilbert bundle on {Ui }, then there exists a Hilbert bundle H on X with H|Ui ≈ Hi . Proof. This is an easy induction, using Lemma A.26. Corollary A.31. Suppose that X is a groupoid, and that {Ui | i = 1 . . . ∞} is a covering of X by open subgroupoids. If Hi is a locally universal Hilbert bundle on {Ui }, then there exists a locally universal Hilbert bundle H on X. LOOP GROUPS AND TWISTED K-THEORY I 793 Lemma A.32. Suppose that X = S//G is a global quotient of a space S by a compact Lie group G. Then the equivariant Hilbert bundle S × L2 (G) ⊗ C1 ⊗ ℓ2 is a locally universal Hilbert bundle on X. Here C1 is the complex Cliﬀord algebra on one (odd) generator. It is there simply to make the odd component of our Hilbert bundle large enough. Proof. As the open (full) subgroupoids of S//G correspond to the G-stable open subsets of S, it suﬃces to show that L2 (G) ⊗ C1 ⊗ ℓ2 is universal. Let V be any Hilbert bundle on S//G, that is, an equivariant Hilbert bundle on S. By Kuiper’s theorem, V is trivial as a (non-equivariant) Hilbert bundle on S. Choose an orthonormal homogeneous basis {ei }, and let ei = ei , − : V → C1 be the corresponding projection operator. By the universal property of L2 (G), each ei lifts uniquely to an equivariant map V −→ L2 (G) ⊗ C1 . Taking the sum of these maps gives an embedding of V in L2 (G) ⊗ C1 ⊗ ℓ2 . Combining Lemma A.32 with Lemma A.30 gives the following. Corollary A.33. Hilbert bundle on X. If X is a local quotient groupoid, then there exists a locally universal Corollary A.34. Suppose that X is a local quotient groupoid, and f : Y → X is a map constructed by pullback from [Y ] → [X]. If H is locally universal on X, then f ∗ H is locally universal on Y . Proof. This is an easy consequence of Lemma A.32 and Corollary A.29. Corollary A.34 is used in the proof of excision in twisted K-theory, and is the reason for our restriction to the class of local quotient groupoids. The following result is well known, but we could not quite find a reference. Our proof is taken from [25, Theorem 1.5], which gives the analogous result for equivariant embeddings of countably infinite-dimensional inner product spaces (and not Hilbert spaces). Of course, the result also follows from Kuiper’s theorem, as the space of embeddings is U (H ⊗ ℓ2 )/U (V ⊥ ). But the contractibility of the space of embeddings is more elementary than the contractibility of the unitary group, so it seemed better to have a proof that does not make use of Kuiper’s theorem. Lemma A.35. Suppose that V and H are Hilbert bundles over a groupoid X, and that there is a unitary embedding V ⊂ H ⊗ ℓ2 . Then the space of embeddings V ֒→ H ⊗ ℓ2 is contractible. Proof. Let f : V ⊂ H ⊗ ℓ2 be a fixed embedding, and write H ⊗ ℓ2 = H ⊕ H ⊕ . . . f = (f1 , f2 , . . .). The contraction is a concatenation of two homotopies. The first takes an embedding g = (g1 , g2 , . . .) 794 D. S. FREED, M. J. HOPKINS AND C. TELEMAN to (0, g1 , 0, g2 , . . .), and then the second is cos(πt/2) · (0, g1 , 0, g2 , . . .) + sin(πt/2) · (f1 , 0, f2 , 0 . . .). It is easier to write down the reverse of the first homotopy. It, in turn, is the concatenation of an infinite sequence of 2-dimensional rotations (0, g1 , 0, g2 , 0, g3 , . . .) −→ (g1 , 0, 0, g2 , 0, g3 , . . .) (g1 , 0, 0, g2 , 0, g3 , . . .) −→ (g1 , g2 , 0, 0, 0, g3 , . . .) 0 t 1/2, 1/2 t 3/4, ··· . One must check that the limit as t → 1 is (g1 , g2 , . . .), and that the path is continuous in the compact-open topology. Both facts are easy and left to the reader. Lemma A.36. Suppose that H is a locally universal Hilbert bundle over a local quotient groupoid X. Then the space of sections Γ(X; U (H)) of the associated bundle of unitary groups is weakly contractible. Proof. This follows easily from Kuiper’s theorem (see [1, Appendix 3]), and Proposition A.19. We conclude this section with a useful criterion for a local quotient stack to be equivalent to a global quotient by a compact Lie group. Proposition A.37. The (locally) universal Hilbert bundle over a compact, local quotient groupoid, splits into a sum of finite-dimensional bundles if and only if the groupoid is equivalent (in the sense of local equivalence) to one of the form X//G, with X compact, and G a compact group. Remark A.38. (i) This implies right away that the extensions of groupoids corresponding to twistings whose invariant in H 3 has infinite order are not quotient stacks: indeed, any 1-eigenbundle for the central T is a projective bundle representative for the twisting, and hence must be infinite-dimensional. (ii) There are simple obstructions to a groupoid being related by a chain of local equivalences to a global quotient by a compact group; for instance, such quotients admit continuous choices of Ad-invariant metrics on the Lie algebra stabilizers that are integral on the coweight lattices. The stack obtained by gluing the boundaries of B(T × T) × [0, 1] via the shearing automorphism of T × T does not carry such metrics. The same is true for the quotient stack A//T ⋉ LT , where T is a torus and A is the space of connections on the trivial T -bundle over the circle. In this case, the stack is fibered over T in B(T × T )-stacks with the tautological shearing holonomies. Hence, the larger stacks A//T ⋉ LG, where G is a compact Lie group and A is the space of connections on the trivial G-bundle over the circle, are not global quotients either. (iii) The result is curiously similar to Totaro’s characterization of smooth quotient stacks as the Artin stacks where coherent sheaves admit resolutions by vector bundles [32]. LOOP GROUPS AND TWISTED K-THEORY I 795 Proof. The ‘if’ part follows from our construction of the universal Hilbert bundle. For the ‘only if part,’ first note that a local quotient groupoid X is weakly equivalent to a groupoid of the form S//G, if and only if there is a principal G-bundle P → X with the property that there is at most one map between any two objects in P . In that case P is equivalent to [P ] (§ A.2.2), and X is weakly equivalent to [P ]//G. This latter condition holds if and only if, for each x ∈ X0 , the map Aut(x) → G associated to P is a monomorphism. Suppose that H is the (locally) universal Hilbert bundle on X, and that we can find an orthogonal decomposition H = ⊕Hα with each Hα of dimension nα < ∞. Take P to be the product of the frame bundles of the Hα , and G to be the product of the unitary groups U (Hα ) ≈ U (nα ). To check that Aut(x) → G is a monomorphism, in this case it suﬃces to check locally near x. The assertion is thus reduced to the case of a global quotient by a compact group, where it follows from our explicit construction. A.5. Fredholm operators and K-theory Our model for twisted K-theory will be based on the Atiyah-Singer spaces of skew-adjoint Fredholm operators [3]. In this section we recall the definition of these spaces, and the modifications made to them by Atiyah and Segal [1]. Let H be a Z/2-graded Hilbert bundle over a groupoid X. We wish to associate to H a bundle of Fredholm operators over X. As mentioned in Remark A.22, we cannot use the norm topology on the space of Fredholm operators here. We are using the compact-open topology on U (H). Were the space of Fredholm operators to be given the norm topology, the conjugation action of U (H) would fail to be continuous. Following Atiyah-Segal [1, Definition 3.2], we make the following definition. Definition A.39 [1]. The space Fred(0) (H) is the space of odd skew-adjoint Fredholm operators A, for which A2 + 1 is compact, topologized as a subspace of B(H) × K(H), with B(H) given the compact-open topology and K(H) the norm topology. 2 + q(z) = 0) denote the complex Cliﬀord algebra associated to the Let Cn = T {Cn }/(z quadratic form q(z) = zi2 . We write ǫi for the ith standard basis element of Cn , regarded as an element of Cn . Following Atiyah and Singer [3], for an operator A ∈ Fred(0) (Cn ⊗ H), with n odd, let ǫ1 . . . ǫn A n ≡ −1 mod 4, w(A) = −1 i ǫ1 . . . ǫn A n ≡ 1 mod 4. The operator A is then even and self-adjoint. Definition A.40 [1]. The space Fred(n) (H) is the subspace of Fred(0) (Cn ⊗ H) consisting of odd operators A, which commute (in the graded sense) with the action of Cn , and for which the essential spectrum of w(A), in case n is odd, contains both positive and negative eigenvalues. Atiyah and Segal [1] show that the ‘identity’ map from Fred(n) (ℓ2 ) in the norm topology to Fred(n) (ℓ2 ) in the above topology is a weak homotopy equivalence. It then follows from Atiyah and Singer [3, Theorem B(k)] that the map Fred(n) (ℓ2 ) −→ Ω′ Fred(n−1) (C1 ⊗ ℓ2 ) A −→ ǫk cos(πt) + A sin(πt) 796 D. S. FREED, M. J. HOPKINS AND C. TELEMAN is a weak homotopy equivalence, where we are making the evident identification Cn ≈ Cn−1 ⊗ C1 , and Ω′ denotes the space of paths from ǫk to −ǫk . Combining these leads to the following simple consequence. Proposition A.41. If X is a local quotient groupoid, and H a Z/2-graded, locally universal Hilbert bundle over X, then the map Γ(X; Fred(n+1) (H)) −→ Ω′ Γ(X; Fred(n) (H)) is a weak homotopy equivalence. Proof. By Proposition A.19, the question is local in X, so we may assume X = S//G, with G a compact Lie group. By our assumption on the existence of locally contractible slices, we may reduce to the case in which S is equivariantly contractible to a fixed point s ∈ S. Finally, as the question is homotopy invariant in X, we reduce to the case S = pt. We are therefore reduced to showing that if H is a universal G-Hilbert space, then the map of G-fixed points Fred(n) (H)G −→ Ω′ Fred(n+1) (H)G (A.42) is a weak equivalence. For each irreducible representation V of G, let HV denote the V -isotypical component of H. Then (A.42) is the product over the irreducible representations V of G, of Fred(n) (HV )G −→ Ω′ Fred(n+1) (HV )G . (A.43) As H is universal, the Hilbert space HV is isomorphic to V ⊗ ℓ2 , and the map Fred(n) (ℓ2 ) −→ Fred(n) (V ⊗ ℓ2 )G T −→ Id ⊗ T is a homeomorphism. The proposition is thus reduced to the result of Atiyah and Singer quoted above. We now assemble the spaces Γ(X; Fred(n) (H)) into a spectrum in the sense of algebraic topology. To do this requires specifying basepoints in Fred(n) (H). As our operators are odd, we cannot take the identity map as a basepoint and a diﬀerent choice must be made. There are some technical diﬃculties that arise in trying to specify consistent choices and we have just chosen to be unspecific on this point. The diﬃculties do not amount to a serious problem as any invertible operator can be taken as a basepoint, and the space of invertible operators is contractible. The reader is referred to [23] for further discussion. We use the symbol ǫ to refer to a chosen basepoint in Fred(n) (H), as well as to the constant section with value ǫ in Γ(X; Fred(n) (H)). Proposition A.41 gives a homotopy equivalence Γ(X; Fred(n+1) (W )) −→ ΩΓ(X; Fred(n) (W )). (A.44) As described in [3], the fact that C2 is a matrix algebra gives a homeomorphism Γ(X Fred(m) (W )) ≈ Γ(X Fred(m+2) (W )). We define the spectrum K(X) by taking Γ(X, Fred(0) (W )) K(X)n = Γ(X, Fred(1) (W )) (A.45) n even, n odd, with structure map K(X)n → ΩK(X)n+1 , to be the map (A.44) when n is odd, and the composite of (A.45) and (A.44) when n is even. The group K n (X) is then defined by K n (X) = π0 K(X)n ≈ πk K(X)n+k . LOOP GROUPS AND TWISTED K-THEORY I 797 Because H is locally universal, when X = S//G, we have K n (X) ≈ [S, Fred(n) (L2 (G) ⊗ ℓ2 )]G . As remarked in § 3.5.4, Fred(n) (L2 (G) ⊗ ℓ2 ) is a classifying space for equivariant K-theory, this latter group can be identified with K n (G)(S). Acknowledgements. This paper has been a long time in preparation and we have benefited from discussion with many people. The authors would like to thank Sir Michael Atiyah and Graeme Segal for making available early drafts of Atiyah and Segal [1, 2]. As will be evident to the reader, our approach to twisted K-theory relies heavily on their ideas. We would like to thank Is Singer for many useful conversations. We would also like to thank Ulrich Bunke and Thomas Schick for their careful study and comments on this work. The report of their seminar appears in [12]. References 1. M. Atiyah and G. Segal, ‘Twisted K-theory’, Ukr. Mat. Visn. 1 (2004) 287–330. MR MR2172633 2. M. Atiyah and G. Segal, ‘Twisted K-theory and cohomology’, Inspired by S. S. Chern, Nankai Tracts Mathematics 11 (World Scientific Publishing, Hackensack, 2006) 5–43. 3. M. F. Atiyah and I. M. Singer, ‘Index theory for skew-adjoint Fredholm operators’, Publ. Math. Inst. Hautes Études Sci. (1969) 5–26. MR 44 #2257 4. R. Bezrukavnikov, M. Finkelberg and I. 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Hopkins Department of Mathematics Harvard University 1 Oxford Street Cambridge, MA 02138 USA dafr@math.utexas.edu mjh@math.harvard.edu Constantin Teleman Department of Mathematics University of California 970 Evans Hall Berkeley, CA 94720-3840 USA teleman@math.berkeley.edu

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