Loop groups and twisted K-theory I
aa r X i v : . [ m a t h . A T ] N ov LOOP GROUPS AND TWISTED K -THEORY I DANIEL S. FREED, MICHAEL J. HOPKINS, AND CONSTANTIN TELEMAN
Contents
Introduction 11. Twisted K -theory by example 62. Twistings of K -theory 132.1. Graded T -bundles 132.2. Graded central extensions 142.3. Twistings 192.4. Examples of twistings 223. Twisted K-groups 233.1. Axioms 233.2. Twisted Hilbert spaces 253.3. Universal Twisted Hilbert Bundles 273.4. Definition of twisted K -groups 273.5. Verification of the axioms 293.6. The Thom isomorphism, pushforward, and the Pontryagin product 313.7. The fundamental spectral sequence 334. Computation of K τG ( G ) 344.1. Notation and assumptions 344.2. The main computation 354.3. The action of W eaff on t K -theory 61References 63 Introduction
Equivariant K -theory focuses a remarkable range of perspectives on the studyof compact Lie groups. One finds tools from topology, analysis, and representationtheory brought together in describing the equivariant K -groups of spaces and themaps between them. In the process all three points of view are illuminated. Our aimin this series of papers [19, 20] is to begin the development of similar relationshipswhen a compact Lie group G is replaced by LG , the infinite dimensional group ofsmooth maps from the circle to G . There are several features special to the representation theory of loop groups.First of all, we will focus only on the representations of LG which have “positiveenergy.” This means that the representation space V admits an action of therotation group of the circle which is (projectively) compatible with the action of LG ,and for which there are no vectors v on which rotation by θ acts by multiplicationby e inθ with n <
0. It turns out that most positive energy representations areprojective, and so V must be regarded as a representation of a central extension LG τ of LG by U (1). The topological class of this central extension is known asthe level . One thing a topological companion to the representation theory of loopgroups must take into account is the level.Next there is the fusion product. Write R τ ( LG ) for the group completion of themonoid of positive energy representations of LG at level τ . In [34], Erik Verlindeintroduced a multiplication on R τ ( LG ) ⊗ C making it into a commutative ring (infact a Frobenius algebra). This multiplication is called fusion , and R τ ( LG ) ⊗ C ,equipped with the fusion product is known as the Verlinde algebra . The fusionproduct also makes R τ ( LG ) into a ring which we will call the Verlinde ring . Agood topological description of R τ ( LG ) should account for the fusion product in anatural way.The positive energy representations of loop groups turn out to be completely re-ducible, and somewhat surprisingly, there are only finitely many irreducible positiveenergy representations at a fixed level. Moreover, an irreducible positive energy rep-resentation is determined by its lowest non-trivial energy eigenspace, V ( n ), whichis an irreducible (projective) representation of G . Thus the positive energy rep-resentations of LG correspond to a subset of the representations of the compactgroup G . This suggests that G -equivariant K -theory might somehow play a rolein describing the representations of LG . In fact this is the case. Here is our maintheorem. Theorem 1.
Let G be a connected compact Lie group and τ a level for the loopgroup. The Grothendieck group R τ ( LG ) at level τ is isomorphic to a twistedform K ζ ( τ ) G ( 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 thePontryagin product. The twisting ζ ( τ ) is given in terms of the level ζ ( τ ) = g + ˇ h + τ, where ˇ h is the “dual Coxeter” twisting. Several aspects of this theorem require clarification. The main new element isthe “twisted form” of K -theory. Twisted K -theory was introduced by Donovanand Karoubi [14] in connection with the Thom isomorphism, and generalized andfurther developed by Rosenberg [29]. Interest in twisted K -theory was rekindled byits appearance in the late 1990’s [26, 35] in string theory. Our results came aboutin 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 twisted K -theory.The twisted forms of G -equivariant K -theory are classified by the nerve of thecategory of invertible modules over the equivariant K -theory spectrum K G . Whatcomes up in geometry though, is only a small subspace, and throughout this pa-per the term “twisting” will refer twistings in this restricted, more geometric class. OOP GROUPS AND TWISTED K -THEORY I 3 These geometric twistings of K G -theory on a G -space X are classified up to iso-morphism by the set(2) H G ( X ; Z / × H G ( X ; Z / × H G ( X ; Z ) . The component in H corresponds to the “degree” of a K -class, and the fact thatthe coefficients are the integers modulo 2 is a reflection of Bott periodicity. In thissense “twistings” refine the notion of “degree,” though when considering twistingsit is important to remember more than just the isomorphism class.The tensor product of K G -modules makes these spaces of twistings into infiniteloop spaces and provides a commutative group structure on the sets of isomorphismclasses. The group structure on (2) is the product of H G ( X ; Z /
2) with with the ex-tension of H G ( X ; Z /
2) by H G ( X ; Z ) with cocycle β ( x ∪ y ), where β is the Bocksteinhomomorphism.A twisting is a form of equivariant K -theory on a space. A level for the loopgroup, on the other hand, corresponds to a central extension of LG . One way ofrelating these two structures is via the classification (2). A central extension of LG has a topological invariant H G ( G ), and so give rise to a twisting, up to isomorphism.When the group G is simple and simply connected this invariant determines thecentral extension up to isomorphism, the group H G ( G ) vanishes, and there is acanonical isomorphism H G ( G ) ≈ Z . In this case an integer can be used to specifyboth a level and a twisting. There is a more refined version of this correspondencedirectly relating twistings to central extension, and the approach to twistings wetake 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 vectorbundle V over X the family of K -modules K ¯ V x , where ¯ V x is the one point com-pactification of the fiber of V over x ∈ X , and for a space S , K S is the K -modulewith π K S = K ( S ). We denote the twisting associated to V by τ V , though whenno confusion is likely to arise we will just use the symbol V . The invariants of τ V in (2) are dim V , w ( V ) and βw ( V ). These twistings are described by Donovanand Karoubi in [14] from the point of view of Clifford algebras.There is also a homomorphism from KO − G ( X ) to the group of twistings ofequivariant K -theory on X . In topological terms it corresponds to the map fromthe stable orthogonal group O to its third Postnikov section O h , . . . , i . It sendsan element of KO − to the twisting whose components are ( σw , σw , x ), where σ : H ∗ ( BO ) → H ∗− ( O ) is the cohomology suspension, and x ∈ H ( O ; Z ) isthe unique element, twice which is the cohomology suspension of p . In terms ofoperator algebras this homomorphism sends a skew-adjoint Fredholm operator toits graded Pfaffian gerbe. We will call this map pfaff .We now describe two natural twistings on G which are equivariant for the adjointaction. The first comes from the adjoint representation of G regarded as an equi-variant vector bundle over a point, and pulled back to G . We’ll write this twistingas g . For the other, first note that the equivariant cohomology group H ∗ G ( G ) isjust H ∗ ( LBG ) . The vector bundle associated to the adjoint representation gives aclass ad ∈ KO ( BG ) which we can transgress to KO − ( LBG ), and then map totwistings by the map pfaff . We’ll call this twisting ˇ h . When G is simple and simplyconnected, the integer corresponding to ˇ h is the dual Coxeter number, and g is justa degree shift. With these definitions, the formula ζ ( τ ) = g + ˇ h + τ DANIEL S. FREED, MICHAEL J. HOPKINS, AND CONSTANTIN TELEMAN in the statement of Theorem 1 should be clear. The twistings g and ˇ h are thosejust described, and τ is the twisting corresponding to the level.Theorem 1 provides a topological description of the group R τ ( LG ) and its fusionproduct when it exists. But it also gives more. The twisted K -group K ζ ( τ ) G ( G ) isdefined for any compact Lie group G , and it makes sense for any level τ . This pointsthe way to a formulation of an analogue of the group R τ ( LG ) for any compactLie group G (even one which is finite). In Parts II and III we take up thesegeneralizations and show that the assertions of Theorem 1 remain true.One thing that emerges from our topological considerations is the need to con-sider Z / -graded central extensions of loop groups. Such extensions are necessarywhen 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 notorientable the dual Coxeter twist makes a non-trivial contribution to H G ( G ; Z / K -group occurs. Inthe 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 generalizationof degree. The “Verlinde ring” in this case is comprised of an even K -group on onecomponent and an odd K -group on another. Such inhomogeneous compositionsare not typically considered when discussing ordinary K -groups.The fusion product on R τ ( LG ) has been defined for simple and simply connected G , and in a few further special cases. The Pontryagin product on K ζ ( τ ) G ( G ) is de-fined exactly when τ is primitive in the sense that its pullback along the multiplica-tion map of G is isomorphic to the sum of its pullbacks along the two projections.This explains, for example, why a fusion product on R τ ( LSO ( n )) exists only athalf of the levels. Using the Pontryagin product we are able to define a fusionproduct on R τ ( LG ) for any G at any primitive level τ . We do not, however, give aconstruction of this product in terms of representation theory.When the fusion product is defined on R τ ( LG ) it is part of a much more elaboratestructure. For one thing, there is a trace map R τ ( LG ) → Z making R τ ( LG ) into aFrobenius algebra. Using twisted K -theory we construct this trace map for generalcompact Lie groups at primitive levels τ which are non-degenerate in the sense thatthe image of τ in H T ( T ; R ) ≈ H ( T ; R ) ⊗ H ( T ; R )is a non-degenerate bilinear form. Again, in the cases when the fusion product hasbeen defined, there are operations on R τ ( LG ) coming from the moduli spaces ofRiemann surfaces with boundary, making R τ ( LG ) part of is often called a topolog-ical conformal field theory . Using topological methods, we are able to construct atopological conformal field theory for any compact Lie group G , at levels τ whichare transgressed from (generalized) cohomology classes on BG and which are non-degenerate. Some of this work appears in [21, 18].Another impact of Theorem 1 is that it brings the computational techniquesof algebraic topology to bear on the representations of loop groups. One veryinteresting approach, for connected G , is to use the Rothenberg-Steenrod spectralsequence relating the equivariant K -theory of Ω G to that of G . In this case onegets a spectral sequence(3) Tor K G ∗ (Ω G ) ( R ( G ) , R ( G )) = ⇒ K Gτ + ∗ ( G ) , OOP GROUPS AND TWISTED K -THEORY I 5 relating the untwisted equivariant K -homology of Ω G , and the representationring of G to the Verlinde algebra. The ring K G ∗ (Ω G ) can be computed usingthe techniques of Bott [5] and Bott-Samelson [6] and has also been described byBezrukavnikov, Finkelberg and Mirkovi´c [4]. The K -groups in the E -term areuntwisted. The twisting appears in the way that the representation ring R ( G ) ismade into an algebra over K G (Ω G ). The equivariant geometry of Ω G has beenextensively studied in connection with the representation theory of LG , and thespectral sequence (3) seems to express yet another relationship. We do not knowof a representation theoretic construction of (3). An analogue of the spectral se-quence (3) has been used by Chris 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 ∆ − K G ∗ (Ω G ) = ∆ − Z [Λ × Π] , where ∆ is the square of the Weyl denominator, Π = π T is the co-weight lattice,and Λ is the weight lattice. When the level τ is non-degenerate there are no higherTor groups, and the spectral sequence degenerates to an isomorphism∆ − K Gτ + ∗ ( G ) ≈ ∆ − R ( G ) /I τ where I τ is the ideal of representations whose characters vanish on certain conjugacyclasses. The main computation of this paper asserts that such an isomorphismholds without inverting ∆ when G is connected, and π G is torsion free. Thedistinguished conjugacy classes are known as Verlinde conjugacy classes , and theideal I τ as the Verlinde ideal . In [21] the ring K ζ ( τ ) G ( G ) ⊗ C is computed using afixed 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 compute the groups K ζG ( G ) for connected G with torsion free fun-damental group, at non-degenerate levels ζ . Our main result is Theorem 4.27. InPart II we introduce a certain family of Dirac operators and our generalization ofR τ ( LG ) to arbitrary compact Lie groups. We construct a map from R τ ( LG ) to K ζ ( τ ) G ( G ) and show that it is an isomorphism when G is connected with torsionfree fundamental group. In Part III we show that our map is an isomorphism forgeneral 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. For one, when workingwith twistings it is important to remember the morphisms between them, and notjust the isomorphism classes. The twistings on a space form a category and spellingout the behavior of this category as the space varies gets a little elaborate. Theother thing has to do with the kind of G -spaces we use. We need to define twistingson G -equivariant K -theory in such a way as to make clear what happens as thegroup G changes, and for the constructions in Part II we need to make the relation-ship between twistings and (graded) central extensions as transparent as possible.We work in this paper with groupoids and define twisted equivariant K -theory forgroupoids. Weakly equivalent groupoids (see Appendix A) have equivalent cate-gories 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.” Acentral extension of G by U (1) defines a twisting of K -theory for X//G . If G is acompact connected Lie group acting on itself by conjugation, and P G denotes the
DANIEL S. FREED, MICHAEL J. HOPKINS, AND CONSTANTIN TELEMAN 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 will define a twisting of a groupoid X to consist of a local equivalence P → X and a graded central extension ˜ P of P .While most of the results we prove reduce, ultimately, to ordinary results aboutcompact Lie groups acting on spaces, not all do. In Part III it becomes necessaryto work with groupoids which are not equivalent to a compact Lie group acting ona space.Nitu Kitchloo [24] has pointed out that the space P G is the universal LG spacefor proper actions. Using this he has described a generalization of our computationto other Kac-Moody groups.At the time we began this work, the paper of Atiyah and Segal [3] was in prepa-ration, and we benefited a great deal from early drafts. Since that time severalother approaches to twisted K -theory have appeared. In addition to [3] we referthe reader to [10, 33]. We have chosen to use “graded central extensions” becauseof the close connection with loop groups and the constructions we wish to makein Part II. Of course our results can be presented from any of the points of viewmentioned above, and the choice of which is a matter of personal preference.We have attempted to organize this paper so that the issues of implementationare independent of the issues of computation. Section 1 is a kind of field guide totwisted K -theory. We describe a series of examples intended to give the reader aworking 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 axiom-atize the theory of twisted K -groups in order to facilitate comparison with othermodels. Our main computation appears in § K -theory relies heavily on their ideas. Wewould like to thank Is Singer for many useful conversations. We would also like tothank Ulrich Bunke and Thomas Schick for their careful study and comments onthis work. The report of their seminar appears in [12].We assume throughout this paper that all spaces are locally contractible, para-compact and completely regular. These assumptions implies the existence of par-titions of unity [13] and locally contractible slices through actions of compact Liegroups [27, 28]. 1. Twisted K -theory by example The K -theory of a space is assembled from data which is local. To give a vectorbundle V on X is equivalent to giving vector bundles V i on the open sets U i of acovering, and isomorphisms λ ij : V i → V j on U i ∩ U j satisfying a compatibility (cocycle) condition on the triple intersections.In terms of K -theory this is expressed by the Mayer-Vietoris (spectral) sequencerelating K ( X ) and the K -groups of the intersections of the U i . In forming twisted OOP GROUPS AND TWISTED K -THEORY I 7 K -theory we modify this descent or gluing datum, by introducing a line bundle L ij on U i ∩ U j , and asking for an isomorphism λ ij : L ij ⊗ V i → V j satisfying a certain cocycle condition. In terms of K -theory, this modifies therestriction maps in the Mayer-Vietoris sequence.In order to formulate the cocycle condition, the L ij must come equipped withan isomorphism L jk ⊗ L ij → L ik on the triple intersections, satisfying an evident compatibility relation on the quadru-ple intersections. In other words, the { L ij } must form a 1-cocycle with values in thegroupoid of line bundles. Cocycles differing by a 1-cochain give isomorphic twisted K -groups, so, up to isomorphism, we can associate a twisted notion of K ( X ) to anelement τ ∈ H ( X ; { Line Bundles } ) . On good spaces there are isomorphisms H ( X ; { Line Bundles } ) ≈ H ( X ; U (1)) ≈ H ( X ; Z ) , and correspondingly, twisted notions of K -theory associated to an integer valued3-cocycle. In this paper we find we need to allow the L ij to be ± line bundles, soin fact we consider twisted notion of K ( X ) classified by elements τ ∈ H ( X ; {± Line Bundles } ) ≈ H ( X ; Z ) × H ( X ; Z / . We will 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 Cech1-cocycle on an explicit covering of X . The twisted K -group is then assembledfrom 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-cocycleis manifest in the restriction maps between the K -groups of the open sets. They aremodified on the two-fold intersections by tensoring with the ( ± ) line bundle givenby the 1-cocycle. This “operational definition” suffices to make most computations.See § Example . Suppose that X = S , and that the isomorphism class of τ is n ∈ H ( X ; Z ) ≈ Z . Let U + = X \ (0 , , , −
1) and U − = X \ (0 , , , U + ∩ U − ∼ S , and τ is represented by the 1-cocycle whose value on U + ∩ U − is L n , with L thetautological line bundle. The Mayer-Vietoris sequence for K τ ( X ) takes the form · · · → 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 − ) → · · · . The set of isomorphism classes of twistings has a group structure induced from the tensorproduct of graded line bundles. While there is, as indicated, a set-theoretic factorization of theisomorphism classes of twistings, the group structure is not, in general, the product.
DANIEL S. FREED, MICHAEL J. HOPKINS, AND CONSTANTIN TELEMAN
Since the restriction of τ to U ± is isomorphic to zero, we have K τ +0 ( U ± ) ≈ K ( U ± ) ≈ Z K τ +1 ( U ± ) ≈ K ( U ± ) ≈ K τ +1 ( U + ∩ U − ) ≈ K ( S ) ≈ , and K τ +0 ( U + ∩ U − ) ≈ K ( S ) ≈ Z ⊕ Z . with basis the trivial bundle 1, and the tautological line bundle L . The Mayer-Vietoris sequence reduces to the exact sequence0 → K τ +0 ( X ) → Z ⊕ Z → Z ⊕ Z → K τ +1 ( X ) → . In ordinary (untwisted) K -theory, the middle map is (cid:18) −
10 0 (cid:19) : Z ⊕ Z → Z ⊕ Z . In twisted K -theory, with suitable conventions, the middle map becomes(1.5) (cid:18) n − − n (cid:19) : Z ⊕ Z → Z ⊕ Z , and so K τ + n ( S ) = ( n = 0 Z /n n = 1 . In the language of twistings, the map (1.5) is accounted for as follows. To identifythe twisted K -groups with ordinary twisted K -groups we have to choose isomor-phisms t ± : τ | U ± → . If we use the t + to trivialize τ on U + ∩ U − , then the following diagram commutes K τ + ∗ ( U + ) t + / / restr. (cid:15) (cid:15) K ∗ ( U + ) restr. (cid:15) (cid:15) K τ + ∗ ( U + ∩ U − ) t + / / K ∗ ( 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 − ) ◦ t + so the restriction map in twisted K -theory is identified with the restriction mapin untwisted K -theory, followed by the map ( t − t − ). By definition of τ , this mapis given by multiplication by L n . This accounts for the second column of (1.5). Example . Now consider the twisted K -theory of U (1) acting trivially on itself.In this case the twistings are classified by H U (1) ( U (1); Z ) × H U (1) ( U (1); Z / ≈ Z ⊕ Z / . We consider twisted K -theory, twisted by τ = ( n, ǫ ). Regard U (1) as the unit circlein the complex plane, and set U + = U (1) \ {− } U − = U (1) \ { +1 } . OOP GROUPS AND TWISTED K -THEORY I 9 The twisting τ restricts to zero on both U + and U − . Write K U (1) = R ( ) = Z [ L, L − ] . Then the Mayer-Vietoris sequence becomes0 → K τ +0 U (1) ( U (1)) → Z [ L ± ] ⊕ Z [ L ± ] → Z [ L ± ] ⊕ Z [ L ± ] → K τ +1 U (1) ( U (1)) → . The 1-cocycle representing τ can be taken to be the equivariant vector bundlewhose fiber over − i is the trivial representation of U (1) and whose fiber over + i is( − ǫ L n . With suitable conventions, the middle map becomes (cid:18) − ( − ǫ L n − (cid:19) : Z [ L ± ] → Z [ L ± ] . It follows that K τ + kU (1) ( U (1)) = ( k = 0 Z [ L ± ] / (( − ǫ L n − k = 1 . When ǫ = 0 this coincides the Grothendieck group of representations of the Heisen-berg extension of Z × U (1) of level n , and in turn with the Grothendieck group ofpositive energy representations the loop group of U (1) at level n . Example . Consider the twisted K -theory of SU (2) acting on itself by conjuga-tion. The group H SU (2) ( SU (2); Z /
2) vanishes, while H SU (2) ( SU (2); Z ) = Z , so a twistings τ in this case is given by an integer n ∈ H SU (2) ( SU (2); Z ) = Z . Set U + = SU (2) \ {− } U − = SU (2) \ { +1 } . The spaces U + and U are equivariantly contractible, while U + ∩ U − is equivariantlyhomotopy equivalent to S = SU (2) /T , where T = U (1) is a maximal torus. Therestrictions of τ to U + and U − are isomorphic to zero. We have K SU (2) ( U ± ) ≈ K SU (2) (pt) = R ( SU (2)) = Z [ L, L − ] W with the Weyl group W ≈ Z / L and L − , and K SU (2) ( U + ∩ U − ) ≈ K SU (2) ( SU (2) /T ) ≈ K T (pt) ≈ Z [ L, L − ] . The ring R ( SU (2)) has an additive basis consisting of the irreducible representa-tions, ρ k = L k + L k − + · · · + L − k , k ≥ ρ l ρ k = ρ k + l + ρ k + l − + · · · + ρ k − l k ≥ l. As in our other example, the Mayer-Vietoris sequence is short exact0 → K τ +0 SU (2) ( SU (2)) → R ( SU (2)) ⊕ R ( SU (2)) → Z [ L ± ] → K τ +1 SU (2) ( SU (2)) → . The 1-cocycle representing the difference between the two trivializations of therestriction of τ to U + ∩ U − can be taken to be the element L n ∈ K SU (2) ( SU (2) /T ) ≈ R ( T ). The sequence is a sequence of R ( SU (2))-modules. With suitable conventions,the middle map is(1.8) (cid:0) − L n (cid:1) : R ( SU (2)) → R ( T ) . 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 { , L } . It follows from the identity L n = L ρ n − − ρ n − that (1.8) is represented by the matrix (cid:18) ρ n − − ρ n − (cid:19) , and that K τ + kSU (2) ( SU (2)) = ( k = 0 R ( SU (2)) / ( ρ n − ) k = 1 . This coincides the Grothendieck group of positive energy representations the loopgroup of SU (2) at level ( n − K -theory andthe representations of loop groups. In both cases the Grothendieck group of positiveenergy representations of the loop group of a compact Lie group G is described bythe twisted equivariant K -group of G acting on itself by conjugation. Two minordiscrepancies appear in this relationship. On one hand, the interesting K -group isin degree k = 1. As explained in the introduction, the representations of the loopgroup at level τ correspond to twisted K -theory at the twisting ζ ( τ ) = g + ˇ h + τ .The shift in K -group to k = 1 corresponds to the term g . In both examples theadjoint representation is Spin c and so contributes only its dimension to ζ ( τ ). Thisterm could be gotten rid of by working with twisted equivariant K -homology ratherthan K -cohomology. We have chosen to work with K -cohomology in order to makebetter contact with our geometric constructions in Parts II and III. The otherdiscrepancy is the shift in level in Example 1.7: twisted equivariant K -theory atlevel n corresponds to the representations of the loop group at level ( n − h in our formula for ζ ( τ ).We now give a series of examples describing other ways in which twistings of K -theory arise. Example . Let V be a vector bundle of dimension n over a space X , and write X V for the Thom complex of V . Then ˜ K n + k ( X V ) is a twisted form of K k ( X ). Toidentify the twisting, choose local Spin c structures µ i on the restrictions V i = V | U i of V to the sets in an open cover of X . The K -theory Thom classes associated tothe µ i allow one to identify ˜ K n + k ( U V i i ) ≈ K k ( U i )The difference between the two identifications on U i ∩ U j is given by multiplicationby the graded line bundle representing the difference between µ i and µ j . Figu-ratively, the cocycle representing the twisting is µ j µ − i and the cohomology classis ( w ( V ) , W ( V )) ∈ H ( X ; Z / × H ( X ; Z ), where W = βw . This is one ofthe original examples of twisted K -theory, described by Donovan and Karoubi [14]from the point of view of Clifford algebras. We will review their description in § OOP GROUPS AND TWISTED K -THEORY I 11 Example . Let G be a compact Lie group. The central extensions T → ˜ G τ −→ G of G by T = U (1) are classified by H G ( { pt } ; Z ) = H ( BG ; Z ). The Grothendieckgroup R τ ( G ), of representations of ˜ G on which T acts according to its definingrepresentation, can be thought of as a twisted form of R( G ). In this case, ourdefinition of equivariant twisted K -theory gives K τ + kG (pt) = ( R τ ( G ) k = 00 k = 1More generally, if S is a G -space, and τ ∈ H G ( S ) is pulled back from H G (pt), then K τ + kG ( S ) is the summand of K k ˜ G ( S ) corresponding to ˜ G -equivariant vector bundleson which T acts according to its defining character. Example . 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 Grothendieck groupof representations of G whose restriction to H is V -isotypical, forms a twistedversion of the Grothendieck group of representations of Q . When H is central, equalto T , and V is the defining representation, this is the situation of 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 therepresentation dual to V . Let ˜ G denote the group of pairs( g, f ) ∈ G × hom( V ∗ , V ∗ )for which f is unitary, and satisfies f ( hv ) = ghg − f ( v ) h ∈ H. Since V is irreducible, and (ad g ) ∗ V ≈ V , the same is true of V ∗ , and the map˜ G → G ( g, f ) g is surjective, with kernel T . The inclusion H ⊂ ˜ Gh ( h, action of h )is normal, and lifts the inclusion of H into G . We define˜ Q = ˜ G/H.
The group ˜ Q is a central extension of Q by T , which we denote˜ Q τ −→ Q. We now describe an equivalence of categories between V -isotypical G -representations,and τ -projective representation of Q (representations of ˜ Q on which T acts accord-ing to its defining character). By definition, the representations V and V ∗ of H come equipped with extensionsto unitary representation of ˜ G . Given a V -isotypical representation W of G , we let M denote the H -invariant part of V ∗ ⊗ W : M = ( V ∗ ⊗ W ) H . The action of ˜ G on V ∗ ⊗ W factors through an action of ˜ Q on M . This defines afunctor from V -isotypical representations of G to τ -projective representations of Q .Conversely, suppose M is a τ -projective representation of ˜ Q . Let ˜ G act on M through the projection ˜ G → ˜ Q , and form W = V ⊗ M. The central T of ˜ G acts trivially on W , giving W a G -action. This defines afunctor 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 anequivalence of categories. Example . 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 bestable under conjugation by G . Write G = { g ∈ G | (ad g ) ∗ V ≈ V } , and Q = G /H . Let S be the set of isomorphism classes of irreducible repre-sentations of H of the form (ad g ) ∗ V . The conjugation action of G on S factorsthrough Q , and we have an identification S = Q/Q . Let’s call a representation of G S -typical if its restriction to H involves only the irreducible representations in S . One easily checks that “induction” and “passage to the V -isotypical part of therestriction” give an equivalence of categories { S -typical representations of G } ↔ { V -isotypical representations of G } , and therefore an isomorphism of the Grothendieck group R S ( G ) of S -typical rep-resentations of G with R τ ( Q ) ≈ K τQ (pt) . We can formulate this isomorphism a little more cleanly in the language of groupoids.For each α ∈ S , choose an irreducible H -representation V α representing α . Considerthe groupoid S//Q , with set of objects S , and in which a morphism α → β is anelement 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 equivalences classes of pairs ( g, φ ) ∈ G × hom( V ∗ α ∗ ,V β ),with φ unitary, and satisfying φ ( h v ) = ghg − φ ( v )(so that, among other things, (ad g ) ∗ α = β ). The equivalence relation is generatedby ( 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 ˜ Q of Q .An easy generalization of the construction of Example 1.11 gives an equivalence ofcategories { τ − projective representations of P } ↔ { S − typical representations of G } . OOP GROUPS AND TWISTED K -THEORY I 13 Central extensions of
S//Q are classified by H ( S//Q ; Z ) = H Q ( S ; Z ) ≈ H Q (pt; Z ) , and so represent twistings of K -theory. Our definition of twisted K -theory ofgroupoids 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 isomorphismR S ( G ) ≈ K τ +0 ( S//Q ) = K τ +0 Q ( S ) . Example . Now let S denote the set of isomorphism classes of all irreduciblerepresentations of H . Decomposing S into orbit types, and using the constructionof Example 1.12 gives a central extension τ : P → ( S//Q ), and an isomorphismR( G ) ≈ K τ +0 ( S//Q ) = K τ +0 Q ( S ) . More generally, if X is a space with a Q -action there is an isomorphism K kG ( X ) ≈ K τ + kQ ( X × S ) , in which τ is the Q -equivariant twisting of X × S pulled back from the Q -equivarianttwisting 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 termi-nology derives from the situation of Example 1.10 in which a central extension ofa group gives rise to a twisted notion of equivariant K -theory. By working withgraded central extensions of groupoids (rather that groups) we are able to includein a single point of view both the twistings that come from 1-cocycles with valuesin the group of Z / T -bundles and T -torsors instead of “line bundles,” where T is the group U (1). We begin with a formal discussion of ( Z / 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 / T -bundle ( P, ǫ ) even (resp. odd ) if ǫ is the constant func-tion function 1 (resp. − T -bundles forms a symmetricmonoidal groupoid. A map of graded T -bundles ( P , ǫ ) → ( P , ǫ ) exists onlywhen ǫ = ǫ , in which case it is a map of principal bundles P → P . The tensorstructure is given by ( P , ǫ ) ⊗ ( P , ǫ ) = ( P ⊗ P , ǫ + ǫ ) , in which P ⊗ P is the usual “tensor product” of principal T -bundles:( P ⊗ P ) x = ( P ) x × ( P ) x / ( vλ, w ) ∼ ( v, wλ ) . It is easiest to describe the symmetry transformation T : ( P , ǫ ) ⊗ ( P , ǫ ) → ( P , ǫ ) ⊗ ( P , ǫ )fiberwise. In the fiber over a point x ∈ X it is( v, w ) ( w, vǫ ( x ) ǫ ( x )) . We will write B T ± for the contravariant functor which associates to a space X the category of graded T -bundles over X , and for B T the functor “category of T -bundles”. We will also write H ( X ; T ± ) for the group of isomorphism classes in B T ± ( X ), and H ( X ; T ± ) for the group of automorphisms of any object. There isan exact sequence(2.2) B T → B T ± → Z/ ǫ : X → Z / T -bundle ǫ := ( X × T , ǫ ) . This splitting is compatible with the monoidal structure, but not with its symmetry.It gives an equivalence(2.3) B T ± ≈ B T × Z / symmetric monoidal categories). It follows thatthe group H ( X ; T ± ), of isomorphism classes of graded T -bundles over X , is iso-morphic to H ( X ; Z / × H ( X ; T ). Since X is assumed to be paracompact, thisin turn is isomorphic to H ( X ; Z / × H ( X ; Z ). The automorphism group of anygraded T -bundle is the group of continuous maps from X to T .2.2. Graded central extensions.
Building on the notion of graded T -bundleswe now turn to graded central extension of groupoids. The reader is referred toAppendix A for our conventions on groupoids, and a recollection of the fundamentalnotions. Unless otherwise stated all groupoids will be assumed to be local quotientgroupoids ( § A.2.2), in the sense that they admit a countable open cover by sub-groupoids, each of which is weakly equivalent to a compact Lie group acting on aHausdorff space.Let X = ( X , X ) be a groupoid. Write B Z / Z / Definition 2.4. A graded groupoid is a groupoid X equipped with a functor ǫ : X → B Z /
2. The map ǫ is called the grading .The collection of gradings on X forms a groupoid, in which a morphism is anatural transformation. Spelled out, a grading of X is a function ǫ : X → Z / ǫ ( g ◦ f ) = ǫ ( g ) + ǫ ( f ), and a morphism from ǫ to ǫ is a continuousfunction η : X → Z / f : x → y ) ∈ X , ǫ ( f ) = ǫ ( f ) + ( η ( y ) − η ( x )) . Example . Suppose that X = S//G , with S a connected topological space. Thena 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, B Z / . Definition 2.6. A graded central extension of X is a graded T -bundle L over X ,together with an isomorphism of graded T -bundles on X λ g,f : L g ⊗ L f → L g ◦ f OOP GROUPS AND TWISTED K -THEORY I 15 satisfying the cocycle condition, that the diagram( L h L g ) L f & & NNNNNNNNNNN ≈ / / L h ( L g L f ) / / L h L g ◦ f (cid:15) (cid:15) L h ◦ g L f / / L h ◦ g ◦ f of graded T -bundles on X commutes.If L → X is a graded central extension of X , then the pair ˜ X = ( X , L ) is agraded groupoid over X , and the functor ˜ X → X represents ˜ X as a graded centralextension of X by T in the evident sense. Our terminology comes from this point ofview. Some constructions are simpler to describe in terms of the graded T -bundles L and others in terms of ˜ X → X .The collection of graded central extensions of X forms a symmetric monoidal2-category which we denote Ext ( X, T ± ) = Ext X . The category of morphisms in Ext X from L → L is the groupoid of graded T -torsors ( η, ǫ ) over X , equippedwith an isomorphism η b L f → L f η a making η c L g L f / / (cid:15) (cid:15) L g η b L f / / L g L f η a (cid:15) (cid:15) η c L g ◦ f / / L g ◦ f η a commute. The tensor product L ⊗ L ′ is the graded central extension L ⊗ L ′ → X with structure map L g ⊗ L ′ g ⊗ L f ⊗ L ′ f ⊗ T ⊗ −−−−−→ L g ⊗ L f ⊗ L ′ g ⊗ L ′ f λ g,f ⊗ λ ′ g,f −−−−−−→ L g ◦ f ⊗ L ′ g ◦ f . The symmetry isomorphism L ⊗ L ′ → L ′ ⊗ L is derived from the symmetry of thetensor product of graded T -bundles.For the purposes of twisted K -groups it suffices to work with the 1-categoryquotient of Ext X . Definition 2.7.
The category
Ext ( X ; T ± ) = Ext X is the category with objectsthe graded central extensions of X , and with morphisms from L to L ′ the set ofisomorphism classes in Ext X ( L, L ′ ) . The symmetric monoidal structure on
Ext X makes Ext X into a symmetric monoidalgroupoid in the evident way. Remark . A 1-automorphisms of L consists of a graded T torsor η over X ,together with an isomorphism η a → η b over X , satisfying the cocycle condition.In this way the category of automorphisms of any twisting can be identified withthe groupoid of graded line bundles over X . A graded line bundle on X definesan element of K ( X ) (even line bundles go to line bundles, and odd line bundlesgo to their negatives). The fundamental property relating twistings and twisted K -theory is that the automorphism η acts on twisted K -theory as multiplicationby the corresponding element of K ( X ) (Proposition 3.3). Remark . (1) The formation of Ext X is functorial in X , in the sense of 2-categories. If F : Y → X is a map of groupoids, and L → X is a graded centralextension of X , then F ∗ L → Y gives a graded central extension, F ∗ L of Y . If η → X is a morphism from L to L , then F ∗ η defines a morphism from F ∗ L to F ∗ L .(2) If T : Y → X is a natural transformation from F to G , then the graded linebundle T ∗ L determines a morphism from F ∗ L to G ∗ L . This is functorial in thesense that given η : L → L , there is a 2-morphism relating the two ways of goingaround F ∗ L F ∗ η (cid:31) (cid:31) ????? T ∗ L (cid:127) (cid:127) (cid:127)(cid:127)(cid:127)(cid:127)(cid:127) F ∗ L T ∗ L (cid:127) (cid:127) (cid:127)(cid:127)(cid:127)(cid:127)(cid:127) G ∗ L G ∗ η (cid:31) (cid:31) ????? = ⇒ G ∗ L . The 2-morphism is the isomorphism( G ∗ η ) ◦ ( T ∗ L ) → ( T ∗ L ) ◦ ( F ∗ η )gotten by pulling back map η : L → L along T . It is given pointwise over y ∈ Y , T y : F y → Gy by ( η Gy )( L T y ) → ( L T y )( η F y ) . (3) It follows that the formation of Ext X is functorial in X , making Ext X a (weak)presheaf of groupoids. Example . Suppose that G is a group, and X = pt //G . Then a graded centralextension of X is just a graded central extension of G by T .Forgetting the T -bundle gives a functor from Ext X to the groupoid of gradingsof X , and the decomposition (2.3) gives a 2-category equivalence(2.11) Ext ( X, T ± ) ≈ Ext ( X, T ) × Hom ( X, B Z / Ext ( X, T )is the 2-category of evenly graded (ie, ordinary) central extensions of X by T .2.2.1. Classification of graded central extensions.
We now turn to the classificationof graded central extensions of a groupoid X . In view of (2.11), it suffices toseparately classify T -central extensions (graded central extensions which are purelyeven) and gradings.For the T -central extensions, first recall that the category of T -torsors on aspace Y is equivalent to the category whose objects are T -valued Cech 1-cocycles,ˇ Z ( Y ; T ), and in which a morphism from z to z is a Cech 0-cochain c ∈ ˇ C ( Y ; T )with the property that δc = z − z . OOP GROUPS AND TWISTED K -THEORY I 17 Now consider the double complex for computing the Cech hyper-cohomology groupsof the nerve X • , with coefficients in T :(2.12) ˇ C ( X ; T ) / / O O ˇ C ( X ; T ) / / O O ˇ C ( X ; T ) / / O O ˇ C ( X ; T ) / / O O ˇ C ( X ; T ) / / O O ˇ C ( X ; T ) / / O O ˇ C ( X ; T ) / / O O ˇ C ( X ; T ) / / O O ˇ C ( X ; T ) / / O O In terms of the Cech cocycle model for T -bundles, the 2-category of T -centralextension of X is equivalent to the category whose objects are cocycles in (2.12), oftotal degree 2, whose component in ˇ C ( X ; T ) is zero. The 1-morphisms are givenby cochains of total degree 1, whose coboundary has the property its componentin ˇ C ( X ; T ) vanishes. The 2-morphisms are given by cochains of total degree 0.Write ˇ H ∗ ( X ) and ˇ H ∗ ( X )for the Cech hyper-cohomology of X , and the Cech cohomology of X respectively.Then the group of isomorphism classes of even graded T -gerbes is given by thekernel of the map ˇ H ( X ; T ) → ˇ H ( X ; T ) , the group of isomorphism classes of 1-automorphisms of any even graded T -gerbeis ˇ H ( X ; T ), and the group of 2-automorphisms of any 1-morphism is ˇ H ( X ; T ).As for the gradings, the group of isomorphism classes of gradings isker (cid:8) ˇ H ( X ; Z / → ˇ H ( X ; Z / (cid:9) and the automorphism group of any grading isˇ H ( X ; Z / . For convenience, writeˇ H t rel ( X ; A ) = ker (cid:8) ˇ H t ( X ; A ) → ˇ H t ( X ; A ) (cid:9) . Proposition 2.13.
The group π Ext X of isomorphism classes of graded centralextension of X is given by the set-theoretically split extension (2.14) ˇ H rel ( X ; T ) → π Ext X → ˇ H rel ( X ; Z / with cocycle c ( ǫ, µ ) = β ( ǫ ∪ µ ) , where β : Z / {± } ⊂ T is the inclusion. The group of isomorphism classes ofautomorphisms of any graded central extension of X is ˇ H ( X ; T ) × ˇ H ( X ; Z / ,and the group of -automorphisms of any morphism of graded central extensions is ˇ H ( X ; T ) .Proof: Most of this result was proved in the discussion leading up to its state-ment. The decomposition (2.11) gives the exact sequence, as well as a set-theoreticsplitting s : ˇ H ( X ; Z / → π Ext X . It remains to identify the cocycle describing the group structure. Suppose that ǫ, µ : X → B Z / X . Then by the discussion leading up to (2.3), s ( ǫ ) s ( µ ) is thegraded central extension given by( s ( ǫ ) s ( µ )) f = ǫ ( f ) µ ( f ) ≈ ǫ ( f )+ µ ( f ) , and structure map(2.15) ǫ ( g ) µ ( g ) ǫ ( f ) µ ( f ) → ǫ ( g ) ǫ ( f ) µ ( g ) µ ( f ) → ǫ ( g ◦ f ) µ ( g ◦ f ) . Using the canonical identifications a b = a + 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( − ǫ ( f ) µ ( g ) of the trivialized graded line ǫ ( g )+ µ ( g )+ ǫ ( f )+ µ ( f ) ≈ ǫ ( g ◦ f )+ µ ( g ◦ f ) . Similarly, the structure map of s ( ǫ + µ ) can be identified with the identity map ofthe same trivialized graded line. It follows that s ( ǫ ) s ( µ ) = c ( ǫ, µ ) s ( ǫ + µ ), where c ( ǫ, µ ) is graded central extension with with L f = , and λ g,f = ( − ǫ ( f ) µ ( g ) . Now the 2-cocycle ǫ ( f ) µ ( g ) is precisely the Alexander-Whitney formula for the cupproduct ǫ ∪ µ ∈ Z ( X ; Z / (cid:3) One easy, but very useful consequence of Proposition 2.13 is the stacky natureof the morphism categories in
Ext X . Corollary 2.16.
Let
P, Q ∈ Ext X , and f : Y → X be a local equivalence. Thenthe functor f ∗ : Ext X ( P, Q ) → Ext Y ( f ∗ P, f ∗ Q ) is an equivalence of categories, and so f ∗ : Ext X ( P, Q ) → Ext Y ( f ∗ P, f ∗ Q ) is a bijection of sets. (cid:3) Another useful, though somewhat technical consequence of the classification is
Corollary 2.17.
Suppose that X is a groupoid with the property that the maps ˇ H ( X ; T ) → ˇ H ( X ; T )ˇ H ( X ; Z / → ˇ H ( X ; Z / are zero. If Y → X is a local equivalence, then the maps ˇ H ( Y ; T ) → ˇ H ( Y ; T )ˇ H ( Y ; Z / → ˇ H ( Y ; Z / OOP GROUPS AND TWISTED K -THEORY I 19 are zero, and Ext X → Ext Y , and Ext X → Ext Y are equivalences.Proof: The first assertion is a simple diagram chase. It has the consequence thatthe maps ˇ H ( X ; T ) → ˇ H ( X ; T )ˇ H ( X ; Z / → ˇ H ( X ; Z / H ( Y ; T ) → ˇ H ( Y ; T )ˇ H ( Y ; Z / → ˇ H ( Y ; Z / (cid:3) For later reference we note the following additional consequence of Proposi-tion 2.13
Corollary 2.18.
Suppose that X is a local quotient groupoid, and that τ is a twist-ing of X represented by a local equivalence P → X and a graded central extension ˜ P → P . Then ˜ P is a local quotient groupoid.Proof: Since the property of being a local quotient groupoid is an invariant oflocal equivalence, we know that P is a local quotient groupoid. The question is alsolocal in P , so we may assume 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. Workingstill more locally in S we may assume S is contractible. But then Proposition 2.13implies that τ is given by a (graded) central extension of ˜ G → G , and ˜ P = S// ˜ G . (cid:3) Twistings.
We now describe the category
Twist X of K -theory twistings ona local quotient groupoid X ( § A.2.2). The objects of
Twist X 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 = ( P , L ) to b = ( P , L ) is defined to be the colimit Twist X ( a, b ) = lim −→ p : P → P Ext P ( p ∗ π ∗ a, p ∗ π ∗ b ) , where P = P × X P , the limit is taken over Cov P and our notation refers tothe diagram P p (cid:15) (cid:15) P P π / / π o o P . We leave to the reader to check that
Ext P ( a, b ) does indeed define a functor on the1-category quotient Cov P .The colimit appearing in the definition of Twist X ( a, b ) is present in order thatthe definition be independent of any extraneous choices. In fact the colimit isattained at any stage. Lemma 2.19.
For any local equivalence (2.20) p : P → P the map Ext P ( p ∗ π ∗ a, p ∗ π ∗ b ) → Twist X ( a, b ) is an isomorphism. Proof:
By Corollary 2.16, for any P ′ → P in Cov P , the map Ext P ( a, b ) → Ext P ′ ( a, b )is an isomorphism. The result now follows from Corollary A.12. (cid:3) For the composition law, suppose we are given three twistings a = ( P , L ) , b = ( P , L ) , and c = ( P , L ) . Find a P ∈ Cov X and maps P p } } zzzzzzzz p (cid:15) (cid:15) p ! ! DDDDDDDD P P P (for example one could take P to be the (2-category) fiber product P × X P × X P , and p i projection to the i th factor). By Lemma 2.19, the maps Ext P ( p ∗ a, p ∗ b ) → Twist X ( a, b ) Ext P ( p ∗ a, p ∗ c ) → Twist X ( a, c ) Ext P ( p ∗ b, p ∗ c ) → Twist X ( b, c )are bijections. With these identifications, the composition law in Twist X is formedfrom that in Ext P . We leave to the reader to check that this is well-defined.The formation of Twist X is functorial in X. Given f : Y → X , and a = ( P, L ) ∈ Twist X form Y × X P π −−−−→ P y y Y −−−−→ f X and set f ∗ a = ( f ∗ P, π ∗ L ) . Proposition 2.21.
The association X Twist X is a weak presheaf of groupoids.If Y → X is a local equivalence then Twist Y → Twist X is an equivalence of cate-gories. (cid:3) There is an evident functor
Ext X → Twist X . Proposition 2.22.
When X satisfies the condition of Corollary 2.17, the functor (2.23) Ext X → Twist X is an equivalence of categories.Proof: Lemma 2.19 shows that (2.23) is fully faithful. Essential surjectivity is aconsequence of Corollary 2.17. (cid:3)
OOP GROUPS AND TWISTED K -THEORY I 21 Corollary 2.24. If Y → X is a local equivalence, and Y satisfies the conditionsof Corollary 2.17 then the functors Twist X → Twist Y ← Ext Y are equivalences of groupoids. (cid:3) Combining this with Proposition 2.13 gives
Corollary 2.25.
The group π Twist X of isomorphism classes of twistings on X is the set-theoretically split extension ˇ H ( X ; T ) → π Twist X → ˇ H ( X ; Z / with cocycle c ( ǫ, µ ) = β ( ǫ ∪ µ ) . The group of automorphisms of any twisting is ˇ H ( X ; T ) × ˇ H ( X ; Z / . (cid:3) We now switch to the point of view of “fibered categories” in order to more easilydescribe the functorial properties of twisted K -groups.Let Ext denote the category whose objects are pairs (
X, L ) consisting of agroupoid X and a graded central extension L of X . A morphism ( X, L ) → ( Y, M )is a functor f : X → Y , and an isomorphism L → f ∗ M in Ext X . We identifymorphisms f, g : ( X, L ) → ( Y, M )if there is a natural transformation T : f → g for which the following diagramcommutes: f ∗ M η T (cid:15) (cid:15) L oooooo ' ' OOOOOO 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 ∈ Twist X , and morphisms ( X, a ) → ( Y, b ) to beequivalence classes of pairs consisting of a functor f : X → Y and an isomorphism a → f ∗ b in Twist X .There is an inclusion Ext → Twist corresponding to the inclusion
Ext X → Twist X . Corollary 2.24 immediately implies 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. Thenthere is a factorization Ext (cid:15) (cid:15) F / / C Twist F ′ < < zzzzz Moreover any two such factorizations are naturally isomorphic by a unique naturalisomorphism.
Examples of twistings.
Example . Suppose that X is a space, P → X a principal G bundle, and˜ G → GG ǫ −→ Z / G . Then P//G → X is a local equivalence, P// ˜ G isa graded central extension of P//G , and (
P// ˜ G, P//G ) represents a twisting of X . Example . As a special case, we note that any double cover P → X defines atwisting. In this case G = Z /
2, ˜ G → G is the trivial bundle, and ǫ : G → Z / Example . Suppose that X = pt //G , with G a compact Lie group. In this caseevery local equivalence ˜ X → X admits a section. Moreover the inclusion { Id X } → Cov X of the trivial category consisting of the identity map of X into Cov X is an equiva-lence. It follows that Ext (pt //G, T ± ) → Twist X is an equivalence of categories, and so twistings of X in this case are just gradedcentral extensions of G . Using Corollary 2.25, one can draw the same conclusionfor S//G when S is contractible.We now describe the main example of twistings used in this paper. Example . Suppose that G is a connected compact Lie group, and consider thepath-loop fibration(2.31) Ω G → P G → G. We regard
P G as a principal bundle over G with structure group Ω G . The group G acts on everything by conjugation. Write LG for the group Map( S , G ) of smoothmaps from S to G. The homomorphism “evaluation at 1” : LG → G is split bythe inclusion of the constant loops. This exhibits LG as a semidirect product LG ≈ Ω G ⋊ 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 ˜ LG → LG then defines a twisting of G//G .We will write τ for a typical twisting of X , and write the monoidal structureadditively: τ + τ . We will use the notations ( ˜ P τ , P τ ) and ( L τ , P τ ) for typicalrepresenting graded central extensions. This is consistent with writing the monoidalstructure additively: L τ + τ = L τ ⊗ L τ . OOP GROUPS AND TWISTED K -THEORY I 23 Example . 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 isinvariant under this action (again, in the 2-category sense). To see this it is easiestto replace X by the weakly equivalent groupoid X ′ = ( S × G/H ) //G, and factor X → Y as X i −→ X ′ f ′ −→ Y. Since i ∗ : Twist X ′ → Twist X is an equivalence of 2-categories, it suffices to exhibitan action of G/H on f ′∗ τ . The obvious left action of G/H on S × G/H commuteswith the right action of G , giving an action of G/H on X ′ commuting with f ′ . Theaction of G/H on f ′∗ τ is then a consequence of naturality. Example . By way of illustration, consider the situation of Example 2.32 inwhich H is commutative, S = { pt } , and τ is given by a central extension T ˜ G → G. Write T ˜ H → H, for the restriction of τ to H and assume, in addition, that ˜ H is commutative. Thenthe action of G/H on f ∗ τ constructed in Example 2.32 works out to be the naturalaction of G/H on ˜ H given by conjugation.3. Twisted K-groups
Axioms.
Before turning to the definition of twisted K -theory, we list somegeneral properties describing it as a cohomology theory on the category Twist oflocal quotient groupoids equipped with a twisting. These properties almost uniquelydetermine twisted K -theory, and suffice to make our main computation in Section 4.Twisted K -theory is going to be homotopy invariant, so we need to define thenotion of homotopy Definition 3.1. A homotopy between two maps f, g : ( X, τ X ) → ( Y, τ Y )is a map ( X × [0 , , π ∗ τ X ) → ( Y, τ Y )( π : X × [0 , → X is the projection) whose restriction to X × { } is f , and to X × { } is g .Twisted K -theory is also a cohomology theory. To state this properly involvesdefining the relative twisted K theory of a triple ( X, A, τ ) consisting of a localquotient groupoid X , a sub-groupoid A , and a twisting of X . We form a categoryof the triples ( X, A, τ ) in the same way we formed
Twist . We’ll call this the categoryof 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 § ( 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 ) is an isomorphism. iii) If ( X, A, τ ) = ` α ( X α , A α , τ α ) , then the natural map K τ + n ( X, A ) → Y α K τ α + n ( X α , A α ) is an isomorphism. The combination of excision and the long exact sequence of a pair gives theMayer-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 sub-groupoids 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 oftwistings coming from Proposition 3.2. ii) Suppose that η : τ → τ is a -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 ( 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 compactLie group, and τ a twisting given by a graded central extension T → G τ → Gǫ : G → Z / . i) If ǫ = 0 then then K τ + n ( X ) is the summand K nG τ ( S )(1) ⊂ K nG τ ( S ) on which T acts via its standard (defining) representation. This isomorphism iscompatible with the product structure. OOP GROUPS AND TWISTED K -THEORY I 25 ii) For general ǫ , K τ + n ( X ) is isomorphic to K n +1 G τ ( S × ( R ( ǫ ) , R ( ǫ ) \ { } ))(1) , in which the symbol R ( ǫ ) denotes the -dimensional representation ( − ǫ of G τ . In part ii), When ǫ = 0, then R ( ǫ ) is the trivial representation, and the isomor-phism can be composed with the suspension isomorphism to give the isomorphismof i).When τ = 0, so that G τ ≈ G × T , Proposition 3.5 reduces to an isomorphism K τ + n ( X ) ≈ K nG ( S ) . In view of this, we’ll often write K τ + nG ( S ) = K τ + n ( X )in case X = S//G and ǫ = 0. Of course there is also a relative version of Proposi-tion 3.5.The reader is referred to Section 4 of [21] for a more in depth discussion of thetwistings of equivariant K -theory, and interpretation of “ ǫ ” part the twisting interms of graded representations.Using the Mayer-Vietoris sequence one can easily check that result of part i)of Proposition 3.5 holds for any local quotient groupoid X . If the twisting τ isrepresented by a central extension P → X , then the restriction mapping is anisomorphism 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 termsof Fredholm operators on a twisted bundle of Hilbert spaces. In this section wedescribe how one associates to a graded central extension of a groupoid, a twistednotion of Hilbert bundle. We refer the reader to Appendix A, § A.4 for our notationand conventions on bundles over groupoids, and to § A.4 for a discussion of Hilbertspace bundles.Let X be a groupoid, and τ : ˜ X → X a graded central extension, whose associ-ated graded T -bundle we denote L τ → X . As in Appendix A.3, we will use a f −→ b and a f −→ b g −→ c to refer to generic points of X and X respectively, and so, for example, in acontext describing bundles over X , the symbol H b will refer to the pullback of X along the map X → X ( a → b → c ) b. Definition 3.6. A τ -twisted Hilbert bundle on X consists of a Z / H on X , together with an isomorphism (on X ) L τf ⊗ H a → H b satisfying the cocycle condition that L τg (cid:16) L τf H a (cid:17) o o / / (cid:15) (cid:15) (cid:16) L τg L τf (cid:17) H a (cid:15) (cid:15) L τg H b / / H c L τg ◦ f H a o o commutes on X . Remark . Phrased differently, a twisted Hilbert bundle is just a graded Hilbertbundle over ˜ X , with the property that the map H a → H b induced by ( a f −→ b ) ∈ ˜ X has degree ǫ ( f ), and for which the central T acts according to its defining character. Example . Suppose that X is of the form P//G , and that our twisting corre-sponds to a central extension of G τ → G . Then a projective unitary representationof G (meaning a representation of G τ on which the central T acts according to itsdefining character) defines a twisted Hilbert bundle over X .Suppose that τ and µ are graded central extensions of X , with associated graded T -torsors L τ and L µ . If H is a τ -twisted Hilbert bundle over X and W is a µ -twistedHilbert bundle, then the graded tensor product H ⊗ W is a ( τ + µ )-twisted Hilbertbundle, with structure map L τ + µf ⊗ H a ⊗ W a = L τf ⊗ L µf ⊗ H a ⊗ W a → L τf ⊗ H a ⊗ L µf ⊗ W a → H b ⊗ W b . Now suppose that H and H are τ -twisted, graded Hilbert bundles over X . Definition 3.9. A linear transformation T : H → H consists of a linear trans-formation of Hilbert bundles T : H → H on X for which the following diagramcommutes on X : L f H a −−−−→ H b ⊗ T y y T L f H a −−−−→ H b . If L τ is a graded central extension of X , we’ll write U τX (or just U τ if X isunderstood) for the category in which the objects are τ -twisted Z/ embeddings. If f : Y → X is amap, there is an evident functor f ∗ : U τX → U f ∗ τY . A natural transformation T : f → g of functors X → Y gives a natural transfor-mation T ∗ : f ∗ → g ∗ . Using Remark 3.7 and descent, one easily checks that f ∗ isan equivalence of categories when f is a local equivalence.The category U τX is also functorial in τ . Indeed, suppose that η : τ → σ is amorphism, given by a graded T -bundle η , and an isomorphism η b ⊗ τ f → σ f ⊗ η a . If H is a τ -twisted Hilbert bundle, then H ⊗ η is a σ -twisted Hilbert bundle. Oneeasily checks that H H ⊗ η gives an equivalence of categories U τX → U σX , withinverse H H ⊗ η − . The 2-morphisms η → η give natural isomorphisms offunctors. OOP GROUPS AND TWISTED K -THEORY I 27 The tensor product of Hilbert spaces gives a natural tensor product U τX × U µX → U τ + µX . Universal Twisted Hilbert Bundles.
We now turn to the existence ofspecial kinds of τ -twisted Hilbert bundles, following the discussion of § A.4. Wekeep the notation of § Definition 3.10. A τ -twisted Hilbert bundle H on X isi) universal if for every τ -twisted Hilbert space bundle V there is a unitary em-bedding V → H ;ii) locally universal if H | U is universal for every open sub-groupoid U ⊂ X ;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 sub-groupoid U ⊂ X .As in Appendix A.4, if H is (locally) universal, then H is automatically absorb-ing. Lemma 3.11.
Suppose that ˜ X → X is a graded central extension and H is agraded Hilbert bundle on ˜ X . 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 Hilbertbundle on X which is (locally) universal if H is. (cid:3) Lemma 3.12. If X is a local quotient groupoid, and τ : ˜ X → X is a graded centralextension then there exists a locally universal τ -twisted Hilbert bundle H on X . Thebundle H is unique up to unitary equivalence.Proof: By Corollary 2.18, ˜ X is a local quotient groupoid which, by Corol-lary A.33, admits a locally universal Hilbert bundle. The result now follows fromLemma 3.11. (cid:3) Definition of twisted K -groups. Our task is to define twisted K -groupsfor 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 toisomorphisms. We will do this by using spaces of Fredholm operators to constructa spectrum K τ ( X, A ) and defining K τ + n ( X, A ) = π − n K τ ( X, A ). The reader isreferred 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 § H is given by a Hilbertbundle H over X , equipped with an isomorphism(3.13) L τf ⊗ H a → H b over f : a → b ∈ X , satisfying the cocycle condition. The map T Id ⊗ T is a homeomorphism between the spaces of Fredholm operators (See § A.5) Fred ( n ) ( H a )and Fred ( n ) ( L τf ⊗ H a ) compatible with the structure maps (3.13). The spacesFred ( n ) ( H a ) therefore form a fiber bundle over Fred ( n ) ( H ) over X . We define spaces K τ ( X ) n by K τ ( X ) n = ( Γ( X ; Fred (0) ( H )) n evenΓ( X ; Fred (1) ( H )) n odd , By an obvious modification of the arguments of Atiyah-Singer [1], the resultsdescribed in § A.5 hold for the bundle Fred ( n ) ( H ) over X . In particular, themaps (A.44) and the homeomorphism (A.45) give weak homotopy equivalences K τ ( X ) n → Ω K τ ( X ) n +1 , making the collection of spaces K τ ( X ) = { K τ ( X ) n } into a spectrum. Definition 3.14.
Suppose that (
X, τ ) is a local quotient groupoid equipped witha graded central extension τ , and H is a locally universal, τ -twisted Hilbert bundleover 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 thenotation K τ ( X ). The value of the twisted K -group is, in the end, independent ofthis choice. See Remark 3.17 below.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 H X be a τ -twisted, locally universal Hilbert bundle over X , and H Y an f ∗ τ -twisted, locallyuniversal Hilbert space bundle over Y . Since H Y is universal, there is a unitaryembedding f ∗ H X ⊂ H Y . Pick one. There is then an induced map f ∗ Fred ( n ) ( H X ) → Fred ( n ) ( H Y ) T T ⊕ ǫ, ( ǫ is the base point) 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 agraded T -bundle η over X , and isomorphism η b ⊗ σ f → τ f ⊗ η a . 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 ⊗ Id η then gives a homeomorphism Fred ( n ) ( H ) → Fred ( n ) ( H ⊗ η ), and so an isomorphismof spectra η ∗ : K σ ( X ) → K τ ( X ) . Since automorphisms of η commute with the identity map, 2-morphisms of twistingshave no effect on η ∗ . In this way the association τ K τ ( X ) can be made into afunctor on Ext X .Now we come to an important point. Suppose Y → X is the inclusion of a(full) subgroupoid of a local quotient groupoid, and H X is locally universal. ByCorollary A.34, we may then take H Y to be f ∗ H X . The bundle of spectra K τ ( Y ) isthen just the restriction of K τ ( X ). This would not be true for general groupoids and OOP GROUPS AND TWISTED K -THEORY I 29 is the reason for our restriction to local quotient groupoids. We use this restrictionproperty in the definition of the twisted K -theory of a pair. While this could beavoided, the restriction property plays a key role in the proof of excision, and doesnot appear to be easily avoided there. Definition 3.15.
Suppose that A ⊂ X is a sub-groupoid of a local quotientgroupoid, and that τ is a graded central extension of X . The twisted K -theory spec-trum 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 )) , where N is the mapping cylinder of A ⊂ XN = X ∐ A × [0 , / ∼ . Definition 3.16.
The twisted K -group K τ + n ( X, A ) is the group π − n K τ ( X, A ) = π K τ ( X, A ) n . Remark . There are several unspecified choices that go into the definition ofthe spectra K τ ( X, A ), and the induced maps between them as X , A and τ vary.It follows from Propositions A.35 and A.36 that these choices are parameterizedby (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.
Proof of Proposition 3.2: functoriality.
Most of this result was proved inthe process of defining the groups K τ + n ( X, A ). Functoriality in
Ext follows fromthe discussion of § H is a locally universal τ -twisted Hilbert bundle over ( X, A ), then π ∗ H is a locallyuniversal 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 twoendpoints. The two restriction maps are thus homotopic, and homotopy invariancefollows easily.The assertion about local equivalences is an immediate consequence of descent.As remarked at the beginning of § Twist .3.5.2.
Proof of Proposition 3.3: cohomological properties.
The long exact sequenceof a pair (assertion i)is just the long exact sequence in homotopy groups associatedto the fibration of spectra K τ ( X, A ) → K τ ( X ) → K τ ( A ) , The “wedge axiom” (part iii) is immediate from the definition. More significantis excision (part ii). In describing the proof, we will 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 thediagram of fibrations K τ ( X, A ) −−−−→ K τ ( X ) −−−−→ K τ ( A ) y y y 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 ′ denote 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 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 ofeach of these groupoids is defined in terms of sections of the bundle pulled backfrom Fred ( n ) ( H ⊗ C ). To simplify the notation a little, let’s denote all of thesepulled back bundles Fred ( n ) . Now consider the diagramΓ( N, A ; Fred ( n ) ) −−−−→ Γ( N ′ , U ∩ A ; Fred ( n ) ) y y K τ ( N, A ) n −−−−→ K τ ( N ′ , U ∩ A ) n . We are to show that the bottom row is a weak equivalence. But the top row is ahomeomorphism, and the vertical arrows are weak equivalences since 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.
The multiplication is derived fromthe pairing Fred ( n ) ( H ) × Fred ( m ) ( H ) → Fred ( n + m ) ( H ⊗ H )( S, T ) S ∗ T = S ⊗ Id + Id ⊗ T, the tensor structure on the category of twisted Hilbert space bundles describedin § Z / ℓ ( R n ) ⊗ C ℓ ( R m ) ≈ C ℓ ( R n + m Verification of part 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 elementof untwisted K -theory, with one of twisted K -theory. For example if V is a vectorbundle over X , we can choose a Hermitian metric on V , regard V as a bundle of OOP GROUPS AND TWISTED K -THEORY I 31 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 virtualdifference V − W of K ( X ) can be represented by the odd, skew-adjoint Fredholmoperator S = 0 on the graded Hilbert space whose even part is V and whose oddpart is W , and S ∗ T represents the product of V − W with the class represented by T . The assertion of Part 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 globalquotient, and τ a twisting given by a graded central extension G τ of G , and a homo-morphism ǫ : G → Z /
2. Replacing X with X × ( R ( ǫ ) , R ( ǫ ) \ { } ) and using (3.19),if necessary, we may reduce to the case ǫ = 0. Write V (1) for the summand of V = C ⊗ L ( G τ ) ⊗ ℓ 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 ) ( C n ⊗ V (1))] G which is the summand of [ S, Fred ( n ) ( C n ⊗ V )] G τ corresponding to the defining representation of T . So the result follows from thefact that Fred ( n ) ( C n ⊗ V ) is a classifying space for K − nG τ . While this is certainly well-known, we were unable to find an explicit statement in the literature. It followseasily from the case in which G is trivial. Indeed, the universal index bundle isclassified by a map to any classifying space for equivariant K -theory, and it sufficesto show that this map is a weak equivalence on the fixed point spaces for the closedsubgroups H of G . The assertion for the fixed point spaces easily reduces to themain result of [1].3.6. The Thom isomorphism, pushforward, and the Pontryagin product.
We begin with a general discussion. Let E = { E n t n −→ Ω E n +1 } ∞ n =0 be a spectrum.For a real vector space V , equipped with a positive definite metric let Ω V ( E n )denote the space of maps from the unit ball B ( V ) to E n , sending the unit sphere S ( V ) to the base point. The collection of spaces Ω V E n forms a spectrum Ω V E . Anisomorphism V ≈ R k gives an identification Ω V E n ≈ E n − k , and of Ω V E with thespectrum derived from E by simply shifting the indices. Such a spectrum is calleda “shift desuspension” of E (see [25]). Some careful organization is required toavoid encountering signs by moving loop coordinates past each other. The readeris referred to [25] for more details. Of course, for a space X one has (cid:0) Ω V E (cid:1) n ( X ) ≈ E n ( X × ( V, V \ { } )) ≈ E − k + n ( X ) . Now suppose that V is a vector bundle of dimension k over a space X . Theconstruction described above can be formed fiberwise to form a bundleΩ V E = { Ω V E n } of spectra over X . The group of vertical homotopy classes of sections(3.18) π Γ( X, Ω V E n )can then be thought of as a twisted form of [ X, E − k + n ] = E − k + n ( X ). We denotethis twisted (generalized) cohomology group E − τ V + n ( X ) . Now the group (3.18) is the group of pointed homotopy classes of maps [ X V , E n ]from the Thom complex of V to E n . This gives a tautological Thom isomorphism˜ E n ( X V ) = E n ( B ( V ) , S ( V )) ≈ ˜ E − τ V + n ( X ) . The more usual Thom isomorphisms arise when a geometric construction is used totrivialize the bundle Ω V E . Such a trivialization is usually called an “ E -orientationof V .”We now return to the case E = K , with the aim of identifying the twisting τ V with type defined in §
2. The main point is that the action of the orthogonal group O ( k ) on Ω k Fred ( n ) lifts through the Atiyah-Singer map Fred ( k + n ) → Ω k Fred ( n ) .Our discussion of this matter is inspired by the Stoltz-Teichner [31] description ofSpin-structures, and, of course Donovan-Karoubi [14].Let X be a local quotient groupoid, and V a real vector bundle over X ofdimension k , and C ℓ ( V ) the associated bundle of Clifford algebras. The bundleC ℓ ( V ) ⊗ H is a locally universal C ℓ ( V )-module. The Atiyah-Singer construction [1]gives a map Fred C ℓ ( V ) (C ℓ ( V ) ⊗ H ) → Ω V Fred (0) (C ℓ ( V ) ⊗ H )which is a weak equivalence on global sections. We can therefore trivialize thebundle of spectra Ω V K by trivializing the bundle of Clifford algebras C ℓ ( V ).Of course something weaker will also trivialize Ω V K . We don’t really need abundle isomorphism C ℓ ( V ) ≈ X × C k . We just need a way of going back andforth between C ℓ ( V )-modules and C k modules. It is enough to have a bundle ofirreducible C ℓ ( V ) − C k bimodules giving a Morita equivalence.Let M = C k , regarded as a C ℓ ( R k ) − C k -bimodule. We equip M with theHermitian metric in which the monomials in the ǫ i are orthonormal. Consider thegroup Pin c ( k ) of pairs ( t, f ) in which t : R k → R k is an orthogonal map, and f : t ∗ M → M is a unitary bimodule isomorphism. The group Pin c ( k ) is a graded central extensionof O ( k ), graded by the sign of the determinant.We now identify the twisting τ V in the terms of § E → X be the bundleof orthonormal frames in V . Thus E → X is a principal bundle with structuregroup O ( k ). Write P = E//O ( k ), ˜ P = E//
Pin c ( k ). Then P → X is a local equivalence. and ˜ P → P is a graded central extension, defining a twisting τ . Over ˜ P we can form the bundle of bimodules˜ M = ( E × M ) // Pin c ( k ) , giving a Morita equivalence between bundles of C ℓ ( V )-modules and bundles of C k -modules. In particular, H ′ = hom C ℓ ( V ) ( ˜ M , C ℓ ( V ) ⊗ H )is a locally universal τ -twisted C k -module, and the mapΓ(Fred C ℓ ( V ) (C ℓ ( V ) ⊗ H )) → Γ(Fred ( k ) ( H ′ )) T T ◦ ( − ) OOP GROUPS AND TWISTED K -THEORY I 33 is a homeomorphism. Thus the group K − τ V + n ( X ) is isomorphic to the twisted K -group K − τ + n ( X ), and, as in Donovan-Karoubi [14] we have a tautological Thomisomorphism K n ( X V ) ≈ K − τ + n ( X ) . More generally, the same construction leads to a tautological Thom isomorphism(3.19) K σ + n ( B ( V ) , S ( V )) ≈ K − τ + σ + n ( X ) , when V is a vector bundle over a groupoid X .With the Thom isomorphism in hand, one can define the pushforward, or umkehrmap in the usual way. Let f : X → Y be a map of smooth manifolds, or a mapof groupoids forming a bundle of smooth manifolds, T = T X/Y the correspondingrelative (stable) tangent bundle, and τ the twisting on X corresponding to T .Given a twisting τ on Y , and an isomorphism f ∗ τ ≈ τ one can combine thePontryagin-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 (bothwith the adjoint action) and X → Y is the multiplication map µ . In this case thetwisting τ can be taken to be the twisting we denoted g in the introduction. Since g is pulled back from pt //G , there are canonical isomorphisms µ ∗ g ≈ p ∗ g ≈ p ∗ g . We’ll just write g for any of these twistings. Suppose σ is any twisting of G//G whichis “primitive” in the sense that it comes equipped with an associative isomorphism µ ∗ σ ≈ p ∗ σ + p ∗ σ . Then the group K σ + g G ( G ) acquires a Pontryagin product K σ + g G ( G ) ⊗ K σ + g G ( G ) → K µ ∗ σ +2 g G ( G × G ) µ ! −→ K σ + g G ( G ) , making it into an algebra over K G (pt) = R ( G ).3.7. The fundamental spectral sequence.
Our basic technique of computationwill be based on a variation of the Atiyah-Hirzebruch spectral sequence, which isconstructed using the technique of Segal [30]. The identification of the E -termdepends only on the properties listed in § X is a local quotient groupoid, and write ˇ K τ + t for the presheaf on[ X ] given by ˇ K τ + t ( U ) = K τ + t ( X U ) . Write K τ + t = sh ˇ K τ + t for the associated sheaf. The limit of the Mayer-Vietoris spectral sequences associ-ated to the (hyper-)covers of [ X ] is a spectral sequence H s ([ X ] ; K τ + t ) = ⇒ K τ + s + t ( X ) . Since X admits locally contractible slices the stalk of K τ + t at a point c ∈ [ X ] is K τ + t ( X c ) ≈ ( t oddR τ ( G x ) t even , where x ∈ X is a representative of c , and G x = X ( x, x ). There is also a relative version. Suppose that A ⊂ X is a pair of groupoids, andwrite K τ + t rel for the sheaf on [ X ] associated to the presheaf U K τ + t ( X U , X [ A ] ∩ U ) . Then the limit of the Mayer-Vietoris spectral sequences associated to the hyper-covers of [ X ] gives H s ([ X ] ; K τ + t rel ) = ⇒ K τ + s + t ( X, A ) . This spectral sequence is most useful when A ⊂ X is closed, and has the propertythat for all sufficiently small U ⊂ [ X ], the map K τ + t ( X U ) → K τ + t ( X [ A ] ∩ U ) issurjective. In that case there is (for sufficiently small U ) a short exact sequence K τ + t ( X U , X [ A ] ∩ U ) → K τ + t ( X U ) → K τ + t ( X [ A ] ∩ U )and the sheaf K τ + t rel can be identified with the extension of i ∗ K τ + t by zero K τ + t rel = i ! ( K τ + t ) , where i : V ⊂ [ X ] is the inclusion of the complement of A . We will make use ofthis situation in the proof of Proposition 4.41.4. Computation of K τG ( G )The aim of this section is to compute the groups K τ + ∗ G ( G ) for non-degenerate τ . We’ll begin by considering general twistings, and adopt the non-degeneracyhypothesis as necessary. Our main results are Theorem 4.27, Corollary 4.38 andCorollary 4.39.4.1. Notation and assumptions.
We first fix some notation. Let • G be a compact connected Lie group; • g the Lie algebra of G ; • T a fixed maximal torus of G ; • t the Lie algebra of T ; • N the normalizer of T ; • W = N/T the Weyl group; • Π = ker exp : t → T ; • Λ = hom(Π , Z ), the character group of T ; • N eaff = Π ⋊ N T • W eaff = Π ⋊ W = N eaff /T , the extended affine Weyl group;The group W eaff can be identified the group of symmetries of t generated by trans-lations in Π and the reflections in W . When G is connected, the exponential map,from the orbit space t / W eaff to the space of conjugacy classes in G , is a homeomor-phism.We will make our computation for groups satisfying the equivalent conditions ofthe 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 π G is torsion free; iii) G is connected and any central extension T → G τ → G splits. OOP GROUPS AND TWISTED K -THEORY I 35 Proof:
The equivalence of (ii) and (iii) is elementary: Since G is connected,its classifying space BG is simply connected, and from the Hurewicz theorem andthe universal coefficient theorem the torsion subgroup of π G is isomorphic to thetorsion subgroup of H ( BG ), and so to the torsion subgroup of H ( BG ; Z ). Butfor any compact Lie group the odd dimensional cohomology of the classifying spaceis torsion—the real cohomology of the classifying space is in even degrees (and isgiven by invariant polynomials on the Lie algebra).The implication (ii) = ⇒ (i) is [7, (3.5)]. For the converse (i) = ⇒ (ii) we note firstthat G = Z ( e ) is connected by hypothesis. Let G ′ ⊂ G denote the derived subgroupof G , the connected Lie subgroup generated by commutators in G , and Z ⊂ G the connected component of the center of G . Set A = Z ( G ′ ) ∩ Z . Then from theprincipal fiber bundle G ′ → G → Z /A we deduce that the torsion subgroup of π G is π G ′ . We must show the latter vanishes. Now the inclusion π Z ( g ) → π G issurjective for any g ∈ G , since any centralizer contains a maximal torus T of G andthe inclusion π T → π G is surjective—the flag manifold G/T is simply connected.It follows that Z ( g ) is connected if and only if the conjugacy class G/Z ( g ) is simplyconnected. Furthermore, the conjugacy class in G of an element of G ′ equals itsconjugacy class in G ′ , from which we deduce that all conjugacy classes in G ′ areconnected and simply connected. Let f G ′ denote the simply connected (finite) coverof G ′ . Then the set of conjugacy classes in f G ′ may be identified as a bounded convexpolytope in the Lie algebra of a maximal torus, and furthermore π G ′ acts on it byaffine transformations with quotient f G ′ /G ′ ; see [16, § π G ′ actsfreely on the corresponding conjugacy class of f G ′ with quotient a conjugacy class in G ′ . Since the former is connected and the latter simply connected, it follows that π G ′ is trivial, as desired. (cid:3) The main computation.
Let X = G//G be the groupoid formed from G acting on itself by conjugation. We will compute K τ + ∗ ( X ) = K τ + ∗ G ( G ) using thespectral sequence described in § H s ( G/G ; K τ + t ) = ⇒ K τ + s + tG ( G ) . The orbit space
G/G is the space of conjugacy classes in G , which is homeomor-phic via the exponential map to t / W eaff . Our first task is to identify the sheaf K τ + t on G/G ≈ t / W eaff .Since G//G admits locally contractible slices, the stalk of K τ + t at a conjugacyclass c ∈ G/G is the twisted equivariant K -group K τ + tG ( c ) . A choice of point g ∈ c gives an identification c = G/Z ( g ), and an isomorphism(4.3) K τ + tG ( c ) ≈ K τ g + tZ ( g ) ( { g } ) ≈ ( R τ g ( Z ( g )) t even0 t oddWe have denoted by τ g the restriction of τ to { g } //Z ( g ), in order to emphasize thedependence on the choice of g . Among other things, this proves that K τ +odd = 0 . The twisting τ g corresponds to a graded central extension(4.4) T → ˜ Z ( g ) → Z ( g ) . The group Z ( g ) has T for a maximal torus, and is connected when G satisfies theequivalent conditions of Lemma 4.1. Denote(4.5) T → ˜ T → T the restriction of (4.4) to T . Then ˜ T is a maximal torus in ˜ Z ( g ). The map fromthe Weyl group of ˜ Z ( g ) to the Weyl group W g of Z ( g ) is an isomorphism, andR τ g ( Z ( g )) → R τ g ( T ) W g is an isomorphism. We can therefore re-write (4.3) as(4.6) K τ + tG ( c ) ≈ ( R τ g ( T ) W g t even0 t oddWe 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, butit doesn’t eliminate the dependence of τ g 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 toidentify the twisting τ g with τ . This amounts to considering the set of elements of t which exponentiate into c . This set admits a transitive action of W eaff , and thestabilizer of an element v is canonically isomorphic to W g where g = exp( v ).We are thus led to look at the groupoid t //T , and the action of W eaff . Writing itthis way, however, does not conveniently display the action of W eaff on the twisting τ . Following Example 2.32, we work instead with the weakly equivalent groupoid( W eaff × t ) // N eaff .Consider the map K τ + tG ( G ) → K τ + t N eaff ( W eaff × t )induced by ( W eaff × t ) // N eaff projection −−−−−−→ t // N eaff exp −−→ G//G
Since the right action of W eaff = N eaff /T commutes with the diagonal left action of N eaff on W eaff × t , the group W eaff acts on the groupoid W eaff × t // N eaff . The twisting τ is fixed by this action since it is pulled back from G//G . The left action of W eaff on ( W eaff × t ) // N eaff therefore induces a right action of W eaff on K τ + t N eaff ( t ), and theimage of K τ + tG ( G ) is invariant:(4.7) K τ + tG ( G ) → K τ + t N eaff ( W eaff × t ) W eaff . Since W eaff = N eaff /T , the map t //T −→ ( W eaff × t ) // N eaff is a local equivalence, and so gives an isomorphism K τ + t N eaff ( W eaff × t ) ≈ K τ + tT ( t ) . There is therefore an action of W eaff on K τ + tT ( t ), and we may re-write (4.7) as K τ + tG ( G ) → K τ + tT ( t ) W eaff . OOP GROUPS AND TWISTED K -THEORY I 37 For c ∈ G/G ≈ t / W eaff , let S c = { s ∈ t | exp( s ) ∈ c } be the corresponding W eaff -orbit in t . A similar discussion gives a map(4.8) K τ + tG ( c ) → K τ + tN ( W eaff × S c ) W eaff ≈ K τ + tT ( S c ) W eaff . Proposition 4.9. If G satisfies the conditions of Lemma 4.1 then the map K τ + tG ( c ) → K τ + tT ( S c ) W eaff constructed above is an isomorphism.Proof: Choose v ∈ S c , and let W v ⊂ W eaff be the stabilizer of v . We then havean identification S c ≈ W eaff /W v , and so an isomorphism K τ + tT ( S c ) W eaff ≈ K τ + tT ( { v } ) W v . Write g = exp( v ). The restriction of N eaff → G identifies W v with the Weyl groupof Z ( g ), { v } //T with { g } //T , and the restriction of τ to { v } //T with τ g . By Exam-ple 2.33 action of W v on K τ + tT ( { v } ) coincides with the action W v by conjugation.The result then follows from (4.6). (cid:3) We now identify the sheaf K τ + t . Since { } → t is an equivariant homotopyequivalence, the restriction map K τ + tT ( S c × t ) → K τ + tT ( S c × { } )is an isomorphism. Next note that the aggregate of the restriction maps to thepoints of S c gives a map from K τ + tT ( S c × t ) W eaff to the set of W eaff -equivariant maps S c → K τ + tT ( t ) , which, using the fact that W eaff acts transitively on S c , is easily checked to be anisomorphism. Write p : t → t / W eaff for the projection, and for an open U ⊂ G/G = t / W eaff set S U = p − ( U ) . Let F τ + t be the presheaf which associates to U ⊂ G/G the set of locally constant W eaff -equivariant maps S U → K τ + tT ( t ) . There is then a map of presheavesˇ K τ + t → F τ + t , hence a map of sheaves(4.10) K τ + t → F τ + t . Corollary 4.11.
The map (4.10) is an isomorphism.
Proof:
Proposition 4.9 implies that (4.10) is an isomorphism of stalks, hence anisomorphism. (cid:3)
We now re-interpret the sheaf F in a form more suitable to describing its coho-mology. Since Twist t //T → Twist { } //T is an equivalence of categories, the restriction of τ to t //T corresponds to a gradedcentral extension(4.12) T → T τ → T equipped with an action of W eaff . The W eaff -action fixes T and acts on T throughits quotient W , the Weyl group. Write Λ τ for the set of splittings of (4.12). Notethat Λ τ is a torsor for Λ and inherits a compatible W eaff action from (4.12).By Proposition 3.5 the group K τ +0 T ( t ) ≈ K τ +0 T ( { } )may be identified with with the set of compactly supported functions on Λ τ withvalues in Z . We will see shortly that the action of W eaff is the combination of itsnatural action on Λ τ and an action on Z given by a homomorphism ǫ : W eaff → Z /
2. Writing Z ( ǫ ) for the sign representation associated to ǫ , we then have anisomorphism of W eaff -modules(4.13) K τ +0 T ( { } ) ≈ Hom c (Λ τ , Z ( ǫ )) . To verify the claim about the action first note that the automorphism group ofthe restriction of τ to t //T is H ( BT ; Z ) × H ( BT ; Z / ≈ Λ × Z / ≈ R ( T ) × ≈ K T (pt) × By Part ii) of Proposition 3.4, the factor Λ acts on K τ +0 T ( { } ) through its naturalaction on Λ τ , while the Z / W eaff = Π ⋊ W , and the action of W eaff on K τ +0 T ( { } ) is determined by itsrestriction to Π and W . The group Π acts trivially on T and so it acts on K τ +0 T ( { } )through a homomorphism Π ( b,ǫ Π ) −−−−→ Λ × Z / . The group W does act on T , and so on H ( BT ; Z ) × H ( BT ; Z / ≈ Λ × Z / , by the product of the natural (reflection) action on Λ and the trivial action on Z / W eaff to W is therefore determined by a crossedhomomorphism W → Λcompatible with b , and an ordinary homomorphism ǫ W : W → Z /
2. The maps ǫ Π and ǫ W combine to give the desired map ǫ : W eaff → Z /
2, while the map b : Π → Λand the crossed homomorphism W → Λ correspond to the natural action of W eaff on Λ τ . This verifies the isomorphism (4.13) of W eaff -modules.We can now give a useful description of F . First recall a construction. Suppose X is a space equipped with an action of a group Γ, and that G is an equivariantsheaf on X . Write p : X → X/ Γ for the projection to the orbit space. There isthen a sheaf, ( p ∗ G ) Γ on X/ Γ whose value on an open set V is the set of Γ-invariantelements of G ( p − V ). A very simple situation is when G is the constant sheaf Z . OOP GROUPS AND TWISTED K -THEORY I 39 In that case ( p ∗ G ) Γ is again the constant sheaf Z . This will be useful in the proofof Proposition 4.18 below. Corollary 4.14.
Write ˜ t = t × W eaff Λ τ and let p : t × Λ τ → ˜ t and f : ˜ t → t / W eaff denote the projections. There is a canonical isomorphism F τ +0 ≈ f c ∗ (cid:16) p ∗ Z ( ǫ ) W eaff (cid:17) , where f c ∗ denotes pushforward with proper supports. (cid:3) To go further we need to make an assumption.
Assumption 4.15.
The twisting τ is non-degenerate in the sense that b is amonomorphism. In terms of the classification of twistings, this is equivalent to requiring that theimage of the isomorphism class of τ in H T ( T ; R ) ≈ H ( T ; R ) ⊗ H ( T ; R )is a non-degenerate bilinear form.Next note Lemma 4.16.
The map ǫ W : W → Z / is trivial.Proof: The homomorphism in question corresponds to the element in H W (pt) = H ( BW ; Z /
2) given by restricting the isomorphism class of the twisting along H G ( G ; Z / × H G ( G ; Z ) → H G ( G ; Z / → H G ( { e } ; Z / → H N ( { e } ; Z / ≈ H W ( { e } ; Z / . Since G is assumed to be connected H G (pt) = 0 and the result follows. (cid:3) Corollary 4.17.
There is an isomorphism Z ≈ Z ( ǫ ) of equivariant sheaves on t × Λ τ .Proof: The sheaf Z ( ǫ ) is classified by the element˜ ǫ ∈ H W eaff ( t × Λ τ ; Z / ǫ ∈ H W eaff (pt; Z / ǫ to W is trivial. By Assumption 4.15, the group Π acts freely on Λ τ , so the restrictionof ˜ ǫ to H W eaff ( t × Λ τ ; Z /
2) is also trivial. This proves that ˜ ǫ = 0. (cid:3) Proposition 4.18.
There is an isomorphism F τ +0 ≈ f c ∗ ( Z ) , where f : ˜ t = t × W eaff Λ τ → t /W is the projection and f c ∗ denotes pushforward withproper supports. Proof:
By Corollaries 4.17 and 4.14 there is an isomorphism F τ +0 ≈ f c ∗ (cid:16) p ∗ Z W eaff (cid:17) , so the result follows from the fact that p ∗ Z W eaff ≈ Z . (cid:3) Finally, using the existence of contractible local slices we can describe the coho-mology of K τ +0 ≈ F τ +0 . Lemma 4.19.
The edge homomorphism of the Leray spectral sequence for f is anisomorphism H ∗ ( t / W eaff ; K τ ) ≈ H ∗ c (˜ t ; Z ) , where H ∗ c denotes cohomology with compact supports. (cid:3) To calculate H ∗ c (˜ t ; Z ) we need to understand the structure of ˜ t . This amounts todescribing more carefully the action of W eaff on Λ τ . Lemma 4.20.
There exists an element λ ∈ Λ τ fixed by W .Proof: The inclusion { } //T ⊂ t //T is equivariant for the action of W . Wecan therefore study the action of W on the central extension of T defined by therestriction of τ to { } //T . Now τ started out as a twisting of G//G , so our twistingof { } //T is the restriction of a twisting τ G of { e } //G . Moreover, the action of W is derived from the action of inner automorphisms of G on τ G . Now the twisting τ G corresponds to a central extension T → G τ → G. By our assumptions on G (Lemma 4.1), this central extension splits. Choose asplitting(4.21) G τ → T and let λ ∈ Λ τ be the composition˜ T → G τ → T . Since 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 ˜ G which normalize˜ T . The claim follows. (cid:3) Remark . Any two choices of λ differ by a character of G , so the element λ is unique if an only if the character group of G is trivial. Since we’ve assumed that G is connected and π G is torsion free, this is in turn equivalent to requiring that G be simply connected. Remark . A primitive twisting τ comes equipped with a trivialization of itsrestriction to { e } //G , or in other words a splitting of the graded central extension G τ → G . A primitive twisting therefore comes equipped with a canonical choice of λ .Using a fixed choice of λ , we can identify Λ τ with Λ as a W -space. To sum up,we can make an identification Λ τ ≈ Λ, the action of Π is given by a W -equivarianthomomorphism Π → Λ, and the W -action is the natural one on Λ. OOP GROUPS AND TWISTED K -THEORY I 41 Lemma 4.24.
When τ is non-degenerate the W eaff -set Λ τ admits an (equivariant)embedding in t . There are finitely many W eaff -orbit in Λ τ , and each orbits is of theform W eaff /W c , with ( W c , t ) a finite (affine) reflection group.Proof: Since b is a monomorphism the map t = Π ⊗ R → Λ ⊗ R is an isomor-phism. The first assertion now follows from our identification of Λ τ with Λ. As forthe finiteness of the number of orbits, since b is a monomorphism, the group Λ / Π isfinite, and there are already only finitely many Π-orbits in Λ τ . The remaining as-sertions follow from standard facts about the action of W eaff on t (Propositions 4.46and 4.47 below). (cid:3) Corollary 4.25.
When τ is non-degenerate, there is a homeomorphism ˜ t ≡ a s ∈ S t /W s with S finite, and W s a finite reflection group of isometries of t . Moreover t /W s ≡ R n × [0 , ∞ ) n with n = 0 if and only if W s is trivial.Proof: This is immediate from Lemma 4.24 above and Proposition 4.47 below. (cid:3)
Since H ∗ c ([0 , ∞ ); Z ) = 0 and H ∗ c ( R ; Z ) = ( Z ∗ = 10 otherwise,the Kunneth formula gives H ∗ c ( R n × [0 , ∞ ) n ; Z ) = ( Z n = 0 and ∗ = n . In summary, we have
Proposition 4.26.
The cohomology group H ∗ c (˜ t ; Z ) is zero unless ∗ = n , and H nc (˜ t ; Z ) is isomorphic to the free abelian group on the set of free W eaff -orbits in Λ τ . More functorially, H nc (˜ t ; Z ) ≈ Hom W eaff (Λ τ , H nc ( t ) ⊗ Z ( ǫ )) . (cid:3) Proposition 4.26 implies that the spectral sequence (4.2) collapses, giving
Theorem 4.27.
Suppose that G is a Lie group of rank n satisfying the conditionsof Lemma 4.1, and that τ is a non-degenerate twisting of G//G , classified by [ τ ] ∈ H G ( G ; Z ) × H G ( G ; Z / . The restriction of τ to pt //T determines a central extension (4.28) T → T τ → T with an action of W eaff . Write Λ τ for the set of splittings of (4.28) , ǫ : W eaff → Z / for the map corresponding to the restriction of [ τ ] to H N ( T ; Z / ≈ H ( W eaff ; Z / , and Z ( ǫ ) for the associated sign representation. Then K τ + n +1 G ( G ) = 0 , and thetwisted K -group K τ + nG ( G ) is given by K τ + nG ( G ) ≈ Hom W eaff (Λ τ , H nc ( t ) ⊗ Z ( ǫ )) , which can be identified with the free abelian group on the set of free W eaff -orbits in Λ τ , after choosing a point in each free orbit. This isomorphism is natural in thesense that if i : H ⊂ G is a subgroup of rank n also satisfying the conditions ofLemma 4.1, then the restriction map K τ + nG ( G ) → K τ + nH ( H ) is given by the inclusion Hom W eaff ( G ) (Λ τ , H nc ( t ) ⊗ Z ( ǫ )) ⊂ Hom W eaff ( H ) (Λ τ , H nc ( t ) ⊗ Z ( ǫ )) . (cid:3) Remark . In Theorem 4.27 the group W eaff acts on H nc ( t ) through the action of W on t . The reflections thus act by ( −
1) and a choice of orientation on t identifies H nc ( t ) with the usual sign representation of W on Z .The group K τ + nG ( G ) is a module over R ( G ). Our next goal is to identify thismodule 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 supportedfunctions Hom c (Λ , Z ). The algebra Hom c (Λ , Z ) acts on Hom(Λ τ , H nc ( t ) ⊗ Z ( ǫ ))by convolution, and one easily checks that the W -invariant elements preserve the W eaff -equivariant functions. Proposition 4.30.
Under the identification K τ + nG ( G ) ≈ Hom W eaff (Λ τ , H nc ( t ) ⊗ Z ( ǫ )) the action of R ( G ) ≈ Hom c (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 andusing Theorem 4.27. (cid:3) Proposition 4.30 leads to a very useful description of K τ + nG ( G ). Choose anorientation of t and hence an identification H nc ( t ) ≈ Z of abelian groups. Bydefinition, the elements of Λ τ are characters of T τ , all of which restrict to thedefining character of T . To a function f ∈ Hom Π (Λ τ , H nc ( t ) ⊗ Z ( ǫ )) ≈ Hom Π (Λ τ , Z ( ǫ ))we associate the series(4.31) δ f = X λ ∈ Λ τ f ( λ ) λ − , OOP GROUPS AND TWISTED K -THEORY I 43 which is the Fourier expansion of the distribution on T τ satisfying δ f ( λ ) = f ( λ ).Our next aim is to work out more explicitly which distribution it is, especially when f comes from an element ofHom W eaff (Λ τ , H nc ( t ) ⊗ Z ( ǫ )) ≈ Hom W eaff (Λ τ , Z ( ǫ )) . Since all of the characters λ restrict to the defining character of the central T ,we’ll think of the distribution δ f as acting on the space of functions g : T τ → C satisfying g ( ζ v ) = ζg ( v ) for ζ ∈ T . This space is the space of sections of a suitablecomplex line bundle L τ over T . The character χ of a representation of G is afunction on T , and action of χ on δ f is given by χ · δ f ( g ) = δ f ( g · χ ) . Since Π and Λ are duals, we have hom(Π , Z /
2) = Λ ⊗ Z /
2, and we may regard ǫ Π as an element of Λ / λ ǫ = ǫ Π ∈ Λ / Λ ⊂ Λ ⊗ R / Z . Thinking of Π as the character group of Λ ⊗ R / Z , the function ǫ Π : Π → Z / λ ǫ : ǫ Π ( π ) = π ( λ ǫ ) . Since ǫ Π is W -invariant, so is λ ǫ .From the embedding b : Π ⊂ Λ we get a map b : T = Π ⊗ R / Z → Λ ⊗ R / Z . We’ll write F = Λ / Π for the kernel of this map, and F ǫ for the inverse image of λ ǫ .Set F τ = Λ τ / Π . The elements of F τ can be interpreted as sections of the restriction of L τ to F .Finally, let F ǫ, reg ⊂ F ǫ and F τ reg ⊂ F τ be the subsets consisting of elements onwhich the Weyl group W acts freely. Proposition 4.32.
For f ∈ Hom Π (Λ τ , H nc ( t ) ⊗ Z ( ǫ )) ≈ Hom Π (Λ τ , Z ( ǫ )) The value of the distribution δ f on a section g of L τ is given by (4.33) δ f ( g ) = 1 | F | X ( λ,x ) ∈ F τ × F ǫ f ( λ ) λ − ( x ) g ( x ) When f is W eaff -invariant, then δ f ( g ) = 1 | F | X ( λ,x ) ∈ F τ reg × F ǫ, reg f ( λ ) λ − ( x ) g ( x ) Proof:
Let’s first check that (4.33) is well-defined. Under λ λπ π ∈ Π , the term f ( λ ) λ − ( x ) g ( x ) gets sent to f ( λπ )( λπ ) − ( x ) g ( x ) = ǫ ( π ) f ( λ ) λ − ( x ) π − ( x ) g ( x )= ǫ ( π ) f ( λ ) λ − ( x ) ǫ ( π − ) g ( x )= f ( λ ) λ − ( x ) g ( x )so f ( λ ) λ − ( x ) g ( x ) does indeed depend only on the Π-coset of λ .To establish (4.33), it suffices by linearity to consider the case in which δ f is ofthe form δ f = λ − · δ Π , with λ ∈ Λ τ , δ Π = X π ∈ Π ǫ ( π ) π − , and g is an element of Λ τ . In this case f vanishes off of the Π-orbit through λ , and f ( λ ) = 1. The sum (4.33) is then(4.34) 1 | F | X x ∈ F ǫ λ − ( x ) g ( x ) . By definition, for η ∈ Π, δ Π ( η ) = η ( λ ǫ ) = 1 | F | X x ∈ F ǫ η ( x ) . For η ∈ Λ \ Π, there is an a ∈ F with η ( a ) = 1. In that case1 | F | X x ∈ F ǫ η ( x ) = 1 | F | X x ∈ F ǫ η ( ax ) = η ( a ) 1 | F | X x ∈ F ǫ η ( x ) , so 1 | F | X x ∈ F ǫ η ( x ) = 0 . It follows that for every η ∈ Λ, δ Π ( η ) = 1 | F | X x ∈ F ǫ η ( x ) . Now suppose g ∈ Λ τ . Then δ f ( g ) = δ Π ( λ − g ) = 1 | F | X x ∈ F ǫ λ − ( x ) g ( x ) , which is (4.34). This proves the first assertion of Proposition 4.32.For the second assertion, note that if λ ∈ F τ reg is fixed by an element of W it isfixed by an element w ∈ W which is a reflection. By Weyl-equivariance, we have f ( λ ) = f ( w λ ) = w · f ( λ ) = − f ( λ ), and so f ( λ ) = 0. This gives δ f ( g ) = 1 | F | X ( λ,x ) ∈ F τ reg × F f ( λ ) λ − ( x ) g ( x ) . OOP GROUPS AND TWISTED K -THEORY I 45 If x ∈ F is an element fixed by a reflection w ∈ W then X λ ∈ F τ reg f ( λ ) λ − ( x ) g ( x ) = X λ ∈ F τ reg f ( λ ) λ − ( w · x ) g ( x )= X λ ∈ F τ reg f ( λ )( λ w ) − ( x ) g ( x )= X λ ∈ F τ reg f ( λ w ) λ − ( x ) g ( x )= − X λ ∈ F τ reg f ( λ ) λ − ( x ) g ( x )so the terms involving such an x sum to zero, and δ f ( g ) = 1 | F | X ( λ,x ) ∈ F τ reg × F ǫ, reg f ( λ ) λ − ( x ) g ( x ) (cid:3) Let I τ ⊂ R ( G ) be the ideal consisting of virtual representations whose charactervanishes on the elements of F ǫ, reg . Corollary 4.35.
The ideal I τ annihilates K τ + ∗ G ( G ) .Proof: Write χ for the character of an element of I τ . For f ∈ Hom W eaff (Λ τ , H nc ( t ) ⊗ Z ( ǫ )) we have, by Proposition 4.32 χ δ f ( g ) = δ f ( g · χ ) = X ( λ,x ) ∈ F τ reg × F reg f ( λ ) λ − ( x ) g ( x ) χ ( x ) = 0 . (cid:3) Remark . The conjugacy classes in G of the elements in F ǫ, reg are known asthe Verlinde conjugacy classes , and the ideal I τ as the Verlinde ideal . Proposition 4.37.
The R ( G ) -module K τ + nG ( G ) is cyclic.Proof: Using Lemma 4.20 choose a W eaff -equivariant isomorphism Λ τ ≈ Λ,and an orientation of t giving an isomorphism H nc ( t ) ≈ Z . We can then identify K τ + nG ( G ) with Hom W eaff (Λ , Z ( ǫ )) , though we remind the reader that W eaff acts on Z through its sign representation.We’ll continue the convention of writing elements f ∈ Hom W eaff (Λ , Z )as Fourier series X f ( λ ) λ − . Set δ Π = X π ∈ Π ǫ ( π ) π − , and for λ ∈ Λ write a ( λ ) = X w ∈ W ( − w w · λ. Then the elements a ( λ ) ∗ δ Π span Hom W eaff (Λ , Z ( ǫ )). Since π G is torsion-free, there is an exact sequence G ′ → G → J where J is a torus, and G ′ is simply connected. The character group of J is thesubgroup Λ W of Weyl-invariant elements of Λ, and the weight lattice for G ′ is thequotient Λ / Λ W . Choose a Weyl chamber for G and let ρ ∈ Λ ⊗ Q be 1 / G (which we will write as a product of square roots of elementsin our Fourier series notation). Since J is a torus, the image ρ ′ of ρ in Λ / Λ W ⊗ Q is 1 / G ′ , which since G ′ is simply connected, liesin Λ / Λ W . Let ˜ ρ ∈ Λ be any element congruent to ρ ′ modulo Λ W . Claim: for any λ ∈ Λ, the ratio a ( λ ) a (˜ ρ )is the character of a (virtual) representation. The claim shows that the class cor-responding to a (˜ ρ ) · δ Π is an R ( G )-module generator of K τ + nG ( G ). For the claim,first note that the element µ = ρ/ ˜ ρ is W -invariant (and is in fact the square rootof a character of J ). It follows from the Weyl character formula that a (( λ ˜ ρ − ) · ρ ) a ( ρ )is, up to sign, the character of an irreducible representation. But then a ( λ ˜ ρ − ρ ) a ( ρ ) = a ( λ µ ) a ( ρ ) = µ a ( λ ) a ( ρ )= a ( λ ) a ( µ − ρ ) = a ( λ ) a (˜ ρ ) . (cid:3) Corollary 4.38.
Let U ∈ K τ + nG ( G ) be the class corresponding to a (˜ ρ ) · δ Π . Themap “multiplication by U ” is an isomorphism R ( G ) /I τ → K τ + nG ( G ) of R ( G ) -modules.Proof: That the map factors through the quotient by I τ is Corollary 4.35, andthat it is surjective is Proposition 4.37. The result now follows from the fact thatboth sides are free of rank equal to the number of free W -orbits in A ǫ, reg (ie, thenumber of Verlinde conjugacy classes). (cid:3) As described at the end of § τ is primitive the R ( G )-module K τ + nG ( G )acquires the structure of an R ( G )-algebra Corollary 4.39.
When τ is primitive, there is a canonical algebra isomorphism K τ + nG ( G ) ≈ R ( G ) /I τ . (cid:3) OOP GROUPS AND TWISTED K -THEORY I 47 Remark . When τ is primitive, the pushforward map K τG ( e ) → K τ + nG is a ringhomomorphism. Being primitive, the restriction of τ to { e } //G comes equippedwith a trivialization and so K τG ( e ) ≈ R ( G ).The isomorphisms of Corollaries 4.38 and 4.39 are proved after tensoring withthe complex numbers in [21], where the distributions δ f are related to the Kacnumerator at q = 1. We refer the reader to § § G = T is a torus of dimension n .The group K τT ( { e } ) is the free abelian group on Λ τ , and the pushforward map K τT ( { e } ) → K τ + nT ( T ) is defined. Proposition 4.41.
The pushforward map i ! : K τT ( { e } ) → K τ + nT ( T ) sends the class corresponding to λ ∈ Λ τ to the class in K τ + nT ( T ) ≈ Hom Π (Λ τ , Z ( ǫ )) corresponding to the distribution with Fourier expansion λ − X π ∈ Π ǫ ( π ) π − . Proof:
The pushforward map is the composition of the Thom isomorphism K τ +0 T ( { e } ) → K τ + nT ( t , t \ { } ) ≈ K τ + nT ( T, T \ { e } )with the restriction map(4.42) K τ + nT ( T, T \ { e } ) → K τ + nT ( T ) . We wish to compute these maps using the spectral sequence for relative twisted K -theory described at the end of § T \ { e } and t \ { } by the smaller T \ B e and t \ { B } , where B e and B are small open balls containing e and 0 respectively. This puts us in the situationdescribed at the end of of § K τ + t rel works out to be the extensionby zero of the restriction of K τ + t to B e .Applying the spectral sequence argument of this section to the pairs ( t , t \ B )and ( T, T \ B e ) gives isomorphisms(4.43) K τ + nT ( T, T \ B e ) ≈ hom Π (Λ τ , H nc ( t , t \ B Π ) ⊗ Z ( ǫ )) K τ + nT ( t , t \ B ) ≈ hom c (Λ τ , H nc ( t , t \ B ) ⊗ Z ( ǫ )) , where B Π ⊂ t is the inverse image of B e under the exponential map. The sameargument identifies the restriction mapping (4.42) with the map induced by H nc ( t , t \ B Π ) → H nc ( t ) , and the isomorphism K τ + nT ( T, T \ B e ) ≈ K τ + nT ( t , t \ B )with the mapHom Π (Λ τ , H nc ( t , t \ B Π ) ⊗ Z ( ǫ )) → Hom c (Λ τ , H nc ( t , t \ B ) ⊗ Z ( ǫ ))which first forgets the Π-action and then uses H nc ( t , t \ B Π ) → H nc ( ¯ B , B ) ≈ H nc ( t , t \ B ) . Finally, the isomorphism K τ + nT ( t , t \ B ) ≈ K τT (pt) ⊗ K n ( t , t \ B )shows that the Thom isomorphism is simply the tensor product of the identity mapwith suspension isomorphism K (pt) → K n ( t , \ B ) (which uses the orientation of t ). In terms of (4.43) this means that the Thom isomorphism K τT (pt) ≈ Hom c (Λ τ , Z ( ǫ )) → K τT ( t , t \ B ) ≈ hom c (Λ τ , H nc ( t , t \ B ) ⊗ Z ( ǫ )) , is simply the map derived from the suspension isomorphism H (pt) ≈ H nc ( t , t \ B ) . The result follows easily from this. (cid:3)
The action of W eaff on t . We summarize here some standard facts aboutabout 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 charactergroup of T and R for the set of roots. Following Bourbaki, write N ( T, R ) for thesubgroup of t consisting of elements on which the roots vanish modulo 2 π Z . Thereis a short exact sequence N ( T, R ) Π ։ π G. Let H be the set of hyperplanes forming the diagram of G . Thus H = { H k,α | k ∈ Z , α ∈ R } , where H k,α = { x ∈ t | α ( x ) = 2 π k } . The collection H is locally finite in the sense that each s ∈ t has a neighborhoodmeeting only finitely many hyperplanes in H . The affine Weyl group is the group W aff be the group generated by reflections in the hyperplanes H k,α ∈ H . It hasthe structure N ( T, R ) ⋊ W. Proposition 4.44.
Let x ∈ t . The stabilizer of x in W aff is the finite reflectiongroup generated by reflections through the hyperplanes H k,α containing x . (cid:3) Write W eaff = L ⋊ W . There is a short exact sequence(4.45) W aff W eaff ։ π G. Proposition 4.46.
Let x ∈ t . If π G is torsion free, then the stabilizer of x inW eaff coincides with the stabilizer of x in W aff . It is therefore the finite reflectiongroup generated by reflections through the hyperplanes H k,α containing x .Proof: Write W x for the stabilizer of x in W eaff . The image of W x in t ⋊ W isconjugate to a subgroup of W , and so W x is finite. By assumption π G has no non-trivial finite subgroups. The exact sequence (4.45) then shows that W x ⊂ W aff .The result then follows from Proposition 4.44. (cid:3) Write R ≥ = [0 , ∞ ). Proposition 4.47.
Suppose that ( W, V ) is a finite reflection group. The orbit space V /W is homeomorphic to R n × R n ≥ . The group W is generated by n reflections.In particular, if W is non-trivial, then n = 0 . OOP GROUPS AND TWISTED K -THEORY I 49 Proof:
This follows immediately from the Theorems on pages 20 and 24 of [11]. (cid:3)
Appendix A. Groupoids
We remind the reader that we are assuming throughout this paper that, unlessotherwise specified, all spaces are locally contractible, paracompact and completelyregular. These assumptions implies the existence of partitions of unity [13] andlocally contractible slices through actions of compact Lie groups [27, 28].A.1.
Definition and First Properties. A groupoid is a category in which allmorphisms are isomorphism. We will consider groupoids in the category of topo-logical spaces. Thus a groupoid X = ( X , X ) consists of a space X of objects, aspace X of morphisms, and map “identity map” X → X , a pair of maps “do-main” and “range” X → X , an associative composition law X × X X → X ,and an “inverse” map X → X . Write X n = X × X · · · × X X for the space of n -tuples of composeable maps. Then the collection { X n } is a simplicial space. The i th face map d i : X n → X n − is given by d i ( f , . . . f n ) = ( f , . . . , f n ) i = 0( f , . . . , f i ◦ f i +1 , . . . , f n ) 0 < i < n ( f , . . . , f n − ) i = n Even though a groupoid is a special kind of simplicial space, we’ll refer to thesimplicial space as the nerve of X = ( X , X ) and write X • . Finally, we let | X | = a n X n × ∆ n / ∼ denote the geometric realization of X • . Example
A.1 . (cf Segal [30]) Suppose that G is a topological group acting on aspace X . Then the pair ( G, X ) forms a groupoid with space of objects X and inwhich a morphism from x to y is an element of g for which g · x = y . In this case X = X and X = G × X . The composition law is given by the multiplication in G . We will write X//G for this groupoid.
Example
A.2 . (Segal [30]) Suppose that X is a space and U = { U i } is a coveringof X . The nerve of the covering U is the nerve of a groupoid. Indeed, let N U bethe category whose objects are pairs ( U i , x ) with U i ∈ U and x ∈ U i and in whicha morphism from ( U i , x ) to ( U j , y ) is an element w ∈ U i × X U j whose projectionto U i is x and whose projection to U j is y . Then N U is a groupoid. If U i and U j are open subsets of X then such a map exists if and only if x = y , in which case itis unique. Writing X = ` U i , then X n = X × X · · · × X X , and the nerve of thisgroupoid is just the nerve of the covering U . Definition A.3.
A map of groupoids F : X → Y is an equivalence if it is fullyfaithful and essentially surjective; that is, if every object y ∈ Y is isomorphic toone of the form F x , and if for every a, b ∈ X the map F : X ( a, b ) → Y ( F a, F b )is a homeomorphism.
An equivalence of ordinary categories automatically admits an inverse (up tonatural isomorphism). Examples A.6 and A.8 below show that the same is notnecessarily true of an equivalence of topological categories, or groupoids. Requiringthe 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 withthe additional property that each y ∈ Y has a neighborhood U admitting a lift inthe diagram ˜ X / / (cid:15) (cid:15) X (cid:15) (cid:15) Y / / domain (cid:15) (cid:15) Y U G G / / Y in which the square is Cartesian. Remark
A.5 . The term local equivalence derives from thinking of a groupoid X as defining a sheaf U X ( U ) on the category of topological spaces. An equiva-lence X → Y has a globally defined inverse if and only if for every space U themap X ( U ) → Y ( U ) is an equivalence. As one easily checks, a map X → Y isa local equivalence if and only if it is an equivalence on stalks. We will say thattwo groupoids X and Y as being weakly equivalent , if there is a diagram of localequivalences X ← Z → Y. Example
A.6 . If U is an open covering of a space X , then the map N U → X is a local equivalence. More generally, if U → V is a map of coverings of X , then N U → N V 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 , themap lim −→ U X ( U ) → lim −→ U Y ( U )is an equivalence of groupoids, where U ranges over all coverings of S . Stated moresuccinctly, a map of groupoids is a local equivalence if and only if the correspondingmap 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, the map of groupoidspt //H → ( G/H ) //G is a local equivalence. OOP GROUPS AND TWISTED K -THEORY I 51 The fiber product of functors P i (cid:24) (cid:24) Q j (cid:6) (cid:6) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X 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 a functors S p −→ P, S q −→ Q, S x −→ X and natural isomorphisms i ◦ p → x ← j ◦ q. The groupoid P × X Q is usually called fiber product of P and Q over X , eventhough strictly speaking it is a kind of homotopy fiber product and not the cate-gorical fiber product. We will also say that the morphism P × X Q → Q is obtainedfrom P → X by change of base along j : Q → X . A natural transformation T : j → j gives a natural isomorphism between the groupoids obtained by changeof base along j and j .One easily checks that the class of local equivalences is stable under compositionand change of base. Consequently, if P → X and Q → X are both local equiv-alences, so is P × X Q → X . Using Example A.7 one easily checks that if two ofthree maps in a composition are local equivalences so is the third. Definition A.10.
The 2-category Cov X is the category whose objects are localequivalences p : P → X and in which a 1-morphism from p : P → X to p : P → X consists of a functor F : P → P and a natural transformation T : p → p ◦ F making P F / / p (cid:25) (cid:25) P p (cid:5) (cid:5) (cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10) T = ⇒ X commute. A 2-morphism ( F , T ) → ( F , T ) is a natural transformation η : F → F for which T = p η ◦ T .We will denote by Cov X the 1-category quotient of Cov X . The objects of Cov X are those of Cov X , and Cov X ( a, b ) is the set of isomorphism classes in Cov X ( a, b ).We will see that Cov X and Cov X are not that different from each other. Lemma A.11.
For every a, b ∈ Cov X , the category Cov X ( a, b ) is a codiscretegroupoid: there is a unique morphism between any two objects. Proof:
Write a = ( F , T ) and b = ( F , T ) P F ,F / / p (cid:26) (cid:26) P p (cid:4) (cid:4) (cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9) T ,T = ⇒ X .
A morphism (natural transformation) η ∈ Cov X ( a, b ) associates to x ∈ P a map η x : F x → F x whose image p η x : p F x → p F x is prescribed to fit into the diagram of isomorphisms p x T x (cid:127) (cid:127) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) T x (cid:31) (cid:31) ??????? p F x p η x / / p F x. The map p η x is therefore forced to be ( T x ) ◦ ( T x ) − , and so η x is uniquelydetermined since P → X is an equivalence. (cid:3) Corollary A.12.
The -category quotient Cov X is a (co-)directed class.Proof: Suppose that p i : P i → X , i = 1 , X . Thegroupoid P = P × X P comes equipped with maps P → P and P → P .If f, g : P → Q are two morphisms in Cov X there is, by Lemma A.11, a unique2-morphism relating them, and so in fact f = g in Cov X (cid:3) A.2.
Further Properties of Groupoids.
We now turn to several constructionswhich 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 underlyingstack.A.2.1.
Point set topology of groupoids.
The orbit space or coarse moduli space of agroupoid X is the space of isomorphism classes of objects, topologized as a quotientspace of X . We denote the coarse moduli space of X by [ X ]. At this level ofgenerality, the space [ X ] can be somewhat pathological, and without some furtherassumptions might not be in our class of locally contractible, paracompact, andcompletely regular spaces. When X = S//G , then [ X ] is the orbit space S/G , andin that case we will 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 X S the full sub-groupoid of X consistingof objects in the isomorphism class of S . There is a one to one correspondencebetween full sub-groupoids A ⊂ X containing every object in their X -isomorphismclass and subspaces [ A ] of [ X ]. With this we transport many notions from the pointset topology of spaces to the context of groupoids. When S is closed (resp. open)we will say that X S is a closed (resp. open) subgroupoid of X . We can speak ofthe interior and closure of a full subgroupoid. By an open covering of a groupoid,we mean an open covering { S α } of [ X ], in which case the collection { X S α } forms acovering of X by open sub-groupoids. OOP GROUPS AND TWISTED K -THEORY I 53 More generally, if f : S → [ X ] is a map, we can form a groupoid X S with objectsthe pairs ( s, x ) ∈ S × X 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 X S is the groupoid whose nerve fits into a pull-back square( X S ) • −−−−→ X • y y S −−−−→ [ X ] . We will say that X S is defined by pullback from the map S → [ X ]. With this we cantransport many of the maneuvers of homotopy theory to the context of groupoids.For instance if S = [ X ] × I , and f is the projection, then X S 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 twosets S , S whose interiors cover. Write U i = X S i . The U i are (full) subgroupoidswhose interiors cover X . Let N denote the groupoid constructed from the mapcyl ( S ← S ∩ S → S ) → [ X ] . It is the double mapping cylinder of U ← U ∩ U → U . Following Segal [30], a partition of unity { φ , φ } subordinate to the covering { S , S } , defines a map [ X ] → cyl ( S ← S ∩ S → S ) . The composite [ X ] cyl ( S ← S ∩ S → S ) → [ X ]is the identity, and so we get functors X → N → X whose composite is the identity. On the other hand, the compositecyl ( S ← S ∩ S → S ) [ X ] → cyl ( S ← S ∩ S → S )is homotopic to the identity, by a homotopy (the obvious linear homotopy) thatpreserves the map to [ X ] (it is a homotopy in the category of spaces over [ X ]). Thishomotopy then defines by pullback, a homotopy N × ∆ → 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 decomposedin a way especially well-suited for constructing sequences of Mayer-Vietoris type.A groupoid X has proper diagonal if the map(A.13) X , range) −−−−−−−−−→ X × X is proper, and [ X ] is Hausdorff. If (A.13) is proper and X is Hausdorff then X isproper. 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 which is related by a chain of localequivalences 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 admittinga countable open cover { U α } with the property that each X U α is weakly equivalentto a groupoid of the form S//G with G a compact Lie group, and S a Hausdorffspace. If Y → X is a local equivalence, then Y is a local quotient groupoid if andonly if X is, so the property of being a local quotient is intrinsic to the underlyingstack.If X is a local quotient groupoid, then [ X ] is paracompact, locally contractibleand completely regular. If X is a local quotient groupoid with the property thatthere is at most one map between any two objects (ie X → X × X is an inclusion),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 groupoidis a local quotient groupoid. In particular, any (full) subgroupoid of a local quotientgroupoid is a local quotient groupoid, and the mapping cylinder of a map X S → X constructed by pullback along a map S → [ X ] to the orbit space of a local quotientgroupoid is a local quotient groupoid. A.3.
Fiber bundles over groupoids and descent.
In this section we define thecategory of fiber bundles over a groupoid, and show that a local equivalence givesan equivalence of categories of fiber bundles (Proposition A.18). Thus the categoryof fiber bundles over a groupoid is intrinsic to the underlying stack.A fiber bundle over a groupoid X = ( X , X ) consists of a fiber bundle P on X together with identifications of certain pullbacks to X n for various n . We introducesome convenient notation for describing these pulled back bundles.Let’s denote a typical point of X n by x f −→ . . . f n −→ x n . Given a bundle P → X we’ll write P x i for the pullback of P along the map X n → X ( x → · · · → x n ) x i . Similarly, if P → X is given, we’ll write P f i for the pullback of P along the map X n → X ( x f −→ . . . f n −→ x n ) ( x i − f i −→ x i ) , and P f i ◦ f i +1 for the pullback along X n → X ( x f −→ . . . f n −→ x n ) ( x i − f i ◦ f i +1 −−−−−→ x i +1 ) , etc. For small values of n we’ll use symbols like( a f −→ b ) ∈ X ( a f −→ b g −→ c ) ∈ X . to denote typical points. OOP GROUPS AND TWISTED K -THEORY I 55 Definition A.15. A fiber bundle on X consists of a fiber bundle P → X , togetherwith an bundle isomorphism(A.16) t f : P a → P b on X , for which t Id = Id, and satisfying the cocycle condition that P a t f / / t g ◦ f AAAAAAAA P bt g ~ ~ }}}}}}} P c commutes on X .This way of describing a fiber bundle is convenient when thinking of X as acategory. The association a → P a is a functor from X to spaces, that is continuousin an appropriate sense. There is a more succinct way of describing a fiber bundleon a groupoid. Namely, a fiber bundle on a groupoid X = ( X , X ) is a groupoid P = ( P , P ) and a functor P → X making P i → X i into fiber bundles, and all ofthe structure maps into maps of fiber bundles (i.e., pullbacks squares).A functor F : Y → X between groupoids defines, in the evident way, a pullbackfunctor F ∗ from the category of fiber bundles over X to the category of fiber bundlesover Y . A natural transformation T : F → G defines a natural transformation T ∗ : F ∗ → G ∗ . Example
A.17 . Let U = { U i } be a covering of a space X . To give a fiber bundleover N U is to give a fiber bundle P i on each U i and the clutching (descent) dataneeded to assemble the P i into a fiber bundle over X . Indeed, pullback along themap N U → X gives an equivalence between the category of fiber bundles over X and the category of fiber bundles over N U .The following generalization of Example A.17 will be referred to as descent forfiber bundles over groupoids. Proposition A.18.
Suppose that F : X → Y is a local equivalence. Then thepullback 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 functorfrom Y to the category of topological spaces. Since Y → X is an equivalence ofcategories, the functor F ∗ has a left adjoint F ∗ , given by F ∗ P ( x ) = lim −→ Y/x P, where Y /x is the category of objects in y ∈ Y equipped with a morphism F y → x .Since Y → X is an equivalence of groupoids, there is a unique map between anytwo objects of Y /x , and so F ∗ P ( x ) is isomorphic to P y for any y ∈ Y /x . For each x ∈ X , choose a neighborhood x ∈ U ⊂ X , a map t : U → Y , and a family ofmorphisms U → X connecting F ◦ t to the inclusion U → X . We topologize [ x ∈ X F ∗ P x by requiring that the canonical map t ∗ P → F ∗ P | U be a homeomorphism. This gives F ∗ P the structure of a fiber bundle over X .Naturality provides F ∗ P with the additional structure required to make it intoa fiber bundle over X . One easily checks that the pair ( F ∗ , F ∗ ) is an adjointequivalence of the category of fiber bundles over X with the category of fiber bundlesover Y . (cid:3) For a fiber bundle p : P → X write Γ( P ) for the space of sectionsΓ( P ) = Γ( X ; P ) = { s : X → P | p ◦ s = Id X } , topologized as a subspace of X P × X P . If f : Y → X is a local equivalence, and P → X is a fiber bundle, then the evidentmap 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 { U α } is acovering of X by open sub-groupoids. Write P α → U α for the restriction of P to U α , and P α ,...,α n for the restriction of P to U α ∩ · · · ∩ U α n , and similarly for Q . Proposition A.19.
If for each non-empty finite collection { α . . . α n } the map Γ( P α ,...,α n ) → Γ( Q α ,...,α n ) 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’sfirst consider the case in which X is covered by just two open sub-groupoids U and V . We form the “double mapping cylinder” C = U ∐ U ∩ V × [0 , ∐ V / ∼ , and consider the functor g : C → X . A choice of partition unity on [ X ] subordi-nate to the covering { [ U ] , [ V ] } gives a section of g making Γ( X ; P ) → Γ( X ; Q ) aretract of Γ( C ; g ∗ P ) → Γ( C ; g ∗ Q ). It therefore suffices, in this case, to show thatΓ( C ; g ∗ P ) → Γ( C ; g ∗ Q ) is a weak equivalence. But Γ( C ; g ∗ P ) fits into a homotopypullback square Γ( C ; g ∗ P ) −−−−→ Γ( U ; P ) y y Γ( V ; g ∗ P ) −−−−→ Γ( U ∩ V ; P ) , and similarly for Γ( C ; g ∗ Q ) (to simplify the diagram we have not distinguishedin notation between P and its restriction to U , V , and U ∩ V ). The result thenfollows from the long exact (Mayer-Vietoris) sequence of homotopy groups. Aneasy induction then gives that the map on spaces of sections of P → Q restrictedto any finite union U α ∪ · · · ∪ U α n is a weak equivalence. For the case the collection OOP GROUPS AND TWISTED K -THEORY I 57 { U α } is countable (to which we are reduced when [ X ] is second countable), orderthe U α and write V n = U ∪ · · · ∪ U n . Form the “infinite mapping cylinder” C = a V i × [ i, i + 1] / ∼ , and consider g : C → X . As before, a partition of unity on [ X ] subordinate tothe covering [ V ] i defines a section of [ C ] → [ X ] and hence of C → X , makingΓ( X ; P ) → Γ( X ; Q ) a retract of(A.20) Γ( C ; g ∗ P ) → Γ( C ; g ∗ Q ) . It therefore suffices to show that (A.20) is a weak equivalence. But (A.20) is thehomotopy inverse limit of the tower(A.21) Γ( V n ; P ) → Γ( V n ; Q )and so its homotopy groups (or sets, in the case of π ) are related to those of (A.21)by a Milnor sequence, and the result follows.Alternatively, following Segal [30], one can avoid the countability hypothesisand the induction by using for C the nerve of the covering { U α } and the homotopyspectral sequences of Bousfield-Kan and Bousfield [9, 8]. (cid:3) A.4.
Hilbert bundles. A Hilbert bundle over a groupoid X is a fiber bundle whosefibers have the structure of a separable Z / Remark
A.22 . There is a tricky issue in the point set topology here. In definingHilbert bundles as special kinds of fiber bundles, we’re implicitly using the compactopen topology on U ( H ) and not the norm topology. This causes trouble when weform the associated bundle of Fredholm operators ( § A.5), since we cannot then usethe norm topology on the space of Fredholm operators. This issue is raised andresolved by Atiyah-Segal [3], 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 ⊗ ℓ ≈ 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 ⊗ ℓ ⊕ V ≈ H ⊗ ℓ , to conclude that H is absorbing. (cid:3) Definition A.25.
A Hilbert bundle H over X is locally universal if for every opensub-groupoid X U ⊂ X the restriction of H to X U is universal. Lemma A.26. If H and H ′ are universal Hilbert bundles on X , then there is aunitary equivalence H ≈ H ′ . (cid:3) Remark
A.27 . Since 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 localissue. Lemma A.28.
Suppose that X is a groupoid, and that { U i | i = 1 . . . ∞} is a covering of X by open sub-groupoids. If H is a Hilbert bundle with the propertythat H i = H | U i is universal, then H ⊗ ℓ is universal.Proof: Let V be a Hilbert-space bundle on X . Choose a partition of unity { λ i } on [ X ] subordinate to the open cover [ U ] i . For each i choose an embedding r i : V | U i ֒ → H i . The map V → H ⊕ H ⊕ · · · = H ⊗ ℓ with components λ i r i is then an embedding of V in H ⊗ ℓ . (cid:3) Corollary A.29.
In the situation of Lemma A.28, if H | U i is locally universal, then H ⊗ ℓ is locally universal. (cid:3) Lemma A.30.
Suppose that X is a groupoid, and that { U i | i = 1 . . . ∞} is a covering of X by open sub-groupoids. If H i is a locally universal Hilbert bundleon { U i } , then there exists a Hilbert bundle H on X with H | U i ≈ H i .Proof: This is an easy induction, using Lemma A.26. (cid:3)
Corollary A.31.
Suppose that X is a groupoid, and that { U i | i = 1 . . . ∞} is a covering of X by open sub-groupoids. If H i is a locally universal Hilbert bundleon { U i } , then there exists a locally universal Hilbert bundle H on X . (cid:3) Lemma A.32.
Suppose that X = S//G is a global quotient of a space S by acompact Lie group G . Then the equivariant Hilbert bundle S × L ( G ) ⊗ C ⊗ ℓ isa locally universal Hilbert bundle on X . OOP GROUPS AND TWISTED K -THEORY I 59 Here C is the complex Clifford algebra on one (odd) generator. It is there simplyto make the odd component of our Hilbert bundle large enough. Proof:
Since the open (full) subgroupoids of
S//G correspond to the G -stableopen subsets of S it suffices to show that that L ( G ) ⊗ C ⊗ ℓ is universal. Let V be any Hilbert bundle on S//G , ie an equivariant Hilbert bundle on S . ByKuiper’s theorem, V is trivial as a (non-equivariant) Hilbert bundle on S . Choosean orthonormal homogeneous basis { e i } , and let e i = h e i , − i : V → C be thecorresponding projection operator. By the universal property of L ( G ), each e i lifts uniquely to an equivariant map V → L ( G ) ⊗ C . Taking the sum of these maps gives an embedding of V in L ( G ) ⊗ C ⊗ ℓ . (cid:3) Combining Lemma A.32 with Lemma A.30 gives:
Corollary A.33. If X is a local quotient groupoid, then there exists a locallyuniversal Hilbert bundle on X . (cid:3) Corollary A.34.
Suppose that X is a local quotient groupoid, f : Y → X is a mapconstructed 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. (cid:3)
Corollary A.34 is needed is the proof of excision in twisted K -theory, and is thereason 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 forequivariant embeddings of countably infinite dimensional inner product spaces (andnot Hilbert spaces). Of course the result also follows from Kuiper’s theorem, sincethe space of embeddings is U ( H ⊗ ℓ ) /U ( V ⊥ ). But the contractibility of the spaceof embeddings is more elementary than the contractibility of the unitary group, soit seemed better to have proof that doesn’t 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 ⊗ ℓ . Then the space of embeddings V ֒ → H ⊗ ℓ is contractible.Proof: Let f : V ⊂ H ⊗ ℓ be a fixed embedding, and write H ⊗ ℓ = H ⊕ H ⊕ · · · f = ( f , f , . . . ) . The contraction is a concatenation of two homotopies. The first takes an embedding g = ( g , g , . . . )to (0 , g , , g , . . . )and then the second iscos( πt/ · (0 , g , , g , . . . ) + sin( πt/ · ( f , , f , . . . ) . It is easier to write down the reverse of the first homotopy. It, in turn, is theconcatenation of an infinite sequence of 2-dimensional rotations(0 , g , , g , , g , . . . ) ( g , , , g , , g , . . . ) 0 ≤ t ≤ / g , , , g , , g , . . . ) ( g , g , , , , g , . . . ) 1 / ≤ t ≤ / · · · . One must check that the limit as t g , g , . . . ), and the that path is contin-uous in the compact-open topology. Both facts are easy and left to the reader. (cid:3) Lemma A.36.
Suppose that H is a locally universal Hilbert bundle over a localquotient groupoid X . Then the space of sections Γ( X ; U ( H )) of the associatedbundle of unitary groups is weakly contractible.Proof: This follows easily from Kuiper’s theorem (see Appendix 3 of [3]), andProposition A.19. (cid:3)
We conclude this section with a useful criterion for a local-quotient stack to beequivalent 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 iff the groupoidis 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 corre-sponding to twistings whose invariant in H has infinite order are not quotientstacks: indeed, any 1-eigenbundle for the central T is a projective bundle represen-tative for the twisting, and hence must be infinite-dimensional.(ii) There are simple obstructions to a groupoid being related by a chain of localequivalences to a global quotient by a compact group; for instance, such quotientsadmit continuous choices of Ad-invariant metrics on the Lie algebra stabilizerswhich are integral on the co-weight lattices. The stack obtained by gluing theboundaries of B ( T × T ) × [0 ,
1] via the shearing automorphism of T × T does notcarry such metrics. The same is true for the quotient stack A // T ⋉ LT , where T isa 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 tautologicalshearing holonomies. Hence, the larger stacks A // T ⋉ LG where G is a compactLie group and A is the space of connections on the trivial G -bundle over the circleare not global quotients either.(iii) The result is curiously similar to Totaro’s characterisation of smooth quotientstacks as the Artin stacks where coherent sheaves admit resolutions by vector bun-dles [32]. 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 equivalentto 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 . Inthat 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 ∈ X the map Aut( x ) → G associated to P is a monomorphism. Suppose that H is the (locally) universal OOP GROUPS AND TWISTED K -THEORY I 61 Hilbert bundle on X , and that we can find an orthogonal decomposition H = ⊕ H α with each H α of dimension n α < ∞ . Take P to be the product of the frame bundlesof the H α , and G to be the product of the unitary groups U ( H α ) ≈ U ( n α ). Tocheck that Aut( x ) → G is a monomorphism in this case it suffices to check locallynear x . The assertion is thus reduced to the case of a global quotient by a compactgroup, where it follows from our explicit construction. (cid:3) A.5.
Fredholm operators and K -theory. We will build our model of twisted K -theory using the “skew-adjoint Fredholm” model of Atiyah-Singer [1]. In thissection we recall this theory, and the modifications described in Atiyah-Segal [3]Let H be a Z / X . We wish to asso-ciate to H a bundle of Fredholm operators over X . As mentioned in Remark A.22,we cannot just use the norm topology on the space of Fredholm operators here.We have used the compact-open topology on U ( H ), and the (conjugation) actionof U ( H ) in the compact-open topology on Fredholm in the norm topology is notcontinuous. Following Atiyah-Segal [3, Definition 3.2], we make the following defi-nition. Definition A.39. ([3]) The space Fred (0) ( H ) is the space of odd skew-adjointFredholm operators A , for which A + 1 is compact, topologized as a subspace of B ( H ) × K ( H ), with B ( H ) given the compact-open topology and K ( H ) the normtopology.Let C n = T { C n } / ( z + q ( z ) = 0) denote the complex Clifford algebra associatedto the quadratic form q ( z ) = P z i . We write ǫ i for the i th standard basis elementof C n , regarded as an element of C n . Following Atiyah-Singer [1], for an operator A ∈ Fred (() C n ⊗ H ), with n odd, let w ( A ) = ( ǫ . . . ǫ n A n ≡ − i − ǫ . . . ǫ n A n ≡ . The operator A is then even and self-adjoint. Definition A.40. ([3]) The space Fred ( n ) ( H ) is the subspace of Fred (0) ( C n ⊗ H )consisting of odd operators A which commute (in the graded sense) with the actionof C n , and for which the essential spectrum of w ( A ), in case n is odd, contains bothpositive and negative eigenvalues.Atiyah and Segal [3] show that the “identity” map from Fred ( n ) ( ℓ ) in the normtopology to Fred ( n ) ( ℓ ) in the above topology is a weak homotopy equivalence. Itthen follows from Atiyah-Singer [1, Theorem B(k)] that the mapFred ( n ) ( ℓ ) → Ω ′ Fred ( n − ( C ⊗ ℓ ) A ǫ k cos( πt ) + A sin( πt )is weak homotopy equivalence, where we are making the evident identification C n ≈ C n − ⊗ C , and Ω ′ denotes the space of paths from ǫ k to − ǫ k . Combining theseleads to the following simple consequence. Proposition A.41. If X is a local quotient groupoid, and H a Z / -graded, locallyuniversal Hilbert bundle over X , 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 oflocally contractible slices, we may reduce to the case in which S is equivariantlycontractible to a fixed point s ∈ S . Finally, since the question is homotopy invariantin 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(A.42) Fred ( n ) ( H ) G → Ω ′ Fred ( n +1) ( H ) G is a weak equivalence. For each irreducible representation V of G , let H V denotethe V -isotypical component of H . Then (A.42) is the product over the irreduciblerepresentations V of G , of(A.43) Fred ( n ) ( H V ) G → Ω ′ Fred ( n +1) ( H V ) G Since H is universal, the Hilbert space H V is isomorphic to V ⊗ ℓ , and the mapFred ( n ) ( ℓ ) → Fred ( n ) ( V ⊗ ℓ ) G T Id ⊗ T is a homeomorphism. The Proposition is thus reduced to the result of Atiyah-Singerquoted above. (cid:3) We now assemble the spaces Γ( X ; Fred ( n ) ( H )) into a spectrum in the sense ofalgebraic topology. To do this requires specifying basepoints in Fred ( n ) ( H ). Sinceour operators are odd, we can’t take the identity map as a basepoint and a differentchoice must be made. There are some technical difficulties that arise in trying tospecify consistent choices and we have just chosen to be unspecific on this point.The difficulties don’t amount to a serious problem since any invertible operator canbe taken as a basepoint, and the space of invertible operators is contractible. Thereader is referred to [23] for further discussion.We will use the symbol ǫ to refer to a chosen basepoint in Fred ( n ) ( H ), as well asto the constant section with value ǫ in Γ( X ; Fred ( n ) ( H )).Proposition A.41 gives a homotopy equivalence(A.44) Γ( X ; Fred ( n +1) ( W )) → ΩΓ( X ; Fred ( n ) ( W ))As described in [1], the fact that C is a matrix algebra gives a homeomorphism(A.45) Γ( X Fred ( m ) ( W )) ≈ Γ( X Fred ( m +2) ( W )) . We the spectrum K ( X ) by taking K ( X ) n = ( Γ( X, Fred (0) ( W )) n evenΓ( X, Fred (1) ( W )) n oddwith structure map K ( X ) n → Ω K ( 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 thendefined by K n ( X ) = π K ( X ) n ≈ π k K ( X ) n + k . Because H is locally universal, when X = S//G , we have K n ( X ) ≈ [ S, Fred ( n ) ( L ( G ) ⊗ ℓ )] G . OOP GROUPS AND TWISTED K -THEORY I 63 Since, as remarked in § ( n ) ( L ( G ) ⊗ ℓ ) is a classifying space for equivariant K -theory, this latter group can be identified with K n ( G )( S ) . References [1] M. F. Atiyah and I. M. Singer,
Index theory for skew-adjoint Fredholm operators , Inst. Hautes´Etudes Sci. Publ. Math. (1969), no. 37, 5–26. MR 44
Twisted K-theory and cohomology .[3] ,
Twisted K -theory , Ukr. Mat. Visn. (2004), no. 3, 287–330. MR MR2172633[4] Roman Bezrukavnikov, Michael Finkelberg, and Ivan Mirkovi´c, Equivariant homology and K -theory of affine Grassmannians and Toda lattices , Compos. Math. (2005), no. 3,746–768. MR MR2135527 (2006e:19005)[5] Raoul Bott, The space of loops on a Lie group , Michigan Math. J. (1958), 35–61.MR MR0102803 (21 Applications of the theory of Morse to symmetric spaces ,Amer. J. Math. (1958), 964–1029. MR MR0105694 (21 ´El´ements de math´ematique: groupes et alg`ebres de Lie , Masson, Paris,1982, Chapitre 9. Groupes de Lie r´eels compacts. [Chapter 9. Compact real Lie groups].MR 84i:22001[8] A. K. Bousfield, Homotopy spectral sequences and obstructions , Israel J. Math. (1989),no. 1-3, 54–104. MR MR1017155 (91a:55027)[9] A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations , LectureNotes in Mathematics, no. 304, Springer–Verlag, Berlin, 1972.[10] Peter Bouwknegt, Alan L. Carey, Varghese Mathai, Michael K. Murray, and Danny Stevenson,
Twisted K -theory and K -theory of bundle gerbes , Comm. Math. Phys. (2002), no. 1, 17–45. MR 1 911 247[11] Kenneth S. Brown, Buildings , Springer-Verlag, New York, 1989. MR 90e:20001[12] U. Bunke and I. Schr¨oder,
Twisted K -theory and TQFT , Mathematisches Institut,Georg-August-Universit¨at G¨ottingen: Seminars Winter Term 2004/2005, Universit¨atsdruckeG¨ottingen, G¨ottingen, 2005, pp. 33–80. MR MR2206878 (2007c:19008)[13] Albrecht Dold, Partitions of unity in the theory of fibrations , Ann. of Math. (2) (1963),223–255. MR 27 Graded Brauer groups and K -theory with local coefficients , Inst.Hautes ´Etudes Sci. Publ. Math. (1970), no. 38, 5–25. MR 43 On the twisted K -homology of simple Lie groups , Topology (2006), no. 6, 955–988. MR MR2263220[16] J. J. Duistermaat and J. A. C. Kolk, Lie groups , Universitext, Springer-Verlag, Berlin, 2000.MR MR1738431 (2001j:22008)[17] Daniel S. Freed,
Higher algebraic structures and quantization , Comm. Math. Phys. (1994), no. 2, 343–398. MR MR1256993 (95c:58034)[18] Daniel S. Freed, Michael J. Hopkins, and Constantin Teleman,
Consistent orientation ofmoduli spaces .[19] ,
Loop Groups and Twisted K-Theory II , arXiv:math.AT/0511232.[20] ,
Loop Groups and Twisted K-theory III , arXiv:math.AT/0312155.[21] ,
Twisted equivariant K-theory with complex coefficients , arXiv:math.AT/0206257,accepted for publication by Topology.[22] Sigurdur Helgason,
Differential geometry, Lie groups, and symmetric spaces , Graduate Stud-ies in Mathematics, vol. 34, American Mathematical Society, Providence, RI, 2001, Correctedreprint of the 1978 original. MR 2002b:53081[23] Michael Joachim,
A symmetric ring spectrum representing K O -theory , Topology (2001),no. 2, 299–308. MR MR1808222 (2001k:55011)[24] Nitu Kitchloo, Dominant K -theory and highest weight representations of Kac-Moody groups ,preprint.[25] L. G. Lewis, J. P. May, and M. Steinberger, Equivariant stable homotopy theory , LectureNotes in Mathematics, vol. 1213, Springer–Verlag, New York, 1986. [26] Ruben Minasian and Gregory Moore, K -theory and Ramond-Ramond charge , J. High EnergyPhys. (1997), no. 11, Paper 2, 7 pp. (electronic). MR 2000a:81190[27] G. D. Mostow, Equivariant embeddings in Euclidean space , Ann. of Math. (2) (1957),432–446. MR MR0087037 (19,291c)[28] Richard S. Palais, On the existence of slices for actions of non-compact Lie groups , Ann. ofMath. (2) (1961), 295–323. MR MR0126506 (23 Continuous-trace algebras from the bundle theoretic point of view , J.Austral. Math. Soc. Ser. A (1989), no. 3, 368–381. MR MR1018964 (91d:46090)[30] G. Segal, Classifying spaces and spectral sequences , Inst. Hautes ´Etudes Sci. Publ. Math. (1968), 105–112.[31] Stephan Stolz and Peter Teichner, What is an elliptic object? , Topology, geometry and quan-tum field theory, London Math. Soc. Lecture Note Ser., vol. 308, Cambridge Univ. Press,Cambridge, 2004, pp. 247–343. MR MR2079378 (2005m:58048)[32] Burt Totaro,
The resolution property for schemes and stacks , J. Reine Angew. Math. (2004), 1–22. MR MR2108211 (2005j:14002)[33] Jean-Louis Tu, Ping Xu, and Camille Laurent-Gengoux,
Twisted K -theory of differen-tiable stacks , Ann. Sci. ´Ecole Norm. Sup. (4) (2004), no. 6, 841–910. MR MR2119241(2005k:58037)[34] Erik Verlinde, Fusion rules and modular transformations in D conformal field theory , Nu-clear Phys. B (1988), no. 3, 360–376. MR 89h:81238[35] Edward Witten,
D-branes and K -theory , J. High Energy Phys. (1998), no. 12, Paper 19, 41pp. (electronic). MR 2000e:81151 Department of Mathematics, University of Texas, Austin, TX
E-mail address : [email protected] Department of Mathematics, Massachusetts Institute of Technology, Cambridge,MA 02139-4307
E-mail address : [email protected] Cambridge, Cambridge
E-mail address ::