Loop groups and twisted K-theory I

LOOP GR OUPS AND TWIST ED K -THEOR Y I DANIEL S. FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEMAN Contents In tro duction 1 1. Twisted K -theory by example 6 2. Twistings of K -theory 13 2.1. Graded T -bundles 13 2.2. Graded cen tral extensions 14 2.3. Twistings 19 2.4. Examples of t wistings 22 3. Twisted K-gro ups 23 3.1. Axioms 23 3.2. Twisted Hilbert spaces 25 3.3. Univ ersal Twisted Hilbert Bundles 27 3.4. Definition of t wisted K -groups 27 3.5. V erification of the axioms 29 3.6. The Thom isomorphism, pushforw ard, and the Pontry agin pro duct 31 3.7. The fundamen tal spec tr al sequence 33 4. Computation of K τ G ( G ) 34 4.1. Notation and assumptions 34 4.2. The main computation 35 4.3. The action of W e aff on t 48 Appendix A. Groupo ids 49 A.1. Definition and First Proper ties 49 A.2. F urther Prop erties of Gro upoids 52 A.3. Fiber bundles over gro upoids and descen t 54 A.4. Hilbert bundles 57 A.5. F redholm op erators a nd K - theory 61 References 63 Introduction Equiv arian t K -theory fo cuses a r emark able ra nge o f p ersp ectiv es on the study of compact Lie groups. One finds tools from top ology , analysis, and repr esen tation theory br ough t together in describing the equiv a riant K -g roups of spaces and the maps betw een them. In the pro cess all three p oints of view are illuminated. Our a im in this series of pap ers [19, 20] is to b egin the developmen t o f similar relationships when a compa ct Lie gro up G is replaced by LG , the infinite dimensiona l group o f smo oth maps fro m the circle to G . 1 2 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST A NTIN TELEMAN There a re several features s pecial to the represent ation theory o f lo op gro ups. First of all, we will foc us only on the repres en tations of LG whic h hav e “p o sitiv e energy .” This means that the r epresent ation space V admits an a ction of the rotation gro up of the circle which is (pro jectiv ely) compatible with the a ction of LG , and for whic h there ar e no vectors v o n which rotation b y θ acts by multiplication b y e inθ with n < 0. It turns out that most positive energy represen tations are pro jectiv e, a nd so V m ust b e regar ded a s a representation of a central extensio n LG τ of LG by U (1). The top ological class o f this central extension is known as the level . One thing a topo logical companion to the representation theory of lo op groups m ust take int o account is the level. Next there is the fusion pro duct. W rite R τ ( LG ) for the group co mpletion of the monoid of pos itiv e ener g y representations of LG at level τ . In [34], Erik V erlinde in tro duced a multiplication on R τ ( LG ) ⊗ C making it into a commutativ e r ing (in fact a F r obenius a lg ebra). This multiplication is ca lled fusion , and R τ ( LG ) ⊗ C , equipped with the fusion pro duct is known a s the V erli nde algebr a . The fusion pro duct also makes R τ ( LG ) in to a ring whic h we will call the V erlinde ring . A go o d top ological description o f R τ ( LG ) should accoun t fo r the fusion pro duct in a natural w ay . The p ositive energ y represent ations of lo op groups turn out to b e completely re- ducible, and somewha t surprising ly , there are only finitely many irreducible p ositiv e energy r epresen tations at a fixed level. Moreov er, a n irreducible p ositive energy rep- resentation is determined by its lo w est non-trivia l energy eigenspace, V ( n 0 ), whic h is an irreducible (pro jective) r epresent ation o f G . Th us the po sitiv e energ y rep- resentations o f L G corr espond to a subset of the represen tations of the co mpact group G . This sugges ts that G -equiv arian t K -theory might so meho w play a ro le in describing the repres en tations of LG . In fact this is the ca s e. Here is our main theorem. Theorem 1. L et G b e a c onne cte d c omp act Lie gr oup and τ a level for the lo op gr oup. The Gr o thendie c k gr oup R τ ( LG ) at level τ is isomorphic to a t wiste d form K ζ ( τ ) G ( G ) , of t he e quivari ant K -the o ry of G acting on itself by c o njugation. Under this isomorphism the fusion pr o duct, wh en it is define d, c orr e sp onds t o the Pontryagin pr o duct. The twisting ζ ( τ ) is given in terms of t he level ζ ( τ ) = g + ˇ h + τ , wher e ˇ h is the “dual Coxeter” twisting. Several asp ects of this theore m require clarification. The main new element is the “t wisted form” of K -theory . Twisted K -theory was in tro duced b y Donov an and Ka roubi [14] in co nnection with the Tho m isomor phism, and g e ner alized and further developed by Rosenberg [29]. Interest in t wisted K -theor y was r ekindled by its app earance in the la te 1990’s [26, 35] in string theory . Our results ca me ab out in the wake of this reviv al when we realized that the work of the first author [1 7] on Chern- Simons theory for finite g roups could b e in ter preted in terms t wis ted K -theory . The twisted forms of G -equiv aria nt K -theory are class ified by the nerve of the category of invertible mo dules over the equiv a riant K -theory sp ectrum K G . What comes up in geo metr y though, is only a s ma ll subspace, and throug hout this pa - per the term “t wisting” will refer t wistings in this restr icted, more geometric class . LOOP GROUPS AND TWISTED K -THEOR Y I 3 These geometric twistings of K G -theory on a G -space X are classified up t o iso- morphism b y the set (2) H 0 G ( X ; Z / 2) × H 1 G ( X ; Z / 2) × H 3 G ( X ; Z ) . The comp onen t in H 0 corresp onds to the “degree” of a K -class, and the fact that the coefficients are the in tegers modulo 2 is a reflection of Bott perio dicity . In this sense “t wistings” refine the notion of “degree,” though when considering t wistings it is important to remember more than just the iso mo rphism class. The tensor pro duct o f K G -mo dules makes these spaces of t wis tings into infinite lo op spaces and pr o vides a commutativ e group structure on the s e ts of isomor phism classes. The gr oup structure on (2) is the pro duct of H 0 G ( X ; Z / 2) with with the ex- tension of H 1 G ( X ; Z / 2) by H 3 G ( X ; Z ) with co cycle β ( x ∪ y ), wher e β is the Bo ckstein homomorphism. A twisting is a for m of equiv aria n t K -theory on a space. A level for th e lo op group, o n the other hand, corre s ponds to a central extension of LG . One way o f relating these tw o structure s is via the classification (2). A cen tral extension of L G has a top ological inv aria n t H 3 G ( G ), and s o give rise to a twisting, up to iso morphism. When the gr oup G is simple and simply connected this in v ar ia n t determines the cent ral extension up to isomo rphism, the group H 1 G ( G ) v anishes, and there is a canonical isomorphism H 3 G ( G ) ≈ Z . In this case an in teger can b e used to sp ecify bo th a level and a t wisting. Ther e is a more re fined version of this corresp ondence directly r elating twistings to central extensio n, and the appro ac h to twi stings we take in this pap e r is designed to make this relationship as transparent as p o ssible. There is a map fro m vector bundles to twistings which a ssoc ia tes to a vector bundle V over X the family of K -mo dules K ¯ V x , wher e ¯ V x is the one p oin t com- pactification of the fib er of V over x ∈ X , and for a space S , K S is the K -module with π 0 K S = K 0 ( S ). W e denote the t wisting asso ciated to V b y τ V , though w hen no confusion is likely to ar is e we will just use the symbol V . The inv ariants of τ V in (2) are dim V , w 1 ( V ) and β w 2 ( V ). These twistings a r e describ ed by Donov an and Karoubi in [14] from the p oint o f view o f Clifford a lgebras. There is also a homomorphism fro m K O − 1 G ( X ) to the gro up of twistin gs of equiv ariant K -theory on X . In top ological terms it cor respo nds to the map from the sta ble or thogonal group O to its third Postnik ov section O h 0 , . . . , 3 i . It sends an element o f K O − 1 to the twisting whose comp onents are ( σ w 1 , σ w 2 , x ), where σ : H ∗ ( B O ) → H ∗− 1 ( O ) is the coho mology susp ension, and x ∈ H 3 ( O ; Z ) is the unique element , twice which is the coho mology susp ension of p 1 . In terms of op erator a lgebras this ho momorphism s ends a skew-adjoint F redholm o perato r to its graded Pfaffian gerbe . W e will call this map pfa ff . W e now describ e tw o natural twistings on G whic h a re equiv ar ian t for the adjoint action. The first comes fro m the adjoint representation of G regarded a s an equi- v ariant vector bundle ov er a point, and pulled back to G . W e’ll wr ite this twisting as g . F or the other, firs t note that the equiv ariant cohomolog y group H ∗ G ( G ) is just H ∗ ( LB G ) . The vector bundle asso ciated to the adjoin t representation gives a class ad ∈ K O 0 ( B G ) which w e can transgress to K O − 1 ( LB G ), a nd then map to t wistings by the map pfaff . W e’ll call this t wisting ˇ h . W hen G is simple and simply connected, the integer cor respo nding to ˇ h is the dual Coxeter num b er, a nd g is just a degree shift. With these definit ions, the formula ζ ( τ ) = g + ˇ h + τ 4 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN in the statemen t o f Theor e m 1 should b e clear. The t wistings g and ˇ h are those just describ e d, and τ is the twisting cor respo nding to the level. Theorem 1 provides a topo logical description of the group R τ ( LG ) and its fusion pro duct when it exists. But it a ls o gives more. The twist ed K -group K ζ ( τ ) G ( G ) is defined for a n y compact Lie g roup G , a nd it makes sens e for any level τ . This p oints the w ay to a formulation o f a n analogue of th e gro up R τ ( LG ) for any compact Lie gr oup G (even one which is finite). In Parts II a nd I II we take up these generalizatio ns and sho w that the assertio ns of Theorem 1 remain true. One thing that emerges fro m our topo logical considera tio ns is the need to con- sider Z / 2 -gr ade d ce ntral extensio ns of lo op groups. Such extensions are necessary when working with a group like S O (3) whose adjoint representation is not Spin . Another interesting case is that of O (2). When the adjoint r epresen tation is not orientable the dual Coxeter t wist makes a non-trivial contribu tion to H 0 G ( G ; Z / 2). One sees an extra c hange in the degree in whic h the in teresting K -group occur s. In the case of O (2) these degree shifts are different on the t wo co nnected compo nen ts, again emphasizing the p oin t that twistings should b e reg arded as a genera lization of degree. The “V erlinde ring ” in this case is compr ised of an even K -group o n one comp onen t a nd an odd K -group o n another. Suc h inhomogeneo us co mpositions are not t ypica lly co nsidered when discussing or dinary K -gro ups. The fusion product on R τ ( LG ) has b een defined for simple and simply connected G , a nd in a few further sp e cial cases. The Pon tr y a gin product on K ζ ( τ ) G ( G ) is de- fined ex a ctly when τ is primitive in the sense that its pullbac k along the m ultiplica- tion map of G is isomorphic to the sum of its pullb acks along the t wo pro jections. This explains, for example, why a fusion pro duct on R τ ( LS O ( n )) exists only at half of the lev els. Using the Pon tryagin product w e are able to define a fusion pro duct on R τ ( LG ) for any G a t any primitiv e lev el τ . W e do not, ho wev er, giv e a construction of this product in terms of representation theory . When the fusion pro duct is defined on R τ ( LG ) it is part of a m uch more elab orate structure. F or one thing, there is a trace map R τ ( LG ) → Z making R τ ( LG ) into a F r obenius alg ebra. Using twisted K -theory w e co nstruct this trace map for gener al compact Lie groups at primitive levels τ which a re non-degener ate in the sense that the image of τ in H 3 T ( T ; R ) ≈ H 1 ( T ; R ) ⊗ H 1 ( T ; R ) is a non-degenerate bilinear form. Again, in the cases when the fusion pro duct has been defined, there ar e op erations on R τ ( LG ) coming fro m the mo duli spa c es o f Riemann sur faces with boundary , ma k ing R τ ( LG ) par t of is o ften called a top olo g- ic al c onformal field the ory . Using top ologica l metho ds, we are able to construct a topo logical conformal field theory for any compact Lie gro up G , at levels τ whic h are transgressed from (generalized) cohomolo gy classes on B G and which are non- degenerate. Some of this w or k app ears in [21, 18]. Another impact of Theo rem 1 is that it brings the computational techniques of algebraic top ology to b ear on the representations of lo op groups. One very in teresting appro ac h, for connected G , is to use the Rothenberg-Steenro d sp ectral sequence relating the e q uiv aria n t K -theory of Ω G to that of G . In this ca se one gets a spectra l sequence (3) T or K G ∗ (Ω G ) ( R ( G ) , R ( G )) = ⇒ K G τ + ∗ ( G ) , LOOP GROUPS AND TWISTED K -THEOR Y I 5 relating the unt wisted equiv ar ian t K -homolo gy o f Ω G , and the representation ring of G to the V erlinde alg ebra. The ring K G ∗ (Ω G ) can b e c o mputed using the tec hniques of Bo tt [5] and Bo tt-Sa melson [6] and ha s also b een descr ibed by Bezruk avnik ov, Fink elb erg and Mirkovi ´ c [4]. The K -groups in the E 2 -term are un twisted. The t wisting a ppear s in the wa y that the representation ring R ( G ) is made into an algebra over K G (Ω G ). The equiv ariant geometry of Ω G has b een extensively studied in connection with the repr esen tation theory of LG , and the sp e c tr al sequence (3) seems to express yet another rela tionship. W e do not know of a repr esen ta tion theoretic construction of (3 ). An ana logue of the sp ectral s e- quence (3) has been used by Chris Doug la s [15] to compute the (non-equiv ariant) t wisted K -groups K τ ( G ) for all simple, simply co nnected G . Using th e Lefschetz fixed point formula one can easily conclude for connected G that ∆ − 1 K G ∗ (Ω G ) = ∆ − 1 Z [Λ × Π] , where ∆ is the squar e of the W eyl denominato r , Π = π 1 T is the co-weigh t lattice, and Λ is the weigh t la ttice. When the lev el τ is non-degenerate there a re no higher T or groups, and the spectral sequence degenerates to an isomorphism ∆ − 1 K G τ + ∗ ( G ) ≈ ∆ − 1 R ( G ) /I τ where I τ is the ideal of r e pr esen tations whose c har acters v anish on cer tain conjugacy classes. The main c o mputation of this pap er asse r ts that suc h an iso morphism holds without inv erting ∆ when G is connected, and π 1 G is torsion free. The distinguished co njuga cy classes a re known as V erlinde c onjugacy classes , and the ideal I τ as the V erlinde ide al . In [21] the ring K ζ ( τ ) G ( G ) ⊗ C is computed using a fixed point form ula, and sho wn to be isomorphic to the V erlinde algebra . The plan of this se ries of pap ers is a s follows. In Part I we define twisted K -groups, and compute the groups K ζ G ( G ) for connected G with torsion free fun- dament al g roup, at non-degener ate lev els ζ . Our main r esult is Theorem 4.27. In Part II we in tro duce a certain family of Dirac op erator s and our g eneralization of R τ ( LG ) to arbitrary compac t Lie groups. W e 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 torsion free fundamen tal group. In Part I II we show that our ma p is an isomor phism for general compact Lie g roups G , and develop some applications. The bulk of this pap er is concerned with setting up twisted equiv a r ian t K -theory . There are tw o things that make this a little complicated. F o r one, w hen working with t wistings it is importa nt to remem b er the morphisms betw een them, and not just the is omorphism class e s . The t wistings on a space form a c ate gory and sp elling out the b ehavior of this catego ry as the spa ce v a ries gets a litt le ela bora te. The other thing has to do with the kind of G -spaces w e use. W e need to define t wistings on G -equiv ar ian t K -theory in such a w ay as to mak e clear what happens as the group G changes, a nd for the constructions in Part II we need to mak e the relation- ship b et ween twistings and (graded) central extensions as transpar en t a s poss ible. W e work in this pa p er with gr oup oids and define twisted equiv ariant K -theory for group oids. W eak ly equiv alent g roupo ids (see Appendix A) hav e equiv alent cate- gories of twistings and isomorphic twisted K - g roups. A group G a cting on a spa c e X forms a sp ecial kind of gr oupoid X/ /G called a “global quotient g roupo id.” A cent ral extension of G by U (1) defines a twisting of K -theor y for X / /G . If G is a compact connected Lie group acting on itself by conjugation, and P G denotes the 6 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN space o f paths in G star ting at the identit y , acted upon by LG b y conjugatio n, then P G/ /LG → G/ /G is a local equiv alence, and a (graded) cen tral extension of LG defines a t wisting of P G/ /LG and hence of G/ /G . In gener al w e will define a t wisting of a gr oupoid X to consist of a lo cal equiv a lence P → X and a graded central extension ˜ P of P . While most of the r esults we prove reduce, ultimately , to ordinary results about compact Lie groups a cting on spa c e s , not all do. In Part I II it beco mes necessary to work with group oids which are not equiv a lent to a co mpact Lie group acting on a space. Nitu Kitc hlo o [24] has p ointed out that the space P G is the universal LG space for prop e r actions. Using this he has descr ibed a genera lization of our computation to other Kac-Mo ody groups. A t the time we began this w ork, the pap er o f A tiyah and Segal [3] w as in prepa- ration, and w e b enefited a great deal from early drafts. Since that time several other approaches to t wisted K -theory have app e a red. In a ddition to [3] we refer the reader to [10, 33]. W e hav e chosen to use “graded central extensions” because of the c lo se connection with lo op g r oups and the c o nstructions w e wish to mak e in Part I I. Of course our res ults can b e presented from a ny of the p oints of view men tioned above, and the choice o f which is a matter o f p ersonal preference. W e hav e a ttempted to orga nize this pap er so that the is sues of implemen tation are indep enden t o f the issues of co mputation. Section 1 is a kind of field guide to t wisted K -theory . W e describ e a series of examples in tended to give the reader a working knowledge of twisted K -theor y sufficient to follow the main computation in § 4. Section 2 con tains our formal discussion of twistings o f K -theory for group oids, and our definition o f t wisted K -gro ups app ears in § 3. W e ha ve attempted to axiom- atize the theory o f twisted K -groups in order to facilitate comparis o n with other mo dels. Our ma in computation a ppears in § 4. This pap er ha s b een a lo ng time in preparatio n a nd we ha ve benefited from discussion with man y p eople. The authors would like to thank Sir Mic hael A tiyah and Gra e me Seg al for making a v ailable early drafts of [3] and [2]. A s will b e evident to the rea der, o ur appro ac h to twisted K -theory r elies heavily on their ideas. W e would like to thank Is Singer for many useful conv ersations. W e would als o like to thank Ulrich Bunke and Thomas Schic k for their careful study and comments o n this w o r k. The report of their seminar appear s in [1 2]. W e assume throug hout this paper that all spaces are lo cally contractible, para- compact a nd completely reg ula r. These assumptions implies the existence o f pa r- titions o f unity [13] a nd lo cally contractible slices through actions of compact Lie groups [27, 28]. 1. Twisted K -theor y by example The K -theor y of a space is assembled fr o m data whic h is lo cal. T o giv e a vector bundle V on X is equiv alent to giving vector bundles V i on the op en sets U i of a cov er ing , and isomorphisms λ ij : V i → V j on U i ∩ U j satisfying a compatibility (coc y cle) condition on the triple intersections. In terms o f K -theory this is expres s ed by the May er-Vietoris (sp ectral) sequence relating K ( X ) and the K - groups of the in ters ections of the U i . In forming twisted LOOP GROUPS AND TWISTED K -THEOR Y I 7 K -theory we modify this descen t or gluing datum, by in tro ducing 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 cer tain co cycle condition. In terms of K -theory , this mo difies the restriction maps in the Mayer-Vietoris sequence. In order to formulate the co cycle condition, the L ij m ust co me equipp ed with an isomorphism L j k ⊗ L ij → L ik on the triple in tersections, sa tisfying an eviden t compatibility relation on the quadru- ple in tersections. In o ther words, the { L ij } must form a 1 -co cycle with v alues in the group oid of line bundles. Cocycles differing b y a 1-co c hain give isomorphic twi sted K -groups, so, up to isomo rphism, w e can ass o ciate a t wisted notion of K ( X ) to an element τ ∈ H 1 ( X ; { Line Bundles } ) . On go o d s paces there a re iso morphisms H 1 ( X ; { Line Bundles } ) ≈ H 2 ( X ; U (1)) ≈ H 3 ( X ; Z ) , and co rresp ondingly , twisted notions of K -theory a sso c ia ted to an integer v alued 3-co cycle. In this pap er we find we need to allow the L ij to b e ± line bundles, so in fact w e consider twisted notion o f K ( X ) classified b y ele ments 1 τ ∈ H 1 ( X ; {± Line Bundles } ) ≈ H 3 ( X ; Z ) × H 1 ( X ; Z / 2) . W e will write K τ + n for the v ersion of K n , t wisted by τ . In pra ctice, to compute twisted K ( X ) one represents the t wisting τ as a Cech 1-co cycle o n an explicit cov e r ing of X . The twisted K -gro up is then assem bled from the May er-Vietoris sequence of this cov ering, inv olving the same (unt wisted) K -groups one w ould encoun ter in computing K ( X ). The presence of the 1-co cycle is manifest in the restriction maps b et ween the K -groups of the op en sets. They a re mo dified on the t wo-fold in ters e ctions b y tensoring with the ( ± ) line bundle given b y the 1-co cycle. This “oper ational definition” suffices to mak e most co mputations. See § 3 for a more ca r eful discuss ion. Here are a few examples. Example 1.4 . Supp ose that X = S 3 , and that the is omorphism class of τ is n ∈ H 3 ( X ; Z ) ≈ Z . Let U + = X \ (0 , 0 , 0 , − 1 ) and U − = X \ (0 , 0 , 0 , 1 ). Then U + ∩ U − ∼ S 2 , and τ is represented by the 1-c o cycle whose v alue on U + ∩ U − is L n , with L the tautological line bundle. The Ma yer-Vietoris sequence for K τ ( X ) tak es 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 − ) → · · · . 1 The set of is omor phis m classes of t wistings has a group structure i nduce d from the tensor product of graded l ine bundles. While there is, as indicated, a set-theoretic factorization of the isomorphism classes of twistings, the group structure is not, in general, the product. 8 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN Since the restriction of τ to U ± is isomorphic to zero, w e hav e K τ +0 ( U ± ) ≈ K 0 ( U ± ) ≈ Z K τ +1 ( U ± ) ≈ K 1 ( U ± ) ≈ 0 K τ +1 ( U + ∩ U − ) ≈ K 1 ( S 2 ) ≈ 0 , and K τ +0 ( U + ∩ U − ) ≈ K 0 ( S 2 ) ≈ Z ⊕ Z . with bas is the trivial bundle 1, and the tautolog ical line bu ndle L . The May er- Vietoris sequence reduces to the exa ct s equence 0 → K τ +0 ( X ) → Z ⊕ Z → Z ⊕ Z → K τ +1 ( X ) → 0 . In ordinary (un twisted) K -theory , the middle map is  1 − 1 0 0  : Z ⊕ Z → Z ⊕ Z . In t wisted K -theory , with suitable conv entions, the middle map beco mes (1.5)  1 n − 1 0 − n  : Z ⊕ Z → Z ⊕ Z , and so K τ + n ( S 3 ) = ( 0 n = 0 Z /n n = 1 . In the language of t wistings, the map (1.5) is accoun ted for as follo ws. T o identif y the twisted K -gr o ups with ordina r y t wisted K -gr oups we hav e to choo se isomor- phisms t ± : τ | U ± → 0 . If w e use the t + to trivialize τ on U + ∩ U − , then the follo wing diagram comm utes K τ + ∗ ( U + ) t + / / restr.   K 0+ ∗ ( U + ) restr.   K τ + ∗ ( U + ∩ U − ) t + / / K 0+ ∗ ( U + ∩ U − ) and we can identify the restriction ma p in twisted K -theory from U + to U + ∩ U − with the r estriction map in unt wisted K -theor y . On U + ∩ U − we hav e t − = ( t − t − 1 + ) ◦ t + so the r estriction map in twisted K -theory is iden tified with the restriction map in unt wisted K -theory , fo llowed b y the map ( t − t − 1 + ). By definition of τ , this ma p is giv en b y m ultiplication by L n . This account s for the second column of (1.5). Example 1 .6 . Now consider the t wis ted K -theory of U (1) acting trivially o n itself. In this case the t wistings are classified by H 3 U (1) ( U (1 ); Z ) × H 1 U (1) ( U (1 ); Z / 2 ) ≈ Z ⊕ Z / 2 . W e consider t wisted K -theory , twisted by τ = ( n, ǫ ). Regard U (1) a s the unit circle in the complex plane, a nd set U + = U (1) \ {− 1 } U − = U (1) \ { +1 } . LOOP GROUPS AND TWISTED K -THEOR Y I 9 The t wisting τ r e s tricts to zero on b oth U + and U − . W r ite K 0 U (1) = R ( 1 ) = Z [ L, L − 1 ] . Then the Ma yer-Vietoris sequence b ecomes 0 → K τ +0 U (1) ( U (1 )) → Z [ L ± 1 ] ⊕ Z [ L ± 1 ] → Z [ L ± 1 ] ⊕ Z [ L ± 1 ] → K τ +1 U (1) ( U (1 )) → 0 . The 1-co cycle r epresent ing τ c a n b e taken to b e the equiv aria n t vector bundle whose fiber ov er − i is the trivial representation of U (1) and whose fib er ov er + i is ( − 1) ǫ L n . With suitable conv entions, the middle map becomes  1 − ( − 1) ǫ L n 1 − 1  : Z [ L ± 1 ] 2 → Z [ L ± 1 ] 2 . It follows that K τ + k U (1) ( U (1 )) = ( 0 k = 0 Z [ L ± 1 ] / (( − 1) ǫ L n − 1) k = 1 . When ǫ = 0 this c o incides the Grothendieck gr oup of representations of the Heisen- ber g extension of Z × U (1) of level n , a nd in turn with the Grothendieck gr oup of po sitiv e energ y representations the lo op group of U (1) at level n . Example 1 .7 . Consider the twisted K - theory of S U (2) acting on itself by conjuga - tion. The group H 1 S U (2) ( S U (2); Z / 2) v anishes, while H 3 S U (2) ( S U (2); Z ) = Z , so a t wistings τ in this case is given by an in teger n ∈ H 3 S U (2) ( S U (2); Z ) = Z . Set U + = S U (2) \ {− 1 } U − = S U (2) \ { +1 } . The spaces U + and U 1 are equiv ar ian tly contractible, while U + ∩ U − is equiv ar ian tly homotopy equiv alent to S 2 = S U (2) /T , where T = U (1) is a ma ximal torus. The restrictions of τ to U + and U − are isomorphic to zero. W e hav e K 0 S U (2) ( U ± ) ≈ K 0 S U (2) (pt) = R ( S U (2)) = Z [ L, L − 1 ] W with the W eyl gr oup W ≈ Z / 2 acting by exchanging L a nd L − 1 , and K 0 S U (2) ( U + ∩ U − ) ≈ K 0 S U (2) ( S U (2) /T ) ≈ K 0 T (pt) ≈ Z [ L, L − 1 ] . The ring R ( S U (2)) has an a dditiv e basis consisting of the irreducible representa- tions, ρ k = L k + L k − 2 + · · · + L − k , k ≥ 0 which multip ly accor ding to the Clebsch-Gordon rule ρ l ρ k = ρ k + l + ρ k + l − 2 + · · · + ρ k − l k ≥ l . As in our other example, the May er-Vietoris sequence is short exact 0 → K τ +0 S U (2) ( S U (2)) → R ( S U (2)) ⊕ R ( S U (2)) → Z [ L ± 1 ] → K τ +1 S U (2) ( S U (2)) → 0 . The 1-co cycle r epresent ing the difference be tw een the tw o trivializations of the restriction o f τ to U + ∩ U − can b e taken to b e the element L n ∈ K S U (2) ( S U (2) /T ) ≈ 10 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN R ( T ). The sequence is a seq uence o f R ( S U (2))-mo dules. With suitable conv entions, the middle map is (1.8)  1 − L n  : R ( S U (2)) 2 → R ( T ) . T o calculate the kernel and cokernel, no te that R ( T ) is a free module of rank 2 over R ( S U (2)). W e give R ( S U (2)) ⊕ R ( S U (2))) the obvious basis, and R ( T ) the ba sis { 1 , L } . It follo ws from the iden tity L n = L ρ n − 1 − ρ n − 2 that (1.8) is represen ted b y the matrix  1 ρ n − 2 0 − ρ n − 1  , and that K τ + k S U (2) ( S U (2)) = ( 0 k = 0 R ( S U (2)) / ( ρ n − 1 ) k = 1 . This coincides the Grothendieck gro up o f pos itiv e energy repr esen ta tions the lo op group of S U (2) at lev el ( n − 2). Examples 1.6 and 1.7 illustrate the relationship be tw een t wisted K -theory and the repr esen ta tions of lo op groups. In bo th cas es the Grothendiec k group of p ositive energy r epresent ations o f the lo op group o f a compact Lie g roup G is described by the twisted equiv ariant K -group of G acting on itself b y conjugation. Two minor discrepancies appea r in th is relationship. O n o ne ha nd, the interesting K -group is in degree k = 1. As explained in the introduction, the repr esen tations of the lo op group at level τ corr espond to twisted K -theory at the twisting ζ ( τ ) = g + ˇ h + τ . The shift in K -group to k = 1 corre s ponds to the term g . In b oth examples the adjoint repres e ntation is Spin c and so con tributes o nly its dimensio n to ζ ( τ ). This term could b e gotten rid of by working with twisted equiv ariant K -homolo g y rather than K -co homology . W e hav e c hosen to w o rk with K -cohomology in o rder to make better contact with our geometric constructions in Parts II and I II. The other discrepancy is the shift in level in E x ample 1 .7: twisted eq uiv aria nt K -theory at level n cor respo nds to the r epresen tations of the lo op gr oup at level ( n − 2). The shift of 2 here corresp onds to the term ˇ h in our formula for ζ ( τ ). W e now give a series of examples descr ibing other ways in which twistings of K -theory arise. Example 1.9 . Let V b e a v ector bundle o f dimension n over a space X , and w r ite X V for the Thom complex of V . Then ˜ K n + k ( X V ) is a t wisted form of K k ( X ). T o iden tify the twistin g, choose local Spin c structures µ i on the restrictio ns V i = V | U i of V to the sets in a n open cov er of X . The K -theory Thom cla sses asso ciated to the µ i allow one to ident ify ˜ K n + k ( U V i i ) ≈ K k ( U i ) The difference b etw een the t wo iden tifica tions on U i ∩ U j is given b y multiplication b y the graded line bundle representing the difference b et ween µ i and µ j . Figu- ratively , the co cycle representing the twisting is µ j µ − 1 i and the co ho mology cla s s is ( w 1 ( V ) , W 3 ( V )) ∈ H 1 ( X ; Z / 2) × H 3 ( X ; Z ), where W 3 = β w 2 . This is one of the or iginal examples of twisted K -theory , describ ed by Donov a n and K a roubi [14] from the p oint o f view of Clifford algebras. W e will review their description in § 3 .6. LOOP GROUPS AND TWISTED K -THEOR Y I 11 Example 1.10 . Let G be a compa ct Lie g roup. The central extensions T → ˜ G τ − → G of G b y T = U (1 ) are classified by H 3 G ( { pt } ; Z ) = H 3 ( B G ; Z ). The Grothendieck group R τ ( G ), of representations of ˜ G on which T acts according to its defining representation, can b e thought of as a t wisted form o f R( G ) . In this case, our definition of equiv ar ia n t twisted K -theory giv es K τ + k G (pt) = ( R τ ( G ) k = 0 0 k = 1 More genera lly , if S is a G -space, a nd τ ∈ H 3 G ( S ) is pulled back from H 3 G (pt), then K τ + k G ( S ) is the summand of K k ˜ G ( S ) c orresp onding to ˜ G -equiv ar ia n t vector bundles on whic h T ac ts a ccording to its defining character. Example 1.11 . Now supp ose that H → G → Q is an extension of g roups, and V is an irreducible r epresen tation of H that is stable, up to iso morphism, under conjugation b y elements o f G . Then Grothendieck gro up of representations of G whose restriction to H is V -iso t ypica l, forms a twisted version of the Gro thendieck group of representations o f Q . When H is central, equal to T , and V is the defining r epresent ation, this is the situation of Example 1.10. W e now describe how to re duce to this case. Fix an H -in v ariant Hermitian metric on V , and write V ∗ = hom( V , C ) for the representation dual to V . Let ˜ G denote the group of pa ir s ( g , f ) ∈ G × hom( V ∗ , V ∗ ) for whic h f is unitary , and satisfies f ( hv ) = g hg − 1 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 ) 7→ g is surjectiv e, with k er nel T . The inclusion H ⊂ ˜ G h 7→ ( h, action of h ) is normal, and lift s the inclusion of H into G . W e define ˜ Q = ˜ G/H . The group ˜ Q is a cen tr a l extension of Q b y T , which we denote ˜ Q τ − → Q. W e now des cribe an equiv alence of catego ries b et ween V -isotypical G -representations, and τ -pro jectiv e repr esen ta tion of Q (representations of ˜ Q on whic h T ac ts accord- ing to its defining c har acter). 12 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN By definition, the r e pr esen tations V a nd V ∗ of H come equipp e d with extensions to unitary represen tation of ˜ G . Giv en a V -isotypical representation W of G , w e let M denote the H -in v ar ian t 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 a functor from V -iso t ypica l representations of G to τ -pro jectiv e representations of Q . Conv ers ely , s upp ose M is a τ -pro jectiv e representation o f ˜ Q . L e t ˜ G act on M through the pro jection ˜ G → ˜ Q , and form W = V ⊗ M . The central T o f ˜ G acts trivially o n W , g iving W a G -action. This defines a functor fro m the categ o ry of τ -pro jecti ve representations of Q to the categ ory of V -is o t ypica l repres en tations of G . O ne ea sily chec ks these tw o functors to for m a n equiv alence of categories. Example 1 .12 . Contin uing with the situation o f Example 1.11 consider an extension H → G → Q and an irreducible r e pr esen tation V of H , whic h this time is not assumed to be stable under conjugation b y G . W rite G 0 = { g ∈ G | (ad g ) ∗ V ≈ V } , and Q 0 = G 0 /H . Let S b e the set of isomorphism classes o f irreducible repre- sentations of H o f the form (ad g ) ∗ V . The co njugation a ction o f G o n S factors through Q , a nd we ha ve a n identification S = Q/Q 0 . Let’s call a r epresent ation of G S -t ypic al if its restriction to H inv olves only the irreducible representations in S . One easily chec ks that “induction” and “passag e to the V -isotypical part o f the restriction” give an equiv alence of categories { S -t ypical representations of G } ↔ { V -isotypical represe ntations of G 0 } , and therefore an isomorphism of the Grothendieck group R S ( G ) of S -t ypical rep- resentations of G with R τ ( Q 0 ) ≈ K τ Q 0 (pt) . W e ca n formulate this isomorphism a little more clea nly in the lang uage of gro up oids. F o r each α ∈ S , c ho ose an ir reducible H -representation V α representing α . Consider the group oid S/ /Q , with set of ob jects S , and in which a morphism α → β is an element g ∈ Q for which (ad g ) ∗ α = β . W e define a new group oid P with ob jects S , and with P ( α, β ) the set of equiv alences classes of pairs ( g , φ ) ∈ G × ho m( V ∗ α ∗ ,V β ), with φ unitary , and satisfying φ ( h v ) = g hg − 1 φ ( v ) (so that, among other things, (ad g ) ∗ α = β ). The equiv alence relation is gener ated b y ( g , φ ) ∼ ( hg , hφ ) h ∈ H . There is an evident funct or τ : P → S/ /Q , r epresent ing P as a cen tral extension of S/ /Q b y T . The automorphism g roup of V in P is the central extensio n ˜ Q 0 of Q 0 . An easy generaliza tion of the construction of Example 1.11 gives an equiv a lence of categories { τ − pro jective repres en tatio ns of P } ↔ { S − typical r epresen tations of G } . LOOP GROUPS AND TWISTED K -THEOR Y I 13 Cent ral extensions of S/ /Q ar e c lassified b y H 3 ( S/ /Q ; Z ) = H 3 Q ( S ; Z ) ≈ H 3 Q 0 (pt; Z ) , and so represent t w is tings of K -theory . Our definition of twisted K -theory o f group oids 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 a ccording to its defining representation. W e ther e fo re hav e an isomorphism R S ( G ) ≈ K τ +0 ( S/ /Q ) = K τ +0 Q ( S ) . Example 1.13 . Now let S denote the set of isomor phism cla sses of all irreducible representations of H . Decompo s ing S into orbit t ype s , and using the construction of Example 1.12 giv es a central extension τ : P → ( S/ /Q ), and an is omorphism R( G ) ≈ K τ +0 ( S/ /Q ) = K τ +0 Q ( S ) . More generally , if X is a space with a Q - action there is a n is o morphism K k G ( X ) ≈ K τ + k Q ( X × S ) , in which τ is the Q -equiv aria nt t wisting of X × S pulled back from the Q -equiv ariant t wisting τ of S , given by P . 2. Twistings of K -theor y W e now turn to a mor e careful discussion of t wistings o f K -theory . Our ter mi- nology derives from the situation of Exa mple 1.10 in which a cen tral ex tension of a group gives rise to a twisted notion of e q uiv aria n t K -theory . By working with graded central ex tensio ns of group oids (rather that groups) w e are able to include in a single p oint of view b oth the t wistings that co me from 1-co cycles with v alues in the gro up of Z / 2 -graded line bundles and the twistings that come from central extensions. In o rder to facilitate this, in the rest of this pap er w e will use the lan- guage of T -bundles a nd T -tor sors instead of “line bundles,” where T is the gr oup U (1). W e b egin with a formal discussion o f ( Z / 2-)graded T -bundles. 2.1. Graded T -bundles. Le t X be a top ological space. Definition 2 .1. A gr ade d T -bund le ov er X consists of a principa l T -bundle P → X , and a locally constant fu nction ǫ : X → Z / 2. W e will ca ll a graded T -bundle ( P, ǫ ) even (resp. o dd ) if ǫ is the constant func- tion function 1 (res p. − 1). The co llection of gr aded T -bundles forms a s y mmetric monoidal gro upoid. A map of graded T -bundles ( P 1 , ǫ 1 ) → ( P 1 , ǫ 2 ) ex is ts only when ǫ 1 = ǫ 2 , in which case it is a map of principal bundles P 1 → P 2 . The tensor structure is giv en b y ( P 1 , ǫ 1 ) ⊗ ( P 2 , ǫ 2 ) = ( P 1 ⊗ P 2 , ǫ 1 + ǫ 2 ) , in whic h P 1 ⊗ P 2 is the usual “tensor product” of principal T -bundles: ( P 1 ⊗ P 2 ) x = ( P 1 ) x × ( P 2 ) x / ( v λ, w ) ∼ ( v , wλ ) . It is easiest to desc r ibe the symmetry transforma tion T : ( P 1 , ǫ 1 ) ⊗ ( P 2 , ǫ 2 ) → ( P 2 , ǫ 2 ) ⊗ ( P 1 , ǫ 1 ) fiber wise. In the fib er ov er a po int x ∈ X it is ( v , w ) 7→ ( w, v ǫ 1 ( x ) ǫ 2 ( x )) . 14 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN W e will write B T ± for the contrav ariant functor which asso ciates to a space X the ca tegory of gr a ded T -bundles ov er X , and for B T the functor “ca teg ory of T - bundles”. W e will also write H 1 ( X ; T ± ) for the group of isomorphism class e s in B T ± ( X ), and H 0 ( X ; T ± ) for the group of automorphisms of any ob ject. There is an exact sequence (2.2) B T → B T ± → Z / 2 in which the rightmost a rrow is “for get everything but the grading.” In fact this sequence can be split b y asso ciating to a locally constan t funct ion ǫ : X → Z / 2 the “trivial” graded T -bundle 1 ǫ := ( X × T , ǫ ) . This splitting is compatible with the monoidal structure , but not with its symmetry . It gives an equiv alence (2.3) B T ± ≈ B T × Z / 2 of monoidal categories (but not of symmetric mo noidal categories). It follo ws that the group H 1 ( X ; T ± ), of isomorphism classes of gr aded T -bundles ov er X , is iso- morphic to H 0 ( X ; Z / 2) × H 1 ( X ; T ). Since X is assumed to b e paracompa c t, this in turn is isomorphic to H 0 ( X ; Z / 2) × H 2 ( X ; Z ). The automo r phism gro up of an y graded T -bundle is the group o f con tinuous maps from X to T . 2.2. Graded cen tral extensions. Building on the notion of gra ded T -bundles we now turn to graded central extension o f group oids. The reader is r eferred to Appendix A for our conv enti ons o n group oids, and a recollection of the fundamental notions. Unless otherwise sta ted all group oids will b e assumed to b e lo c al quotient gr oup oids ( § A.2.2) , in the se ns e that they admit a coun table op en cover by sub- group oids, each of whic h is weakly equiv alent to a compa c t Lie group acting on a Hausdorff space. Let X = ( X 0 , X 1 ) b e a group oid. W rite B Z / 2 for the g roupo id asso ciated to the action of Z / 2 on a p oint. Definition 2.4 . A gr ade d gr oup oid is a gr oupoid X equipp ed with a functor ǫ : X → B Z / 2. The map ǫ is called the gr ading . The collection o f g radings o n X forms a group oid, in whic h a morphism is a natural transformation. Spelled out, a gr ading of X is a f unction ǫ : X 1 → Z / 2 satisfying ǫ ( g ◦ f ) = ǫ ( g ) + ǫ ( f ), and a mo rphism from ǫ 0 to ǫ 1 is a co n tinuous function η : X 0 → Z / 2 sa tisfying, for each ( f : x → y ) ∈ X 1 , ǫ 1 ( f ) = ǫ 0 ( f ) + ( η ( y ) − η ( x )) . Example 2.5 . Supp ose that X = S/ /G , with S a c o nnected top ological spa ce. Then a grading of X is just a homomorphism G → Z / 2, making G into a graded group. W e deno te the groupo id of gra dings of X Hom ( X, B Z / 2) . Definition 2.6. A gr ade d c entr al extension of X is a graded T -bundle L o ver X 1 , together with an isomorphism o f graded T -bundles on X 2 λ g,f : L g ⊗ L f → L g ◦ f LOOP GROUPS AND TWISTED K -THEOR Y I 15 satisfying the cocycle condition, that the diagram ( L h L g ) L f & & N N N N N N N N N N N ≈ / / L h ( L g L f ) / / L h L g ◦ f   L h ◦ g L f / / L h ◦ g ◦ f of graded T -bundles o n X 3 commut es. If L → X 1 is a graded cen tral extensio n of X , then the pa ir ˜ X = ( X 0 , L ) is a graded gr o upoid o ver X , a nd the fu nctor ˜ X → X repr e s en ts ˜ X as a graded central extension o f X by T in the evident sense. Our terminolog y c o mes fro m this p oint of view. Some cons tr uctions are s impler to describe in terms of the graded T -bundles L and others in terms o f ˜ X → X . The collection of gr a ded central extensio ns of X for ms a s ymmetric monoidal 2-catego ry whic h w e denote Ext ( X, T ± ) = Ext X . The categor y of morphisms in Ext X from L 1 → L 2 is the gr oupoid of graded T -tor sors ( η , ǫ ) ov e r X 0 , equipp ed with an isomorphism η b L 1 f → L 2 f η a making η c L 1 g L 1 f / /   L 2 g η b L 1 f / / L 2 g L 2 f η a   η c L 1 g ◦ f / / L 2 g ◦ f η a commut e. The tensor pr o duct L ⊗ L ′ is the graded cen tral extension L ⊗ L ′ → X 1 with structure map L g ⊗ L ′ g ⊗ L f ⊗ L ′ f 1 ⊗ T ⊗ 1 − − − − − → 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 fr o m the symmetry of the tensor product of graded T -bundles. F o r the purp oses of twisted K -groups it suffices to work with the 1-catego ry quotient of Ext X . Definition 2.7 . The category Ext ( X ; T ± ) = Ext X is the catego ry with ob jects the graded central ex tensions of X , and with morphisms from L to L ′ the set of isomorphism classes in Ext X ( L, L ′ ) . The symmetric monoidal structure on Ext X makes Ext X in to a symmetric monoida l group oid in the evident w ay . R emark 2.8 . A 1-a utomorphisms of L consists of a graded T torso r η ov er X 0 , together with an isomor phism η a → η b ov er X 1 , satisfying the co cycle condition. In this wa y the c a tegory o f a uto mo rphisms of any twistin g ca n b e identified with the gro upoid of graded line bun dles o ver X . A graded line bundle on X defines an element of K 0 ( X ) (even line bundles go to line bundles, a nd o dd line bundles go to their nega tiv es ). The fundamen tal prop ert y relating t wistings and twisted K -theory is that the automorphism η acts o n t wisted K -theory as multiplication b y the corresp onding elemen t of K 0 ( X ) (Propo s ition 3.3). 16 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN R emark 2 .9 . (1) The formation of Ext X is functorial in X , in the sense o f 2- categories . If F : Y → X is a map o f gro upoids, and L → X 1 is a graded c en tra l extension o f X , then F ∗ L → Y 1 gives a graded central extension, F ∗ L of Y . If η → X 0 is a mor phism from L 1 to L 2 , then F ∗ η defines a morphism fro m F ∗ L 1 to F ∗ L 2 . (2) If T : Y 0 → X 1 is a natural transforma tion fro m F to G , then the graded line bundle T ∗ L determines a morphism fro m F ∗ L to G ∗ L . This is functorial in the sense that given η : L 1 → L 2 , there is a 2- morphism relating the t wo wa y s of g oing around F ∗ L 1 F ∗ η   ? ? ? ? ? T ∗ L 1        F ∗ L 2 T ∗ L 2        G ∗ L 1 G ∗ η   ? ? ? ? ? = ⇒ G ∗ L 2 . The 2-morphism is the isomorphism ( G ∗ η ) ◦ ( T ∗ L 1 ) → ( T ∗ L 2 ) ◦ ( F ∗ η ) gotten b y pulling bac k map η : L 1 → L 2 along T . It is given p oint wise ov er y ∈ Y , T y : F y → Gy by ( η Gy )( L 1 T y ) → ( L 2 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 group oids. Example 2 .10 . Supp ose that G is a group, and X = pt / /G . Then a gr aded central extension of X is just a graded cen tral extension of G by T . F o rgetting the T -bundle gives a functor from Ext X to the group oid of gra dings of X , a nd the decomposition (2.3) gives a 2- c a tegory equiv alence (2.11) Ext ( X, T ± ) ≈ Ext ( X , T ) × Hom ( X, B Z / 2 ) which is not, in gener a l, compatible with the mo noidal structure. Here Ext ( X, T ) is the 2-category of evenly graded (ie, ordinary) cen tra l extensio ns of X by T . 2.2.1. Classific ation of gr ade d c entr al extensions. W e now turn to the classifica tion of graded cent ral extensions of a gr oupoid X . In view of (2.11), it suffices to separately classify T -cen tra l extensions (graded central extensions which are purely even) and gradings. F o r the T -central extensions, firs t r ecall that the catego ry o f T -tor sors on a space Y is equiv a len t to the ca tegory whose ob jects are T -v alued Cech 1-co cycles, ˇ Z 1 ( Y ; T ), and in which a morphism from z 0 to z 1 is a Cec h 0-c o chain c ∈ ˇ C 0 ( Y ; T ) with the prope r t y that δ c = z 1 − z 0 . LOOP GROUPS AND TWISTED K -THEOR Y I 17 Now consider the double co mplex for computing the Ce ch hyper- c ohomology groups of the nerv e X • , with co e fficients in T : (2.12) ˇ C 2 ( X 0 ; T ) / / O O ˇ C 2 ( X 1 ; T ) / / O O ˇ C 2 ( X 2 ; T ) / / O O ˇ C 1 ( X 0 ; T ) / / O O ˇ C 1 ( X 1 ; T ) / / O O ˇ C 1 ( X 2 ; T ) / / O O ˇ C 0 ( X 0 ; T ) / / O O ˇ C 0 ( X 1 ; T ) / / O O ˇ C 0 ( X 2 ; T ) / / O O In terms of the Cech co cycle mo del for T -bundles, the 2-catego ry o f T -cen tral extension of X is equiv alent to the ca tegory whose ob jects are co cycles in (2.12), o f total degree 2, whose component in ˇ C 2 ( X 0 ; T ) is zero. The 1-morphisms are giv e n b y co c ha ins of total degree 1, whose cob oundary has the prop ert y its comp onent in ˇ C 2 ( X 0 ; T ) v anishes. The 2-mo r phisms ar e given by co chains of total deg ree 0. W r ite ˇ H ∗ ( X ) and ˇ H ∗ ( X 0 ) for the Cech hyper-co homology o f X , and the Cec h cohomolo gy o f X 0 resp ectiv ely . Then the group of isomorphism classes of even gr aded T -gerbes is given b y the kernel of the map ˇ H 2 ( X ; T ) → ˇ H 2 ( X 0 ; T ) , the g roup of isomorphism classes of 1-automorphisms o f any even graded T -gerb e is ˇ H 1 ( X ; T ), and the g roup of 2 -automorphisms of a n y 1-mor phism is ˇ H 0 ( X ; T ). As for the gradings, the group o f isomorphism cla sses of g radings is ker  ˇ H 1 ( X ; Z / 2) → ˇ H 1 ( X 0 ; Z / 2)  and the automorphism group of an y grading is ˇ H 0 ( X ; Z / 2) . F o r con venience, write ˇ H t rel ( X ; A ) = ker  ˇ H t ( X ; A ) → ˇ H t ( X 0 ; A )  . Prop osition 2.13. The gr oup π 0 Ext X of isomorp hism classes of gr ade d c entr al extension of X is given by the set- the or etic al ly split ex t ension (2.14) ˇ H 2 r el ( X ; T ) → π 0 Ext X → ˇ H 1 r el ( X ; Z / 2) with c o cycle c ( ǫ, µ ) = β ( ǫ ∪ µ ) , wher e β : Z / 2 = {± 1 } ⊂ T is the inclusion. The gr oup of isomorphi sm classes of automorphisms of any gr ade d c entr al extension of X is ˇ H 1 ( X ; T ) × ˇ H 0 ( X ; Z / 2) , and the gr oup of 2 -automorphisms of any morphism of gr ade d c entr al extensions is ˇ H 0 ( X ; T ) . Pr o of: Most of this result was proved in the discussion leading up to its state- men t. The decompositio n (2.11) g iv e s the exact sequence, as well as a set-theoretic splitting s : ˇ H 1 rel ( X ; Z / 2) → π 0 Ext X . 18 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN It remains to iden tify the co cycle describing the group structure. Supp ose that ǫ, µ : X → B Z / 2 are t wo gradings of X . Then b y the discussion leading up to (2.3), s ( ǫ ) s ( µ ) is the graded cen tral extension giv en b y ( s ( ǫ ) s ( µ )) f = 1 ǫ ( f ) 1 µ ( f ) ≈ 1 ǫ ( f )+ µ ( f ) , and structure map (2.15) 1 ǫ ( g ) 1 µ ( g ) 1 ǫ ( f ) 1 µ ( f ) → 1 ǫ ( g ) 1 ǫ ( f ) 1 µ ( g ) 1 µ ( f ) → 1 ǫ ( g ◦ f ) 1 µ ( g ◦ f ) . Using the canonical iden tifications 1 a 1 b = 1 a + b , and ǫ ( g ◦ f ) = ǫ ( g ) + ǫ ( f ) µ ( g ◦ f ) = µ ( g ) + µ ( f ) one c hecks that (2.15) can be identified with the a utomorphism ( − 1) ǫ ( f ) µ ( g ) of the trivialized graded line 1 ǫ ( g ) + µ ( g )+ ǫ ( f )+ µ ( f ) ≈ 1 ǫ ( g ◦ f )+ µ ( g ◦ f ) . Similarly , the structure ma p of s ( ǫ + µ ) ca n be iden tified with the identity map of the same trivialized graded line. It follows that s ( ǫ ) s ( µ ) = c ( ǫ, µ ) s ( ǫ + µ ), wher e c ( ǫ, µ ) is graded cen tral extension with with L f = 1 , a nd λ g,f = ( − 1) ǫ ( f ) µ ( g ) . Now the 2-co cycle ǫ ( f ) µ ( g ) is precisely the Alexander-Whitney formula for the cup pro duct ǫ ∪ µ ∈ Z 2 ( X ; Z / 2). This completes the proof.  One easy , but very useful conseq uence of Pro p osition 2.13 is the stacky nature of the morphism categories in Ext X . Corollary 2.1 6. L et P , Q ∈ Ext X , and f : Y → X b e a lo c al e quivalenc e. Then the fun ct or f ∗ : Ext X ( P, Q ) → Ext Y ( f ∗ P, f ∗ Q ) is an e quivalenc e of c ate gories, and so f ∗ : Ext X ( P, Q ) → Ext Y ( f ∗ P, f ∗ Q ) is a bije ction of sets.  Another useful, though somewhat tec hnical consequence of the classification is Corollary 2.17. Supp ose t hat X is a gr oup oid with the pr op ert y that the maps ˇ H 2 ( X ; T ) → ˇ H 2 ( X 0 ; T ) ˇ H 1 ( X ; Z / 2) → ˇ H 1 ( X 0 ; Z / 2) ar e zer o. If Y → X is a lo c al e quivalenc e, then the maps ˇ H 2 ( Y ; T ) → ˇ H 2 ( Y 0 ; T ) ˇ H 1 ( Y ; Z / 2 ) → ˇ H 1 ( Y 0 ; Z / 2) LOOP GROUPS AND TWISTED K -THEOR Y I 19 ar e zer o, and Ext X → Ext Y , and Ext X → Ext Y ar e e quivalenc es. Pr o of: The first assertion is a simple diagram c hase. It ha s the consequence that the maps ˇ H 2 rel ( X ; T ) → ˇ H 2 ( X 0 ; T ) ˇ H 1 rel ( X ; Z / 2) → ˇ H 1 ( X 0 ; Z / 2) ˇ H 2 rel ( Y ; T ) → ˇ H 2 ( Y 0 ; T ) ˇ H 1 rel ( Y ; Z / 2) → ˇ H 1 ( Y 0 ; Z / 2) are isomorphisms, and the second a ssertion follo ws.  F o r later reference we note the following additional consequence of Pro p osi- tion 2.13 Corollary 2.18. Supp ose t hat X is a lo c al quotient gr oup oid, and that τ is a twist- ing of X r epr esente d by a lo c al e quivalenc e P → X and a gr ade d c en tr al extension ˜ P → P . Then ˜ P is a lo c al quotient gr oup oid. Pr o of: Since the prop erty o f b eing a lo cal quotient gro upoid is an inv ariant of lo cal equiv alence, we know that P is a lo cal quotient group oid. The question is a lso lo cal in P , so w e may ass ume P is o f the form S/ /G for a compact Lie gr oup G . By our ass umptions, the action of G on S has loca lly con tractible slices. W or king still mor e lo cally in S we may assume S is contractible. But then Propo sition 2.13 implies that τ is given by a (graded) central e xtension of ˜ G → G , and ˜ P = S/ / ˜ G .  2.3. Twistings . W e now describ e the categor y Twist X of K -theory twistings on a lo cal quotien t gr oupoid X ( § A.2.2). The ob jects of Twist X are pairs a = ( P , L ) consisting of a lo cal equiv alence P → X , and a graded central extension L of P . The set of morphisms from a = ( P 0 , L 0 ) to b = ( P 1 , L 1 ) is defined to b e the colimit Twist X ( a, b ) = lim − → p : P → P 12 Ext P ( p ∗ π ∗ 1 a, p ∗ π ∗ 2 b ) , where P 12 = P 1 × X P 2 , the limit is taken over Cov P 12 and o ur notation refers to the diagram P p   P 1 P 12 π 2 / / π 1 o o P 2 . W e leav e to the reader to ch eck that Ext P ( a, b ) do es indeed define a functor on the 1-catego ry quotien t Cov P 12 . The colimit app earing in the definition of Twist X ( a, b ) is present in order that the definition be indep endent of any extraneous choices. In fact the colimit is attained at an y stage. Lemma 2.19 . F or any lo c al e quivalenc e (2.20) p : P → P 12 the map Ext P ( p ∗ π ∗ 1 a, p ∗ π ∗ 2 b ) → Twist X ( a, b ) is an isomorphism. 20 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN Pr o of: By Coro llary 2.16, for any P ′ → P in Co v P 12 , the map Ext P ( a, b ) → Ext P ′ ( a, b ) is an isomorphism. The result now follows from Corolla ry A.12.  F o r the composition law, supp ose we a re g iv en three t wistings a = ( P 1 , L 1 ) , b = ( P 2 , L 2 ) , a nd c = ( P 3 , L 3 ) . Find a P 123 ∈ Cov X and maps P 123 p 1 } } z z z z z z z z p 2   p 3 ! ! D D D D D D D D P 1 P 2 P 3 (for example one could take P 123 to be the (2-category) fib er pro duct P 1 × X P 2 × X P 3 , and p i pro jection to the i th factor). By Lemma 2.19, the maps Ext P 123 ( p ∗ 1 a, p ∗ 2 b ) → Twist X ( a, b ) Ext P 123 ( p ∗ 1 a, p ∗ 2 c ) → Twist X ( a, c ) Ext P 123 ( p ∗ 1 b, p ∗ 2 c ) → Twist X ( b, c ) are bijections. With these identifications, the composition law in T wist X is formed from that in Ext P 123 . W e leave to the reader to c heck that this is well-defined. The for mation of Twist X is functor ial 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 ) . Prop osition 2 .21. The asso ciation X 7→ Twist X is a we ak pr eshe af of gr oup oids. If Y → X is a lo c al e quivalenc e then Twist Y → T wist X is an e quivalenc e of c ate- gories.  There is an eviden t functor Ext X → T wist X . Prop osition 2.22. When X satisfies the c ondition of Cor ol lary 2.17, t he functor (2.23) Ext X → T wist X is an e quivalenc e of c ate gories. Pr o of: Lemma 2.19 shows that (2.23) is fully faithful. Essential surjectivity is a consequence of Corollary 2.17.  LOOP GROUPS AND TWISTED K -THEOR Y I 21 Corollary 2.24. If Y → X is a lo c al e quivalenc e, and Y satisfies the c onditions of Cor ol lary 2.17 then the fun ctors Twist X → T wist Y ← Ext Y ar e e quivalenc es of gr oup oids.  Combinin g this with Prop osition 2 .13 g iv es Corollary 2.2 5. The gr oup π 0 Twist X of isomorphism classes of twistings on X is the set-the or etic al ly split extension ˇ H 2 ( X ; T ) → π 0 Twist X → ˇ H 1 ( X ; Z / 2) with c o cycle c ( ǫ, µ ) = β ( ǫ ∪ µ ) . The gr oup of automorphisms of any twisting is ˇ H 1 ( X ; T ) × ˇ H 0 ( X ; Z / 2) .  W e now switch to the po in t of v iew of “fib ered categories” in order to more e asily describ e the functor ia l pr operties of t wisted K -groups. Let Ext denote the ca tegory whose ob jects are pair s ( X , L ) consisting of a group oid X and a g raded cen tra l 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 . W e identify morphisms f , g : ( X , L ) → ( Y , M ) if there is a natural transformation T : f → g for whic h the follo wing diagram commut es: f ∗ M η T   L 7 7 o o o o o o ' ' O O O O O O g ∗ M The functor ( X , L ) 7→ X from Ext to group o ids represents Ext a s a fibered catego r y , fiber ed over the categor y of group oids. Similarly , w e define a ca tegory Twist with o b jects ( X , a ) cons is ting of a gro upoid X , and a K -theory t wisting a ∈ Twist X , and mor phisms ( X , a ) → ( Y , b ) to be equiv alence classes of pairs c onsisting of a fu nctor f : X → Y and an isomorphism a → f ∗ b in Twist X . There is an inclusion Ext → Twist cor respo nding to the inclusion Ext X → Twist X . Corollar y 2.24 immediately implies Lemma 2. 26. Su pp ose t hat F : Ext → C is a functor sending every morphism ( f , t ) : ( X , L ) → ( Y , M ) in which f is a lo c al e quivalenc e to an isomorphism. Then ther e is a factorization Ext   F / / C Twist F ′ < < z z z z z Mor e over any two such factorizations ar e natur al ly isomorphic by a unique natur al isomorphi sm. 22 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN 2.4. Examples of twistings. Example 2.27 . Suppo se that X is a space, P → X a principal G bundle, and ˜ G → G G ǫ − → Z / 2 is a graded cen tral extension of G . Then P / /G → X is a lo cal equiv alence, P / / ˜ G is a graded cen tra l ex tensio n o f P / /G , and ( P / / ˜ G, P / /G ) represents a t wis ting of X . Example 2.28 . As a specia l case, we note that any double cov er P → X defines a t wisting. In this case G = Z / 2, ˜ G → G is the trivia l bundle, and ǫ : G → Z / 2 is the iden tit y map. An y co homology theor y can b e twisted by a double cov er, and in fact these are the only t wistings of ordinary cohomology with in teger co efficien ts. Example 2.29 . Suppo se that X = pt / /G , with G a compact Lie gr oup. In this case every lo cal equiv a lence ˜ X → X admits a section. Mor eo ver the inclusion { Id X } → Cov X of the trivial category consisting of the identit y map of X into Co v X is an equiv a - lence. It follo ws that Ext (pt / /G, T ± ) → Twist X is an equiv a lence o f categor ies, and so twistings of X in this ca se a r e just gra ded cent ral ex tensions of G . Using Corollar y 2.2 5 , one can draw the sa me co nclusion for S/ /G when S is contractible. W e now describe the main example of twistings used in this pap er. Example 2.30 . Suppose that G is a connected compact Lie gr oup, and consider the path-lo op fibration (2.31) Ω G → P G → G. W e rega rd P G a s a pr incipal bundle ov er G with structure gro up Ω G . The group G acts on ev e r ything by conjugation. W rite LG for the group Map( S 1 , G ) of smo oth maps from S 1 to G. The homomor phism “ ev aluatio n a t 1” : LG → G is split by the inclusion of the constan t lo ops. This exhibits LG as a semidirect product LG ≈ Ω G ⋊ G. The g roup LG acts on the fibration (2.3 1) by co njugation. The action of LG on G factors throug h the action of G on itself b y conjugation, through the map LG → G . This defines a map o f groupo ids P G/ /LG → G/ /G which is easily c heck ed to be a lo cal equiv alence. A gr aded central extension ˜ LG → LG then defines a t wisting of G/ /G . W e will write τ for a t ypical t wisting of X , and write the mo noidal structure additively: τ 1 + τ 2 . W e will use th e notations ( ˜ P τ , P τ ) a nd ( L τ , P τ ) for t y pical representing gr aded cent ral extensions. This is consistent with wr iting the monoidal structure additiv ely: L τ 1 + τ 2 = L τ 1 ⊗ L τ 2 . LOOP GROUPS AND TWISTED K -THEOR Y I 23 Example 2.32 . Suppo se that Y = S/ /G and H ⊂ G is a normal subgroup. W r ite X = S/ /H , and f : X → Y for the natural map. If τ is a t wisting of Y then f ∗ τ has a natura l action of G/H (in t he 2-categor y sens e), and the map X → Y is in v ariant under this action (again, in the 2- category sense). T o see this it is easie s t to replace X by the weakly e q uiv alent group oid X ′ = ( S × G/H ) / /G, and factor X → Y as X i − → X ′ f ′ − → Y . Since i ∗ : T wist X ′ → T wist X is a n equiv alence of 2-categories, it suffices to exhibit an action of G/H on f ′ ∗ τ . The obvious left a ction of G/H o n S × G/H commutes with the righ t action of G , g iving an a c tio n of G/H on X ′ commut ing with f ′ . The action of G/H on f ′ ∗ τ is then a consequence of naturalit y . Example 2.33 . By wa y of illu stration, co nsider the situation of Exa mple 2.32 in which H is co mm utative, S = { pt } , and τ is given by a central extension T 7→ ˜ G → G. W r ite T 7→ ˜ H → H , for the restriction o f τ to H and assume, in addition, that ˜ H is comm utative. Then the action of G/ H o n f ∗ τ constructed in Example 2.32 w orks out to be the natura l action of G/H on ˜ H given b y conjugation. 3. Twisted K-gr oups 3.1. Axioms. Before turning t o t he definition of twisted K -theor y , w e list some general prop erties describing it as a cohomology theor y on the categ ory Twist o f lo cal quotient g r oupoids equipp ed with a twisting. These prop erties almost uniquely determine twisted K -theory , and suffice to make our main computation in Section 4. Twisted K -theory is going to be homotopy inv ariant, so we need to define the notion of homotop y Definition 3.1. A homotopy betw een tw o maps f , g : ( X , τ X ) → ( Y , τ Y ) is a map ( X × [0 , 1] , π ∗ τ X ) → ( Y , τ Y ) ( π : X × [0 , 1] → X is the pro jection) whose restriction to X × { 0 } is f , and to X × { 1 } is g . Twisted K -theory is also a cohomo logy theory . T o state this prop erly inv o lv e s defining the r elativ e twisted K theor y of a triple ( X , A, τ ) consisting o f a lo cal quotient g roupo id X , a sub-gr oupoid A , and a twisting of X . W e fo r m a category of the triples ( X , A, τ ) in the s ame wa y we for med Twist . W e’ll ca ll this the c ate gory of p airs in Twist . W e now turn to the a xiomatic proper ties o f twisted K -theory . Prop osition 3.2. The a sso ciation ( X, A, τ ) 7→ K τ + n ( X, A ) to b e c onstru cte d in § 3.4 is a c ontr avariant homo topy functor on the c ate gory of p airs ( X , A, τ ) in Twist , taking lo c al e quivalenc es to isomorphisms. 24 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN Prop osition 3.3. The functors K τ + n form a c ohomolo gy the ory: i) t her e is a natu r al long exact se quenc e · · · → 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 (ful l) sub gr oup oid whose closur e is c ontaine d in the interior of A , then the r estriction (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 natur al map K τ + n ( X, A ) → Y α K τ α + n ( X α , A α ) is an isomorphism. The com bination of ex c is ion and the long exa ct sequence of a pair gives the May er- 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 a s the union of tw o sub-gr oupoids whose interiors for m a covering. Prop osition 3.4. i) Ther e is a biline ar p airing K τ + n ( X ) ⊗ K µ + m ( X ) → K τ + µ + n + m ( X ) which is a sso ciative and (gr ade d) c ommutative up to t he natur al isomorphisms o f twistings c oming fr om Pr op osition 3.2. ii) S u pp ose that η : τ → τ is a 1 -morphism, c orr esp onding to a gr ade d line bund le L on X . Then η ∗ = multiplic ation by L : K τ + n ( X ) → K τ + n ( X ) wher e L is r e gar de d as an element of K 0 ( X ) and the multiplic ation is that of i) Twisted K -theory also re duces to equiv a riant K -theory in sp ecial cases. Prop osition 3. 5. L et X = S/ /G b e a glob al quotient gr oup oid, with G a c omp act Lie gr oup, and τ a twisting given by a gr ade d c en t r al extension T → G τ → G ǫ : G → Z / 2 . i) If ǫ = 0 then then K τ + n ( X ) is the summand K n G τ ( S )(1) ⊂ K n G τ ( S ) on which T acts via its standar d (defining) r epr esentation. This isomorph ism is c omp atible with the pr o duct stru ctur e. LOOP GROUPS AND TWISTED K -THEOR Y I 25 ii) F or gener al ǫ , K τ + n ( X ) is isomorphic to K n +1 G τ ( S × ( R ( ǫ ) , R ( ǫ ) \ { 0 } ))(1) , in which the symb ol R ( ǫ ) denotes the 1 -dimensional r epr esentation ( − 1) ǫ of G τ . In part ii), When ǫ = 0, then R ( ǫ ) is the trivial representation, and the isomor- phism can b e compo sed with the suspens io n iso morphism to give the isomorphism of i). When τ = 0, so that G τ ≈ G × T , Pr opos ition 3.5 reduces to an isomorphism K τ + n ( X ) ≈ K n G ( S ) . In view of th is, we’ll often write K τ + n G ( S ) = K τ + n ( X ) in case X = S/ /G and ǫ = 0. Of cour se there is also a relative v er sion of Prop osi- tion 3.5. The r eader is referred to Section 4 o f [21] for a mor e in depth discussio n of the t wistings of equiv ariant K -theory , a nd in terpretation of “ ǫ ” part the t wisting in terms of graded represent ations. Using the Ma yer-Vietoris sequence one can easily c heck th at result of part i) of Propos ition 3.5 holds for any lo cal quotien t groupoid X . If the t wisting τ is represented b y a central extensio n P → X , then the res triction mapping is an isomorphism K τ + n ( X ) ≈ K n ( P )(1) . In this way , once K -theory is defined for group oids, t wisted K -theory is also defined. 3.2. Twisted Hi l b ert spaces. Our definition of twisted K -theory will be in terms of F redholm op e r ators on a t wisted bun dle of Hilbert spaces. In this section we describ e ho w one asso ciates to a graded central extension of a gro upoid, a twisted notion of Hilb ert bundle. W e refer the reader to App endix A, § A.4 for our notation and c o n vent ions o n bundles o ver gr oupoids, and to § A.4 for a discussion o f Hilbert space bundles. Let X be a gro upoid, and τ : ˜ X → X a graded cen tral extension, whose asso ci- ated graded T -bund le we denote L τ → X 1 . As in Appendix A.3, we will use a f − → b and a f − → b g − → c to refer to g e ner ic points o f X 1 and X 2 resp ectiv ely , and so, for example, in a context describing bundles over X 2 , the symbol H b will refer to the pullback of X along the map X 2 → X 0 ( a → b → c ) 7→ b. Definition 3.6. A τ -twisted Hilb ert bundle on X consists of a Z / 2-gr aded Hilbe rt bundle H on X 0 , together with an isomorphism (on X 1 ) L τ f ⊗ H a → H b 26 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN satisfying the cocycle condition that L τ g  L τ f H a  o o / /    L τ g L τ f  H a   L τ g H b / / H c L τ g ◦ f H a o o commut es on X 2 . R emark 3.7 . Phr ased differently , a t wis ted Hilbert bundle is just a gr a ded Hilb ert bundle o ver ˜ X , with the pr oper ty that the map H a → H b induced b y ( a f − → b ) ∈ ˜ X 1 has degree ǫ ( f ), and for whic h the central T acts according to its defining character. Example 3.8 . Supp ose t hat X is of the form P / /G , and tha t our twisting cor re- sp o nds to a cen tral extension of G τ → G . Then a pro jectiv e unitary repr esen tation of G (meaning a representation o f G τ on which the central T acts a ccording to its defining c ha racter) defines a twisted Hilbert bundle ov e r X . Suppos e that τ and µ are graded central extensions of X , with ass o ciated graded T -torsors L τ and L µ . If H is a τ -twisted Hilb ert bundle ov er X a nd W is a µ - t wisted Hilbert bundle, then the graded tensor pr oduct H ⊗ W is a ( τ + µ )-twisted Hilber t bundle, with structure map L τ + µ f ⊗ H a ⊗ W a = L τ f ⊗ L µ f ⊗ H a ⊗ W a → L τ f ⊗ H a ⊗ L µ f ⊗ W a → H b ⊗ W b . Now supp ose that H 1 and H 2 are τ -twisted, graded Hilbert bundles ov e r X . Definition 3.9 . A line ar t ra nsformation T : H 1 → H 2 consists o f a linear trans- formation of Hilber t bundles T : H 1 → H 2 on X 0 for which the following diagram commut es on X 1 : L f H 1 a − − − − → H 1 b 1 ⊗ T   y   y T L f H 2 a − − − − → H 2 b . If L τ is a graded central extension o f X , we’ll write U τ X (or just U τ if X is understo od) for the category in which the ob jects ar e τ -twisted Z / 2-graded Hilb ert bundles, and w ith mo rphisms the linear isometric emb e ddings. If f : Y → X is a map, there is an evident functor f ∗ : U τ X → U f ∗ τ Y . A natural tra nsformation T : f → g of functors X → Y gives a natural transfor- mation T ∗ : f ∗ → g ∗ . Using Remark 3.7 a nd descent, one easily chec ks that f ∗ is an equiv a lence o f categories when f is a lo cal equiv a lence. The catego ry U τ X is also functoria l in τ . Indeed, supp ose that η : τ → σ is a morphism, given by a gr aded T -bundle η , and an isomorphism η b ⊗ τ f → σ f ⊗ η a . If H is a τ -twisted Hilbe r t bundle, then H ⊗ η is a σ -t wisted Hilb ert bundle. One easily chec ks that H 7→ H ⊗ η giv es an equiv alence of catego ries U τ X → U σ X , with in verse H 7→ H ⊗ η − 1 . The 2-morphisms η 1 → η 2 give natura l isomorphisms of functors. LOOP GROUPS AND TWISTED K -THEOR Y I 27 The tensor pro duct o f Hilbert spaces gives a natural tensor pro duct U τ X × U µ X → U τ + µ X . 3.3. Univ ersal Twisted Hilb ert Bundl es. W e now turn to the existence o f sp e c ia l kinds of τ -twisted Hilbert bundles, follo wing the discussion o f § A.4. W e keep the notation of § 3.2. Definition 3.10. A τ - t wisted Hilb ert bundle H on X is i) u niversal if for every τ -twisted Hilb ert space bundle V there is a unitar y em- bedding V → H ; ii) lo c al ly universal if H | U is univ er sal for every open sub-gro upoid U ⊂ X ; iii) absorbing if fo r e very τ -twisted Hilbe r t space bundle V there is a n isomorphism H ⊕ V ≈ H ; iv) lo c al ly absorbing if H | U is absorbing for ev er y op en sub-group oid U ⊂ X . As in Appendix A.4, if H is (locally) universal, then H is automa tically absorb- ing. Lemma 3. 1 1. Su pp ose that ˜ X → X is a gr ade d c entr al ext ension and H is a gr ade d Hilb ert bund le on ˜ X . L et H (1) ⊂ H b e the eigenbund le on which the c entr al T acts ac c or ding to i ts defining r epr esentation. Then H (1) is a τ -twiste d Hilb ert bund le on X which is (lo c al ly) universal if H is.  Lemma 3.12. If X is a lo c al quotient gr oup oid, and τ : ˜ X → X is a gr ade d c entra l extension then ther e exists a lo c al ly universal τ -twiste d Hilb ert bun d le H on X . The bund le H is unique up to unitary e quivalenc e. Pr o of: By Corolla ry 2.18, ˜ X is a lo cal quotient group oid which, by Corol- lary A.33, admits a lo cally universal Hilb ert bundle. The result now follows from Lemma 3.11.  3.4. Definition of twisted K -groups. Our task is to define t wisted K -groups for pairs ( X, A, τ ) in Twist . In view of Lemma 2 .26 it suffices to define f unctors K τ + ∗ ( X, A ) for ( X , A, τ ) in Ext , a nd show that they take lo cal equiv alences to isomorphisms. W e will do this b y using spaces of F redho lm operato rs to construct a spectrum K τ ( X, A ) and defining K τ + n ( X, A ) = π − n K τ ( X, A ). The reader is referred to § A.5 for some ba c kg round discuss ion on spac e s of F redholm op erators. Suppos e then that ( X , τ ) is an ob ject of Ext and H is a lo cally univ er sal, τ - t wisted Hilber t bundle o ver X . With the notation of § 3.2, H is given b y a Hilb ert bundle H ov e r X 0 , equipped with an isomor phism (3.13) L τ f ⊗ H a → H b ov er f : a → b ∈ X 1 , satisfying the coc y cle co ndition. The ma p T 7→ Id ⊗ T is a homeomorphism betw een the spaces of F redholm op erators (See § A.5 ) F red ( n ) ( H a ) and F red ( n ) ( L τ f ⊗ H a ) compatible with the structure ma ps (3.13). The spaces F r ed ( n ) ( H a ) therefore form a fiber bundle ov e r F red ( n ) ( H ) o ver X . 28 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN W e define spa ces K τ ( X ) n b y K τ ( X ) n = ( Γ( X ; F red (0) ( H )) n ev en Γ( X ; F red (1) ( H )) n odd , By an obvious mo dification of the a rgumen ts of Atiy ah-Sing er [1], the results describ ed in § A.5 hold for the bundle F red ( n ) ( H ) ov er X . In particular, the maps (A.44) and the homeomorphism (A.45) g iv e weak homo top y equiv alences K τ ( X ) n → Ω K τ ( X ) n +1 , making the collection o f spaces K τ ( X ) = { K τ ( X ) n } in to a spectrum. Definition 3.14. Suppo s e that ( X , τ ) is a lo c al quotien t gro upoid equipp ed with a graded central extension τ , and H is a lo cally universal, τ -twisted Hilber t bundle ov er X . The twiste d K -the ory s p e ctrum of X is the sp ectrum K τ ( X ) defined above. T o k eep thin gs simple, w e do not indicate the choice of Hilbert bundle H in the notation K τ ( X ). The v alue o f the twisted K -group is, in the e nd, independent of this c hoic e . See Remark 3.17 b elo w. W e no w turn to the funct orial prop erties of X 7→ K τ ( X ). Supp ose that f : Y → X is a map of lo cal quo tien t groupoids, and τ is a t wisting of X . Let H X be a τ -twisted, locally univ ersal Hilbert bundle over X , and H Y an f ∗ τ -twisted, locally univ ersal Hilb ert space bundle o ver Y . Since H Y is universal, there is a unitary embedding f ∗ H X ⊂ H Y . Pick one. There is then an induced map f ∗ F r ed ( n ) ( H X ) → F red ( n ) ( H Y ) T 7→ T ⊕ ǫ, ( ǫ is the base point) a nd s o a map of sp ectra f ∗ : K τ ( X ) → K τ ( Y ) . Suppos e that η : σ → τ is a morphism of central extensio ns o f X , giv en by a graded T - bundle η over X 0 , and isomorphism η b ⊗ σ f → τ f ⊗ η a . If H is a lo cally universal σ -twisted Hilb ert bundle, then H ⊗ η is a lo cally univ er sal τ -twisted Hilbert bundle. The map T 7→ T ⊗ Id η then g ives a homeomorphism F red ( n ) ( H ) → F red ( n ) ( H ⊗ η ), and so an iso morphism of spectra η ∗ : K σ ( X ) → K τ ( X ) . Since automorphisms of η comm ute with the identit y map, 2- morphisms o f twistin gs hav e no effect on η ∗ . In this wa y the a sso c ia tion τ 7→ K τ ( X ) ca n b e made into a functor on Ext X . Now w e come to an imp ortant point. Suppos e Y → X is the inclusion of a (full) subgro upoid of a lo cal quotient gr oupoid, and H X is lo cally uni versal. By Corollar y A.34, w e may then tak e H Y to b e f ∗ H X . The bundle of sp ectra K τ ( Y ) is then just the res triction of K τ ( X ). This would not b e true for g eneral group oids and LOOP GROUPS AND TWISTED K -THEOR Y I 29 is the reason for o ur restriction to lo cal quo tien t group oids. W e us e this restriction prop ert y in the definition of the twisted K -theory of a pair. While this could b e av oided, the restriction prope r t y pla y s a k ey role in the proof of excis ion, and does not appear to b e easily a voided there. Definition 3. 15. Supp ose that A ⊂ X is a sub-g r oupoid of a lo cal q uotien t group oid, and that τ is a graded central extension of X . The twiste d K -the ory sp e c- trum of ( X, A, τ ) is the homotopy fib er K τ ( X, A ) of th e restriction map K τ ( X ) → K τ ( A ). If w e write Γ( X, A ; F r ed ( n ) ( H )) ⊂ Γ( X ; F red ( n ) ( H )) for the subspace of s ections whose restriction to A is the basep oin t ǫ , then K τ ( X, A ) n = Γ( N , A ; F red ( n ) ( H )) , where N is the mapping cylinder o f A ⊂ X N = X ∐ A × [0 , 1] / ∼ . Definition 3 .16. The twisted K -group K τ + n ( X, A ) is the group π − n K τ ( X, A ) = π 0 K τ ( X, A ) n . R emark 3 .1 7 . There are several unspecified choices that go in to the definition of the sp ectra K τ ( X, A ), and the induced maps b e tw een them as X , A a nd τ v a r y . It follows from Propos itions A.35 and A.36 that these c hoices are pa r ameterized b y (weakly) con tractible spaces, a nd so hav e no effect on the homotopy in v ar ian ts (such as twisted K -g r oups, and maps of t wisted K -gr oups) derived from them. 3.5. V e rification of the axioms. 3.5.1. Pr o of of Pr op osition 3.2: fun ct oria lity. Most o f this re s ult w as proved in the pro cess o f defining the gro ups K τ + n ( X, A ). F unctor ialit y in Ext follows from the discuss ion of § 3.4 and Remark 3.17. F or ho motop y inv ar iance, note that if H is a locally universal τ -twisted Hilb ert bundle o ver ( X , A ), then π ∗ H is a lo cally univ ersal Hilbert spa ce bundle over ( X , A ) × I , and so K τ n (( X, A ) × I ) = K τ n ( X, A ) I , and the tw o r estriction maps to K τ n ( X ) cor respo nd to ev aluation of paths at the t wo endpo in ts. The tw o res tr iction maps are thus homotopic, and homotopy in v a riance follows easily . The asse r tion about lo cal equiv alence s is an immediate conse q uence of descent. As r emarked at the b eginning of § 3.4, this, in turn, gives functoriality o n the cate- gory of pairs in Twist . 3.5.2. Pr o of of Pr op osition 3.3: c ohomolo gic al pr op erties. The long exa ct sequence of a pair (as sertion i)is just the long exact sequence in homotopy groups ass o ciated to the fibration of spectra K τ ( X, A ) → K τ ( X ) → K τ ( A ) , The “wedge axiom” (par t iii) is immediate from th e definition. More significa n t is exc is ion (part ii). In describing the pro of, we will freely use, in the context of 30 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN group oids, the ba sic co ns tr uctions of homoto py theory as describ ed in A.2.1. W rite U = X \ Z and let N b e the double mapping cylinder of U ← U ∩ A → A. Then the map N → X is a homotopy eq uiv a lence of group oids, a nd so by the diagram 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 equiv alence. 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 w ea k equiv alence. W e therefore need to show that for each n , the map K τ ( N , A ) n → K τ ( N ′ , U ∩ A ) n is a weak e q uiv alence. Let H b e a lo cally universal τ -t wisted Hilb ert bundle over X . Then the pullbac k of H to each of the (lo cal quotient) group oids N , N ′ A , U , U ′ , U ∩ A is also lo cally universal. It follows that the t wisted K -theory spectr a of each of these groupo ids is defined in ter ms of sections of the bundle pulled bac k from F red ( n ) ( H ⊗ C 1 ). T o simplify the notation a lit tle, let’s denote all of these pulled back bundles F r ed ( n ) . Now consider the diagr am Γ( N , A ; F red ( n ) ) − − − − → Γ( N ′ , U ∩ A ; F re d ( n ) )   y   y K τ ( N , A ) n − − − − → K τ ( N ′ , U ∩ A ) n . W e ar e to show that the b ottom row is a weak equiv alence. But the top row is a homeomorphism, and the vertical arrows are weak equiv a lences since the maps ( N ∪ cyl( A ) , A ) → ( N , A ) ( N ′ ∪ cyl( U ∩ A ) , U ∩ A ) → ( N ′ , U ∩ A ) are relative homotopy equiv a le nce s . 3.5.3. Pr o of of Pr op osition 3.4: Multiplic ation. The m ultiplication is derived fro m the pairing F r ed ( n ) ( H 1 ) × F re d ( m ) ( H 2 ) → F red ( n + m ) ( H 1 ⊗ H 2 ) ( S, T ) 7→ S ∗ T = S ⊗ Id + Id ⊗ T , the tensor s tr ucture on the categor y of t wisted Hilber t space bundles describ ed in § 3.2, and the natural identification of the Z / 2 -graded tensor pro duct C ℓ ( R n ) ⊗ C ℓ ( R m ) ≈ C ℓ ( R n + m V er ification of par t i) is left to the reader . Even if one of S , T is not acting on a loca lly universal Hilb ert bundle the pro duct S ∗ T will. This is particularly useful when des c r ibing the pro duct of an element of un twisted K -theory , with one of t wisted K -theory . F or example if V is a vector bundle ov er X , w e can choos e a Hermitian metric on V , regar d V as a bundle of LOOP GROUPS AND TWISTED K -THEOR Y I 31 finite dimensional g r aded Hilb ert spaces, with o dd comp o nen t 0, a nd take S = 0. Then S ∗ T is just the iden tity map of V tenso red with T . More generally , a vir tual difference V − W of K 0 ( X ) can b e repr esen ted by the odd, skew-adjoin t F redholm op erator S = 0 o n the graded Hilber t spa ce whose even part is V a nd whose o dd part is W , a nd S ∗ T repr esen ts the pro duct of V − W with the class represented by T . The asser tion of Part ii) is the sp ecial case in whic h V is a graded line bundle. 3.5.4. Pr o of of Pr op osition 3.5: Equivariant K -the ory. Let X = S/ /G b e a glo ba l quotient , and τ a t wisting g iv en by a graded central ex tension G τ of G , a nd a ho mo- morphism ǫ : G → Z / 2. Replacing X with X × ( R ( ǫ ) , R ( ǫ ) \ { 0 } ) and using (3.19), if necessary , w e ma y reduce to the ca se ǫ = 0. W rite V (1) for the summand of V = C 1 ⊗ L 2 ( G τ ) ⊗ ℓ 2 on which the central T acts accor ding to its defining character. Then H = S × V (1) is a locally universal Hilb ert bundle. Our definition of K τ ( X ) beco mes K τ − n ( X ) = [ S, F red ( n ) ( C n ⊗ V (1))] G which is the summand of [ S, F red ( n ) ( C n ⊗ V )] G τ corresp onding to the defining repres e ntation of T . So the re s ult follows from the fact that F red ( n ) ( C n ⊗ V ) is a c la ssifying space for K − n G τ . While this is cer tainly well- known, we were unable to find an explicit statement in the literature. It fo llows easily fr o m the ca se in whic h G is trivial. Indeed, the univ er sal index bundle is classified b y a map to any class ifying s pa ce for eq uiv a rian t K -theor y , and it suffices to sho w that this map is a weak equiv a lence on the fixed p oin t spaces for the clo sed subgroups H of G . The a ssertion for the fixed point spaces eas ily reduces to the main result of [1]. 3.6. The Thom isomorphis m, pushforw ard, and the Pon try agin pro duct. W e b egin with a gener al discussion. Let E = { E n t n − → Ω E n +1 } ∞ n =0 be a s pectrum. F o r a real v ector space V , equipp ed with a po s itiv e definite metric let Ω V ( E n ) denote the spac e of ma ps fro m 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 sp ectrum Ω V E . An isomorphism V ≈ R k gives an ident ification Ω V E n ≈ E n − k , a nd of Ω V E with the sp e c tr um derived fro m E by simply shifting the indices. Suc h a sp ectrum is c alled a “shift desuspens io n” o f E (see [25]). Some careful orga nization is require d to av oid enco un tering signs by moving lo op co ordinates past e a c h other. The reader is referred to [25] for mo re details. O f course, for a s pace X one has  Ω V E  n ( X ) ≈ E n ( X × ( V , V \ { 0 } )) ≈ E − k + n ( X ) . Now supp ose that V is a v ector bun dle of dimensio n k over a space X . The construction described ab ov e ca n b e formed fiber wise to form a bundle Ω V E = { Ω V E n } of spectra ov er X . The group of vertical homotop y class es of sections (3.18) π 0 Γ( X, Ω V E n ) can then b e thought of as a twisted form o f [ X, E − k + n ] = E − k + n ( X ). W e denote this t wisted (genera lized) co homology g r oup E − τ V + n ( X ) . 32 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN Now the group (3.18) is the gr oup of p oint ed homotopy classe s of maps [ X V , E n ] from the Thom complex o f V to E n . This giv es a tautological Thom is omorphism ˜ E n ( X V ) = E n ( B ( V ) , S ( V )) ≈ ˜ E − τ V + n ( X ) . The more usua l Thom isomorphisms arise when a geometric construction is used to trivialize the bund le Ω V E . Such a trivia liza tion is usually called an “ E -orient ation of V .” W e now return to the case E = K , with the aim of identifying the twisting τ V with type defined in § 2. The ma in p oin t is that the action of the orthogo nal group O ( k ) o n Ω k F r ed ( n ) lifts through the Atiy a h- Singer map F red ( k + n ) → Ω k F r ed ( n ) . Our discussion o f this ma tter is inspired by the Stoltz-T eichn er [31] description of Spin-structures, and, of course Dono v an- Karoubi [14]. Let X be a lo cal quotient group oid, and V a real vector bundle over X of dimension k , a nd C ℓ ( V ) the asso ciated bundle of Cliff ord algebras. The bundle C ℓ ( V ) ⊗ H is a locally universal C ℓ ( V )-mo dule. The A tiyah-Singer c o nstruction [1] gives a map F r ed C ℓ ( V ) (C ℓ ( V ) ⊗ H ) → Ω V F r ed (0) (C ℓ ( V ) ⊗ H ) which is a weak equiv alence on global sectio ns . W e can therefore trivialize the bundle of spectra Ω V K by trivia lizing the bundle of Clifford algebr as C ℓ ( V ). Of course something w eaker will also trivialize Ω V K . W e don’t rea lly need a bundle isomor phism C ℓ ( V ) ≈ X × C k . W e just need a wa y of going back and forth b etw een C ℓ ( V )-mo dules and C k mo dules. It is enoug h to hav e a bundle of irreducible C ℓ ( V ) − C k bimo dules giving a Morita equiv alence. Let M = C k , regar ded as a C ℓ ( R k ) − C k -bimo dule. W e equip M with the Hermitian metric in which the mono mia ls in the ǫ i are orthonorma l. Co nsider the group Pin c ( k ) of pairs ( t, f ) in whic h t : R k → R k is an orthogonal map, and f : t ∗ M → M is a unitary bimo dule isomorphism. The gro up Pin c ( k ) is a gr aded cent ral extension of O ( k ), graded b y the sign of the determinan t. W e now ident ify the t wisting τ V in the terms of § 2.3. Let E → X b e the bundle of orthonormal frames in V . Thus E → X is a principal bundle with structure group O ( k ). W rite P = E / /O ( k ), ˜ P = E / / Pin c ( k ). Then P → X is a lo cal e q uiv alence. and ˜ P → P is a graded cen tral extension, defining a twisting τ . O ver ˜ P we can form the bundle of bimodules ˜ M = ( E × M ) / / Pin c ( k ) , giving a Morita equiv alence b etw een bundles of C ℓ ( V )-modules and bundles o f C k - mo dules. In pa rticular, H ′ = hom C ℓ ( V ) ( ˜ M , C ℓ ( V ) ⊗ H ) is a locally universal τ -twisted C k -mo dule, a nd the map Γ(F r ed C ℓ ( V ) (C ℓ ( V ) ⊗ H )) → Γ(F red ( k ) ( H ′ )) T 7→ T ◦ ( − ) LOOP GROUPS AND TWISTED K -THEOR Y I 33 is a homeomorphism. Thus the group K − τ V + n ( X ) is isomorphic to the t wis ted K - group K − τ + n ( X ), a nd, as in Donov an-Kar oubi [14] w e hav e a tautolog ical Thom isomorphism K n ( X V ) ≈ K − τ + n ( X ) . More generally , the same construction lea ds to a tautolo gical Thom isomor phism (3.19) K σ + n ( B ( V ) , S ( V )) ≈ K − τ + σ + n ( X ) , when V is a vector bundle ov er a groupo id X . With the Thom isomor phism in hand, o ne can define the pushforward, o r umk ehr map in the usua l wa y . Let f : X → Y be a map of smo oth manifolds, or a map of group oids forming a bundle o f smo oth manifolds, T = T X/ Y the co rresp onding relative (stable) tangent bundle, and τ 0 the twisting on X corresp onding to T . Given a twisting τ on Y , and an isomorphism f ∗ τ ≈ τ 0 one can combine the Pon tryagin-Thom collapse with the Thom-isomorphism to form a pushforw ard map f ! : K f ∗ σ + n ( X ) → K − τ + σ + n ( Y ) where σ is an y t wisting on Y . W e leav e the details to the reader. W e a pply th is to the situation in whic h X = ( G × G ) / /G , Y = G/ /G (both with the adjoint action) and X → Y is the multip lication ma p µ . In this ca s e the t wisting τ 0 can b e taken to b e the twisting we denoted g in the in tro duction. Since g is pulled bac k from pt / /G , there are canonical isomorphisms µ ∗ g ≈ p ∗ 1 g ≈ p ∗ 2 g . W e’ll just write g for any of these twistings. Supp ose σ is a n y twisting of G/ / G which is “primitive” in the sense that it comes equipp ed with an asso ciative isomorphism µ ∗ σ ≈ p ∗ 1 σ + p ∗ 2 σ . Then the group K σ + g G ( G ) acquires a P ontry a gin pro duct 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 in to an algebra ov e r K 0 G (pt) = R ( G ). 3.7. The fundamen tal sp ectral sequence. Our basic technique o f co mputation will b e based o n a v ar iation o f the Atiy ah-Hir zebruc h s pectral s equence, which is constructed using the technique o f Segal [30]. The iden tification of the E 2 -term depends o nly o n the pro perties listed in § 3.1. Suppos e that X is a lo cal quotien t groupo id, a nd write ˇ K τ + t for the pr esheaf on [ X ] giv en b y ˇ K τ + t ( U ) = K τ + t ( X U ) . W r ite K τ + t = sh ˇ K τ + t for the asso ciated sheaf. The limit of the Ma yer-Vietoris sp ectral sequences assoc i- ated to the (h yp er-)cov ers o f [ X ] is a sp ectral sequence H s ([ X ] ; K τ + t ) = ⇒ K τ + s + t ( X ) . Since X admits lo cally con tractible slices the sta lk of K τ + t at a point c ∈ [ X ] is K τ + t ( X c ) ≈ ( 0 t o dd R τ ( G x ) t even , where x ∈ X 0 is a representativ e of c , and G x = X ( x, x ). 34 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN There is also a rela tive version. Su pp ose that A ⊂ X is a pair of g roupo ids, and write K τ + t rel for the sheaf on [ X ] asso ciated to the presheaf U 7→ K τ + t ( X U , X [ A ] ∩ U ) . Then the limit of the Mayer-Vietoris sp ectral sequences ass ociated to the hyper - cov er s of [ X ] giv es H s ([ X ] ; K τ + t rel ) = ⇒ K τ + s + t ( X, A ) . This sp ectral sequence is most useful when A ⊂ X is closed, and has the prop erty that for all sufficient ly small U ⊂ [ X ], the map K τ + t ( X U ) → K τ + t ( X [ A ] ∩ U ) is surjective. In that case there is (for sufficien tly 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 ex tensio n of i ∗ K τ + t b y zero K τ + t rel = i ! ( K τ + t ) , where i : V ⊂ [ X ] is the inclusion of the co mplement of A . W e will make use of this situation in the proof of Prop osition 4.41. 4. Comput a tion o f K τ G ( G ) The aim of this section is to c o mpute the groups K τ + ∗ G ( G ) for non-degenera te τ . W e’ll b egin b y considering genera l twistin gs, and adopt the non-degener acy h yp othesis as necessary . Our main results a re Theore m 4 .27, Corollar y 4.38 and Corollar y 4.39. 4.1. Notation and assum ptions. W e fir s t fix so me no tation. Let • G b e a compact connected Lie group; • g the Lie algebr a of G ; • T a fixed maximal torus of G ; • t the Lie algebr a of T ; • N the normalizer of T ; • W = N /T the W ey l g roup; • Π = k er exp : t → T ; • Λ = hom(Π , Z ), the c har acter gro up of T ; • N e aff = Π ⋊ N T • W e aff = Π ⋊ W = N e aff /T , the extended affine W eyl gro up; The g roup W e aff can be iden tified the group of symmetries of t generated by trans- lations in Π and the reflections in W . Wh en G is connected, the exp onential map, from the orbit space t / W e aff to the space of conjugac y classe s in G , is a ho meomor- phism. W e will make our computation for groups satisfying the equiv alent conditions of the following lemma. Lemma 4.1. F or a Lie gr oup G the fol lowing ar e e quivalent i) F or e ach g ∈ G t he c entr alizer Z ( g ) is c onne cte d; ii) G is c onne cte d and π 1 G is torsion fr e e; iii) G is c onne cte d and any c entr al extension T → G τ → G splits. LOOP GROUPS AND TWISTED K -THEOR Y I 35 Pr o of: The equiv a le nce of (ii) and (iii) is elementary: Since G is connected, its classifying space B G is simply co nnected, and from the Hur e wicz theorem and the univ ersal co efficient theorem the torsion subgroup of π 1 G is isomor phic to the torsion subgro up o f H 2 ( B G ), and so to the to rsion subgroup of H 3 ( B G ; Z ). But for an y co mpact Lie gro up the odd dimensional cohomolog y o f the classifying space is torsio n—the rea l cohomo lo gy of the classifying space is in even degr e e s (and is given by in v aria n t polyno mials on the Lie a lg ebra). The implication (ii) = ⇒ (i) is [7, (3.5)]. F or the co n verse (i) = ⇒ (ii) we no te first that G = Z ( e ) is connected by hypothesis. Let G ′ ⊂ G denote the derived subgroup of G , the co nnected Lie subgroup g enerated by commutators in G , and Z 1 ⊂ G the connected comp onent of the center of G . Set A = Z ( G ′ ) ∩ Z 1 . Then fr om the principal fib er bundle G ′ → G → Z 1 / A we deduce that the tor sion subgroup of π 1 G is π 1 G ′ . W e must show the latter v anishes. Now the inclusio n π 1 Z ( g ) → π 1 G is surjective for any g ∈ G , since an y cen tralizer contains a maximal torus T of G and the inclusion π 1 T → π 1 G is surjective—the flag manifold G/ T is simply connected. It follows that Z ( g ) is c o nnected if and only if the conjugacy class G/ Z ( g ) is simply connected. F urthermore , t he conjugacy class in G o f an element of G ′ equals its conjugacy cla ss in G ′ , from which we deduce that all conjugacy classes in G ′ are connected and simply connected. Let f G ′ denote the simply connected (finite) co ver of G ′ . Then the set of co njugacy class e s in f G ′ may b e ident ified as a b ounded conv ex po lytope in the Lie algebra of a maxima l tor us, and furthermore π 1 G ′ acts on it by affine trans fo rmations with quotient f G ′ /G ′ ; see [16, § 3.9]. The cen ter of mass of the vertices of the p olytop e is a fixe d p oint of the action. The finite group π 1 G ′ acts freely on the corresp onding conjuga cy cla ss of f G ′ with quotien t a conjugacy class in G ′ . Since the former is co nnected and the latter simply connected, it follows that π 1 G ′ is trivial, as desired.  4.2. The main computation. Let X = G/ /G be the group oid formed f rom G acting o n itself by conjugation. W e will compute K τ + ∗ ( X ) = K τ + ∗ G ( G ) using the sp e c tr al sequence des c r ibed in § 3.7. In this case it tak es the form (4.2) H s ( G/G ; K τ + t ) = ⇒ K τ + s + t G ( G ) . The orbit space G/G is the space of conjugacy classes in G , whic h is ho meomor- phic via the expo nen tial map to t / W e aff . Our first task is to iden tify the sheaf K τ + t on G/G ≈ t / W e aff . Since G/ / G a dmits lo cally contractible slice s , the stalk of K τ + t at a co njugacy class c ∈ G/G is the t wisted equiv ariant K -group K τ + t G ( c ) . A c hoice of point g ∈ c giv es an iden tification c = G/ Z ( g ), and an isomorphism (4.3) K τ + t G ( c ) ≈ K τ g + t Z ( g ) ( { g } ) ≈ ( R τ g ( Z ( g )) t even 0 t o dd W e hav e denoted b y τ g the res triction of τ to { g } / / Z ( g ), in o rder to emphasize the dependence o n the choice of g . Among other things, this proves that K τ +o dd = 0 . 36 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN The t wisting τ g corresp onds to a gr aded cen tra l extension (4.4) T → ˜ Z ( g ) → Z ( g ) . The group Z ( g ) has T for a maximal torus, and is connected when G satisfies the equiv alent 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 from the W eyl gro up of ˜ Z ( g ) to the W eyl gr oup W g of Z ( g ) is a n iso morphism, and R τ g ( Z ( g )) → R τ g ( T ) W g is an isomorphism. W e can therefore re-write (4.3) as (4.6) K τ + t G ( c ) ≈ ( R τ g ( T ) W g t ev en 0 t odd W e now refor m ulate these r emarks in order to eliminate the e x plicit c hoice of g ∈ c . W e ca n cut down the size o f c b y requiring that g lie in T . That helps, but it do esn’t eliminate the dep endence of τ g on g . W e can g et rid of the r eference to g by c ho osing a ge o desic in T from each g to the identit y element , and using it to iden tify the t wisting τ g with τ 0 . This amounts to considering the set of elemen ts o f t which exp onentiate into c . This set admits a transitive a ction of W e aff , and the stabilizer of an elemen t v is canonically isomor phic to W g where g = exp( v ). W e are thus led to look at the group oid t / / T , and the action of W e aff . W riting it this wa y , how ever, does not conv eniently displa y the action o f W e aff on the t wisting τ . F o llo wing Ex a mple 2.32, we work instead with the weakly equiv alent group oid ( W e aff × t ) / / N e aff . Consider the map K τ + t G ( G ) → K τ + t N e aff ( W e aff × t ) induced b y ( W e aff × t ) / / N e aff pro jec tion − − − − − − → t / / N e aff exp − − → G/ /G Since the right action of W e aff = N e aff /T commutes with the diagonal left action of N e aff on W e aff × t , the gro up W e aff acts on the gro upoid W e aff × t / / N e aff . The t wisting τ is fixed b y this action since it is pulled bac k from G/ /G . The left action of W e aff on ( W e aff × t ) / / N e aff therefore induces a right a ction o f W e aff on K τ + t N e aff ( t ), and the image of K τ + t G ( G ) is in v a riant : (4.7) K τ + t G ( G ) → K τ + t N e aff ( W e aff × t ) W e aff . Since W e aff = N e aff /T , the map t / / T − → ( W e aff × t ) / / N e aff is a local equiv alence, and so gives an iso mo rphism K τ + t N e aff ( W e aff × t ) ≈ K τ + t T ( t ) . There is therefore an a ction of W e aff on K τ + t T ( t ), and we may re-write (4.7) a s K τ + t G ( G ) → K τ + t T ( t ) W e aff . LOOP GROUPS AND TWISTED K -THEOR Y I 37 F o r c ∈ G/G ≈ t / W e aff , let S c = { s ∈ t | exp( s ) ∈ c } be the co rresp onding W e aff -orbit in t . A similar discussion giv es a map (4.8) K τ + t G ( c ) → K τ + t N ( W e aff × S c ) W e aff ≈ K τ + t T ( S c ) W e aff . Prop osition 4.9. If G satisfies the c onditions of L emma 4.1 then t he map K τ + t G ( c ) → K τ + t T ( S c ) W e aff c onstructe d ab ove is an isomorphism. Pr o of: Cho ose v ∈ S c , a nd let W v ⊂ W e aff be the stabilizer of v . W e then hav e an iden tificatio n S c ≈ W e aff /W v , and so an isomorphism K τ + t T ( S c ) W e aff ≈ K τ + t T ( { v } ) W v . W r ite g = exp( v ). The restriction of N e aff → G identifies W v with the W eyl gro up of 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 + t T ( { v } ) co incides with the action W v b y co njugation. The result then follo ws from (4.6).  W e now iden tify the shea f K τ + t . Since { 0 } → t is an equiv ariant homotopy equiv alence, the restriction map K τ + t T ( S c × t ) → K τ + t T ( S c × { 0 } ) is an isomor phism. Next note that the aggregate o f the restriction maps to the po in ts of S c gives a map from K τ + t T ( S c × t ) W e aff to the set of W e aff -equiv ariant maps S c → K τ + t T ( t ) , which , using the fact that W e aff acts transitively on S c , is easily chec ked to b e an isomorphism. W rite p : t → t / W e aff for the pro jection, and for an ope n U ⊂ G/G = t / W e aff set S U = p − 1 ( U ) . Let F τ + t be the presheaf which asso ciates to U ⊂ G/G the set of lo cally constant W e aff -equiv ariant maps S U → K τ + t T ( t ) . There is then a map o f pres hea ves ˇ 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. 38 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN Pr o of: Prop osition 4.9 impl ies that (4.10) is an isomorphism of stalks, hence an isomorphism.  W e now re-int erpret the sheaf F in a form more suitable to describing its coho- mology . Since Twist t / /T → T wist { 0 } / /T is a n equiv alence of categories, the restriction of τ to t / /T cor respo nds to a graded cent ral extension (4.12) T → T τ → T equipped with an a ction of W e aff . The W e aff -action fixe s T and acts on T through its quotient W , the W eyl gr oup. W rite Λ τ for the set of s plittings of (4 .12). Note that Λ τ is a torsor for Λ and inherits a compa tible W e aff action from (4.12). By Prop osition 3 .5 the group K τ +0 T ( t ) ≈ K τ +0 T ( { 0 } ) may be identified with with the set of compactly supp orted functions o n Λ τ with v alues in Z . W e will see shor tly that the action of W e aff is the combination of its natural action o n Λ τ and an action on Z g iv en by a homomo rphism ǫ : W e aff → Z / 2. W riting Z ( ǫ ) for the sign repr esen tation asso ciated to ǫ , w e then hav e an isomorphism of W e aff -mo dules (4.13) K τ +0 T ( { 0 } ) ≈ Hom c (Λ τ , Z ( ǫ )) . T o verify the claim ab out the action first no te that the automorphism group of the restriction of τ to t / /T is H 2 ( B T ; Z ) × H 0 ( B T ; Z / 2) ≈ Λ × Z / 2 ≈ R ( T ) × ≈ K 0 T (pt) × By P art ii) of Prop osition 3.4, the factor Λ acts on K τ +0 T ( { 0 } ) through its natural action on Λ τ , while the Z / 2 acts b y its sign representation. Since W e aff = Π ⋊ W , and the action of W e aff on K τ +0 T ( { 0 } ) is determined by its restriction to Π and W . The gr oup Π acts trivially on T and so it acts on K τ +0 T ( { 0 } ) through a homomorphism Π ( b,ǫ Π ) − − − − → Λ × Z / 2 . The group W do es act on T , and so on H 2 ( B T ; Z ) × H 0 ( B T ; Z / 2) ≈ Λ × Z / 2 , b y the product of the natural (reflectio n) actio n on Λ and the trivia l action on Z / 2. The restriction of the action of W e aff to W is therefore determined b y a crossed homomorphism W → Λ compatible with b , and an or dinary homomorphism ǫ W : W → Z / 2. The ma ps ǫ Π and ǫ W combine to g iv e the desired map ǫ : W e aff → Z / 2 , w hile the map b : Π → Λ and the crosse d ho momorphism W → Λ corr espond to the natural action of W e aff on Λ τ . This verifies the isomorphism (4.13) of W e aff -mo dules. W e ca n now giv e a useful description of F . First recall a co ns truction. Sup p ose X is a spa ce equipped with an action of a group Γ, a nd that G is an e q uiv aria n t sheaf o n X . W rite p : X → X/ Γ for the pro jection to the orbit space. Ther e is then a shea f, ( p ∗ G ) Γ on X/ Γ whose v alue on an op en set V is the set of Γ-in v ariant element s of G ( p − 1 V ). A very simple situation is when G is the co nstan t sheaf Z . LOOP GROUPS AND TWISTED K -THEOR Y I 39 In that ca se ( p ∗ G ) Γ is ag ain the constant sheaf Z . This will b e useful in the pro of of Prop osition 4 .18 b elow. Corollary 4.14. Write ˜ t = t × W e aff Λ τ and let p : t × Λ τ → ˜ t and f : ˜ t → t / W e aff denote the pr oje ctions. Ther e is a c anonic al isomorphism F τ +0 ≈ f c ∗  p ∗ Z ( ǫ ) W e aff  , wher e f c ∗ denotes pushforwar d with pr op er supp orts.  T o go further w e need to mak e an assumption. Assumption 4.1 5. The twisting τ is non-de gener ate in the sense that b is a monomorphi sm. In terms of the classification of twistings, this is equiv alent to re q uiring that the image of the isomorphism cla ss of τ in H 3 T ( T ; R ) ≈ H 1 ( T ; R ) ⊗ H 1 ( T ; R ) is a non-degenerate bilinear fo rm. Next note Lemma 4.16 . The map ǫ W : W → Z / 2 is trivial. Pr o of: The homo mo rphism in question cor respo nds to the elemen t in H 1 W (pt) = H 1 ( B W ; Z / 2) given b y restric ting the isomor phism class of the t wis ting a long H 1 G ( G ; Z / 2) × H 3 G ( G ; Z ) → H 1 G ( G ; Z / 2) → H 1 G ( { e } ; Z / 2) → H 1 N ( { e } ; Z / 2) ≈ H 1 W ( { e } ; Z / 2) . Since G is assumed to b e connected H 1 G (pt) = 0 and the result follo ws.  Corollary 4.17. Ther e is an isomorph ism Z ≈ Z ( ǫ ) of e qu ivari ant she aves on t × Λ τ . Pr o of: The sheaf Z ( ǫ ) is classified b y the element ˜ ǫ ∈ H 1 W e aff ( t × Λ τ ; Z / 2) pulled back from the ǫ ∈ H 1 W e aff (pt; Z / 2). By Coro llary 4.16 the r e s triction of ˜ ǫ to W is trivia l. By Assumption 4.15, the gro up Π acts freely on Λ τ , so the restrictio n of ˜ ǫ to H 1 W e aff ( t × Λ τ ; Z / 2) is also trivial. This pro ves that ˜ ǫ = 0.  Prop osition 4.18. Ther e is an isomorphism F τ +0 ≈ f c ∗ ( Z ) , wher e f : ˜ t = t × W e aff Λ τ → t /W is the pr oje ction and f c ∗ denotes pushforwar d with pr op er supp orts. 40 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN Pr o of: By Coro llaries 4.17 and 4.14 there is an isomorphism F τ +0 ≈ f c ∗  p ∗ Z W e aff  , so the result follo ws from the fact that p ∗ Z W e aff ≈ Z .  Finally , using the existence of contractible lo cal slices we can describ e the coho- mology of K τ +0 ≈ F τ +0 . Lemma 4.19. The e dge homomorphism of the L er ay sp e ctr al s e quenc e for f is an isomorphi sm H ∗ ( t / W e aff ; K τ ) ≈ H ∗ c ( ˜ t ; Z ) , wher e H ∗ c denotes c ohomolo gy with c omp act su pp orts.  T o calcula te H ∗ c ( ˜ t ; Z ) we need to understand the structure o f ˜ t . This amounts to describing more carefully the action o f W e aff on Λ τ . Lemma 4.20 . Ther e exists an element λ 0 ∈ Λ τ fixe d by W . Pr o of: The inclusio n { 0 } / /T ⊂ t / / T is equiv a riant for the action of W . W e can therefore study the action o f W on the central extension of T defined b y the restriction of τ to { 0 } / / T . Now τ started out as a t wisting of G/ / G , so o ur t wisting of { 0 } / /T is the restriction of a twisting τ G of { e } / /G . Moreov er , the action of W is derived from the action of inner automorphisms of G on τ G . Now the twisting τ G corresp onds to a cen tra l extension T → G τ → G. By our assumptions on G (Lemma 4.1), this central extension splits. Cho ose a splitting (4.21) G τ → T and let λ 0 ∈ Λ τ be the co mposition ˜ T → G τ → T . Since T is a belian, the splitting (4.21) is preserved by inner automorphisms of G . It follows that splitting e is fixe d by the inner automorphisms o f ˜ G which normalize ˜ T . The claim follows.  R emark 4.22 . Any tw o choices of λ 0 differ by a character of G , so the element λ 0 is unique if an only if the character gr oup of G is trivial. Since we’v e a ssumed that G is connected and π 1 G is to r sion free, this is in turn eq uiv alent to requiring that G be simply connected. R emark 4 .23 . A primi tive t wisting τ comes equipped with a trivializatio n of its restriction to { e } / /G , or in other words a splitt ing o f the graded cen tral extension G τ → G . A primitive t wisting therefor e comes equipp ed with a ca nonical choice of λ 0 . Using a fixed c hoice of λ 0 , w e can identify Λ τ with Λ as a W -s pa ce. T o sum up, we can make a n iden tification Λ τ ≈ Λ, the action of Π is given b y a W -equiv ariant homomorphism Π → Λ, and the W -action is the natural one on Λ. LOOP GROUPS AND TWISTED K -THEOR Y I 41 Lemma 4.2 4. When τ is non-de gener ate the W e aff -set Λ τ admits an (e quivariant) emb e dding in t . Ther e ar e finitely many W e aff -orbit in Λ τ , and e ach orbits is of t he form W e aff /W c , with ( W c , t ) a finite (affine) r efle ction gr oup. Pr o of: Since b is a mono morphism the map t = Π ⊗ R → Λ ⊗ R is an isomor- phism. The firs t asser tio n now follows fro m our identification of Λ τ with Λ. As for the finiteness of the n umber of orbits, since b is a monomorphism, the group Λ / Π is finite, and there are alrea dy only finitely many Π-orbits in Λ τ . The remaining as- sertions follo w from standard facts abo ut the action of W e aff on t (Pr opos itio ns 4.46 and 4.47 below).  Corollary 4.25. When τ is n on-de gener ate, ther e is a home omorphism ˜ t ≡ a s ∈ S t /W s with S finite, and W s a fin ite r efle ction gr oup of isometries of t . Mor e over t /W s ≡ R n 1 × [0 , ∞ ) n 2 with n 2 = 0 if and only if W s is trivial. Pr o of: This is immediate from Lemma 4.24 ab ov e and Prop osition 4.4 7 b elo w.  Since H ∗ c ([0 , ∞ ); Z ) = 0 and H ∗ c ( R ; Z ) = ( Z ∗ = 1 0 otherwise, the Kunneth form ula giv es H ∗ c ( R n 1 × [0 , ∞ ) n 2 ; Z ) = ( Z n 2 = 0 and ∗ = n 1 0 otherwise . In summary , w e hav e Prop osition 4. 26. The c ohomolo gy gr oup H ∗ c ( ˜ t ; Z ) is zer o u nless ∗ = n , and H n c ( ˜ t ; Z ) is isomorphic to the fr e e ab elian gr oup on the set of fr e e W e aff -orbits in Λ τ . Mor e functorial ly, H n c ( ˜ t ; Z ) ≈ Hom W e aff (Λ τ , H n c ( t ) ⊗ Z ( ǫ )) .  Prop osition 4.26 implies that the sp ectral sequence (4.2) collapses, giving Theorem 4.27. Supp ose that G is a Lie gr oup of r ank n satisfying the c onditions of L emma 4.1, and that τ is a non-de gener ate twisting of G/ / G , classifie d by [ τ ] ∈ H 3 G ( G ; Z ) × H 1 G ( G ; Z / 2) . The r estriction of τ to pt / /T determines a c entr al extension (4.28) T → T τ → T 42 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN with an action of W e aff . Write Λ τ for t he set of splittings of (4.28) , ǫ : W e aff → Z / 2 for the map c orr esp onding to t he r estriction of [ τ ] to H 1 N ( T ; Z / 2) ≈ H 1 ( W e aff ; Z / 2) , and Z ( ǫ ) for the asso ciate d sign r epr esentation. Then K τ + n +1 G ( G ) = 0 , and the twiste d K -gr oup K τ + n G ( G ) is given by K τ + n G ( G ) ≈ Hom W e aff (Λ τ , H n c ( t ) ⊗ Z ( ǫ )) , which c an b e identifie d with the fr e e ab elian gr oup on the set of fr e e W e aff -orbits in Λ τ , af ter cho osing a p oint i n e ach f r e e orbit. This isomorphi sm is natur al i n t he sense tha t if i : H ⊂ G is a sub gr oup of r ank n a lso satisfying the c onditions of L emma 4.1, then the r estriction map K τ + n G ( G ) → K τ + n H ( H ) is given by t he inclusion Hom W e aff ( G ) (Λ τ , H n c ( t ) ⊗ Z ( ǫ )) ⊂ Hom W e aff ( H ) (Λ τ , H n c ( t ) ⊗ Z ( ǫ )) .  R emark 4.29 . In Theo r em 4.2 7 the group W e aff acts on H n c ( t ) through the a ction of W on t . The reflectio ns th us act by ( − 1 ) and a choice o f or ien tatio n on t iden tifies H n c ( t ) with the usual sign r epresen tation of W on Z . The group K τ + n G ( G ) is a module ov er R ( G ). Our next goa l is to identify this mo dule structure. Because G is connected we c an identify R ( G ) with the ring o f W - in v ariant element s of Z [Λ] or with the conv olution algebra of compactly suppo rted functions Hom c (Λ , Z ). The algebra Hom c (Λ , Z ) a cts on Hom(Λ τ , H n c ( t ) ⊗ Z ( ǫ )) b y conv o lution, a nd o ne easily chec ks that the W -in v a riant elemen ts preser ve the W e aff -equiv ariant functions. Prop osition 4.30. Under the identific ation K τ + n G ( G ) ≈ Hom W e aff (Λ τ , H n c ( t ) ⊗ Z ( ǫ )) the action of R ( G ) ≈ Hom c (Λ , Z ) W c orr esp onds to c onvolution of functions. Pr o of: This is str a igh tforward to c heck in case G is a torus. The c a se of general G is reduced to this ca se by lo oking at the restriction map to a maximal torus and using Theorem 4.27.  Prop osition 4.30 lea ds to a very useful descr iption of K τ + n G ( G ). Cho ose a n orientation of t and hence an iden tification H n c ( t ) ≈ Z o f ab elian gro ups . By definition, the elements of Λ τ are c haracters of T τ , all of which restrict to the defining c ha racter of T . T o a function f ∈ Hom Π (Λ τ , H n c ( t ) ⊗ Z ( ǫ )) ≈ Hom Π (Λ τ , Z ( ǫ )) we asso ciate the series (4.31) δ f = X λ ∈ Λ τ f ( λ ) λ − 1 , LOOP GROUPS AND TWISTED K -THEOR Y I 43 which is the F ourier expansion of the distribution on T τ satisfying δ f ( λ ) = f ( λ ). Our next a im is to w o rk out more e x plicitly whic h distribution it is, especia lly when f comes from an element of Hom W e aff (Λ τ , H n c ( t ) ⊗ Z ( ǫ )) ≈ Hom W e aff (Λ τ , Z ( ǫ )) . Since all of the characters λ restrict to the defining character o f the central T , we’ll think o f the distribution δ f as acting on the space of functions g : T τ → C satisfying g ( ζ v ) = ζ g ( v ) for ζ ∈ T . This space is the spa ce of sections of a suitable complex line bundle L τ ov er T . The c haracter χ of a representation o f G is a function on T , and action of χ on δ f is given by χ · δ f ( g ) = δ f ( g · χ ) . Since Π and Λ ar e duals, we hav e hom(Π , Z / 2) = Λ ⊗ Z / 2 , and w e may regar d ǫ Π as an elemen t of Λ / 2Λ. This deter mines a n element λ ǫ = 1 2 ǫ Π ∈ 1 2 Λ / Λ ⊂ Λ ⊗ R / Z . Thinking o f Π as the character gro up of Λ ⊗ R / Z , the function ǫ Π : Π → Z / 2 is given by ev aluation of c haracter s on λ ǫ : ǫ Π ( π ) = π ( λ ǫ ) . Since ǫ Π is W -inv ar ian t, so is λ ǫ . F r om the e mbedding b : Π ⊂ Λ we get a map b : T = Π ⊗ R / Z → Λ ⊗ R / Z . W e’ll write F = Λ / Π for the k ernel o f this map, a nd F ǫ for the inv erse image of λ ǫ . Set F τ = Λ τ / Π . The elements of F τ can b e in terpreted 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 elemen ts on which the W eyl group W a cts freely . Prop osition 4.32. F or f ∈ Hom Π (Λ τ , H n c ( t ) ⊗ Z ( ǫ )) ≈ Hom Π (Λ τ , Z ( ǫ )) The value of t he distribution δ f on a se ction g of L τ is given by (4.33) δ f ( g ) = 1 | F | X ( λ,x ) ∈ F τ × F ǫ f ( λ ) λ − 1 ( x ) g ( x ) When f is W e aff -invariant, then δ f ( g ) = 1 | F | X ( λ,x ) ∈ F τ re g × F ǫ, r eg f ( λ ) λ − 1 ( x ) g ( x ) Pr o of: Let’s first chec k that (4.33) is w ell- defined. Under λ 7→ λπ π ∈ Π , 44 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN the term f ( λ ) λ − 1 ( x ) g ( x ) gets sen t to f ( λπ )( λπ ) − 1 ( x ) g ( x ) = ǫ ( π ) f ( λ ) λ − 1 ( x ) π − 1 ( x ) g ( x ) = ǫ ( π ) f ( λ ) λ − 1 ( x ) ǫ ( π − 1 ) g ( x ) = f ( λ ) λ − 1 ( x ) g ( x ) so f ( λ ) λ − 1 ( x ) g ( x ) does indeed dep end only on the Π-coset o f λ . T o esta blish (4 .3 3), it suffices by linearity to consider the case in which δ f is of the form δ f = λ − 1 · δ Π , with λ ∈ Λ τ , δ Π = X π ∈ Π ǫ ( π ) π − 1 , and g is an elemen t of Λ τ . In this cas e f v anishes o ff of the Π-orbit through λ , and f ( λ ) = 1. The sum (4.33) is th en (4.34) 1 | F | X x ∈ F ǫ λ − 1 ( x ) g ( x ) . By definition, for η ∈ Π, δ Π ( η ) = η ( λ ǫ ) = 1 | F | X x ∈ F ǫ η ( x ) . F o r η ∈ Λ \ Π, there is an a ∈ F with η ( a ) 6 = 1 . In that case 1 | 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 supp ose g ∈ Λ τ . Then δ f ( g ) = δ Π ( λ − 1 g ) = 1 | F | X x ∈ F ǫ λ − 1 ( x ) g ( x ) , which is (4.34). This prov es the first assertion of Prop o sition 4.32. F o r the second assertion, note that if λ ∈ F τ reg is fixed by an element of W it is fixed by a n element w ∈ W which is a reflection. By W eyl- equiv ar iance, we hav e f ( λ ) = f ( w λ ) = w · f ( λ ) = − f ( λ ), and so f ( λ ) = 0 . This g iv e s δ f ( g ) = 1 | F | X ( λ,x ) ∈ F τ reg × F f ( λ ) λ − 1 ( x ) g ( x ) . LOOP GROUPS AND TWISTED K -THEOR Y I 45 If x ∈ F is an elemen t fixed by a r eflection w ∈ W then X λ ∈ F τ reg f ( λ ) λ − 1 ( x ) g ( x ) = X λ ∈ F τ reg f ( λ ) λ − 1 ( w · x ) g ( x ) = X λ ∈ F τ reg f ( λ )( λ w ) − 1 ( x ) g ( x ) = X λ ∈ F τ reg f ( λ w ) λ − 1 ( x ) g ( x ) = − X λ ∈ F τ reg f ( λ ) λ − 1 ( x ) g ( x ) so the terms in volving suc h an x sum to zero, and δ f ( g ) = 1 | F | X ( λ,x ) ∈ F τ reg × F ǫ, reg f ( λ ) λ − 1 ( x ) g ( x )  Let I τ ⊂ R ( G ) be the ideal c onsisting o f vir tual represent ations whose c haracter v anishes on the elemen ts of F ǫ, reg . Corollary 4.35. The ide al I τ annihilates K τ + ∗ G ( G ) . Pr o of: W rite χ for the character o f an element of I τ . F or f ∈ Hom W e aff (Λ τ , H n c ( t ) ⊗ Z ( ǫ )) w e hav e, b y Pr o position 4.32 χ δ f ( g ) = δ f ( g · χ ) = X ( λ,x ) ∈ F τ reg × F reg f ( λ ) λ − 1 ( x ) g ( x ) χ ( x ) = 0 .  R emark 4 .3 6 . The conjuga cy classe s in G of the elements in F ǫ, reg are known as the V erlinde c onjugacy classes , and the ideal I τ as the V erlinde ide al . Prop osition 4.37. The R ( G ) -mo dule K τ + n G ( G ) is cyclic. Pr o of: Using Lemma 4.2 0 cho o se a W e aff -equiv ariant isomor phism Λ τ ≈ Λ, and a n or ien tatio n of t giving an isomo rphism H n c ( t ) ≈ Z . W e c a n then identify K τ + n G ( G ) with Hom W e aff (Λ , Z ( ǫ )) , though we remind the reader that W e aff acts on Z thro ugh its sign representation. W e’ll co n tinue th e conv ent ion of writing elemen ts f ∈ Hom W e aff (Λ , Z ) as F o urier ser ies X f ( λ ) λ − 1 . Set δ Π = X π ∈ Π ǫ ( π ) π − 1 , 46 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN and for λ ∈ Λ write a ( λ ) = X w ∈ W ( − 1) w w · λ. Then the elemen ts a ( λ ) ∗ δ Π span Hom W e aff (Λ , Z ( ǫ )). Since π 1 G is torsion-free, there is a n exact sequence G ′ → G → J where J is a torus, a nd G ′ is simply connected. The ch aracter group of J is the subgroup Λ W of W e y l-in v a rian t elemen ts o f Λ, and the weigh t lattice for G ′ is the quotient Λ / Λ W . Cho ose a W e y l cham b er for G and let ρ ∈ Λ ⊗ Q b e 1 / 2 the sum of the p ositive ro ots of G (which we will write a s a pro duct of s quare ro ots of elements in our F ourier series notation). Since J is a torus, the image ρ ′ of ρ in Λ / Λ W ⊗ Q is 1 / 2 the sum of the pos itiv e r oots of G ′ , which since G ′ is simply connected, lies in Λ / Λ W . Let ˜ ρ ∈ Λ b e a ny element congruent to ρ ′ mo dulo Λ W . Claim: for a n y λ ∈ Λ, the ratio a ( λ ) a ( ˜ ρ ) is the character of a (vir tual) represe n tation. The claim s ho ws that the clas s cor- resp onding to a ( ˜ ρ ) · δ Π is an R ( G )-module gener ator of K τ + n G ( G ). F or the claim, first note tha t the element µ = ρ/ ˜ ρ is W - in v a r ian t (and is in fa c t the squa re ro ot of a c har acter o f J ). It follows from the W eyl character formul a that a (( λ ˜ ρ − 1 ) · ρ ) a ( ρ ) is, up to sign, the c hara cter of an irreducible representation. But then a ( λ ˜ ρ − 1 ρ ) a ( ρ ) = a ( λ µ ) a ( ρ ) = µ a ( λ ) a ( ρ ) = a ( λ ) a ( µ − 1 ρ ) = a ( λ ) a ( ˜ ρ ) .  Corollary 4.3 8. L et U ∈ K τ + n G ( G ) b e t he class c orr esp onding to a ( ˜ ρ ) · δ Π . The map “mu lt ipl ic ation by U ” is an isomorphism R ( G ) /I τ → K τ + n G ( G ) of R ( G ) -mo dules. Pr o of: That the map factors throug h the quo tien t by I τ is Corollary 4.3 5 , and that it is surjective is P r opo s ition 4.37. The result now follows from the fact tha t bo th sides a r e free of r ank equa l to the num b e r of fr ee W -orbits in A ǫ, reg (ie, the n umber of V erlinde conjugac y class es).  As described at the end o f § 3.6, when τ is primitiv e the R ( G )-mo dule K τ + n G ( G ) acquires the structure of a n R ( G )-algebra Corollary 4.39. When τ is primitive, ther e is a c anonic al algebr a isomorphism K τ + n G ( G ) ≈ R ( G ) /I τ .  LOOP GROUPS AND TWISTED K -THEOR Y I 47 R emark 4.40 . When τ is primitive, the pushforw a rd map K τ G ( e ) → K τ + n G is a ring homomorphism. Being primitiv e, the restriction of τ to { e } / /G comes equipp e d with a trivialization and so K τ G ( e ) ≈ R ( G ). The isomorphisms of Cor ollaries 4 .38 and 4.39 a re prov ed after tenso r ing with the complex num b ers in [21], where the distributions δ f are rela ted to the Kac n umerator a t q = 1. W e refer the reader to § 6 and § 7 of [21] for further discussion. W e conclude with a further computation which will b e used in Part I I. W e consider the situation of this section in whic h G = T is a torus o f dimension n . The group K τ T ( { e } ) is the fre e ab elian group on Λ τ , and the pushforward map K τ T ( { e } ) → K τ + n T ( T ) is defined. Prop osition 4.41. The pushforwar d map i ! : K τ T ( { e } ) → K τ + n T ( T ) sends the cla ss c orr esp onding to λ ∈ Λ τ to t he class in K τ + n T ( T ) ≈ Hom Π (Λ τ , Z ( ǫ )) c orr esp onding to t he distribution with F ourier exp ansion λ − 1 X π ∈ Π ǫ ( π ) π − 1 . Pr o of: The pushforward map is the comp osition of the Thom isomorphism K τ +0 T ( { e } ) → K τ + n T ( t , t \ { 0 } ) ≈ K τ + n T ( T , T \ { e } ) with the restriction map (4.42) K τ + n T ( T , T \ { e } ) → K τ + n T ( T ) . W e wish to compute these maps using the spectral sequence for relative twisted K -theory described at the end of § 3 .7. In order to do so, how ever, w e need to replace T \ { e } and t \ { 0 } by the smaller T \ B e and t \ { B 0 } , wher e B e and B 0 are small op en balls cont aining e and 0 re s pectively . This puts us in the situation describ ed at the end of of § 3.7, where the sheaf K τ + t rel works out to be the ex tension b y zero of the restriction of K τ + t to B e . Applying the sp ectral sequence a rgument of this section to the pair s ( t , t \ B 0 ) and ( T , T \ B e ) gives isomorphisms (4.43) K τ + n T ( T , T \ B e ) ≈ hom Π (Λ τ , H n c ( t , t \ B Π ) ⊗ Z ( ǫ )) K τ + n T ( t , t \ B 0 ) ≈ hom c (Λ τ , H n c ( t , t \ B 0 ) ⊗ Z ( ǫ )) , where B Π ⊂ t is th e inv erse imag e of B e under the exponential map. The sa me argument identifies the r estriction mapping (4.42) with the map induced by H n c ( t , t \ B Π ) → H n c ( t ) , and the isomorphism K τ + n T ( T , T \ B e ) ≈ K τ + n T ( t , t \ B 0 ) with the map Hom Π (Λ τ , H n c ( t , t \ B Π ) ⊗ Z ( ǫ )) → Hom c (Λ τ , H n c ( t , t \ B 0 ) ⊗ Z ( ǫ )) which first forgets the Π-action and then uses H n c ( t , t \ B Π ) → H n c ( ¯ B 0 , B 0 ) ≈ H n c ( t , t \ B 0 ) . 48 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN Finally , the isomorphism K τ + n T ( t , t \ B 0 ) ≈ K τ T (pt) ⊗ K n ( t , t \ B 0 ) shows that the Thom isomorphism is simply the tensor pr oduct of the identit y map with susp ension isomo r phism K 0 (pt) → K n ( t , \ B 0 ) (which uses the orientation of t ). In terms of (4.43) this means that the Thom isomorphism K τ T (pt) ≈ Hom c (Λ τ , Z ( ǫ )) → K τ T ( t , t \ B 0 ) ≈ hom c (Λ τ , H n c ( t , t \ B 0 ) ⊗ Z ( ǫ )) , is simply the map der iv e d from the susp ension isomorphism H 0 (pt) ≈ H n c ( t , t \ B 0 ) . The result follo ws easily from this.  4.3. The action of W e aff on t . W e summarize here some standard fac ts ab out ab out affine W ey l groups and co njugacy classes in G . Our basic references a re [7, 22]. Reca ll that we hav e fixed a maximal torus T of G . W e write Λ for the c ha r acter group of T and R for the s et of ro ots. F o llo wing Bo urbaki, wr ite N ( T , R ) fo r the subgroup of t consisting of element s on which the ro ots v anish mo dulo 2 π Z . There is a short exact s equence N ( T , R ) ֌ Π ։ π 1 G. Let H b e the set of h yp erplanes forming the d iagr am o f 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 lo cally finite in the sense that e a c h s ∈ t has a neighbor hoo d meeting only finitely many hyperplanes in H . The affine Weyl gr oup is the gro up W aff be the group genera ted by reflections in the hyperplanes H k,α ∈ H . It has the structure N ( T , R ) ⋊ W. Prop osition 4.44. L et x ∈ t . The stabilizer of x in W aff is the fin ite r efl e ction gr oup gener ate d by r efle ctions thr ough the hyp erplanes H k,α c ontaining x .  W r ite W e aff = L ⋊ W . There is a short exact s equence (4.45) W aff ֌ W e aff ։ π 1 G. Prop osition 4.4 6 . L et x ∈ t . If π 1 G is torsion fr e e, then the stabilizer of x in W e aff c oincides with the stabilizer of x in W aff . It is ther efor e the finite r efle ction gr oup gener ate d by r efle ctions thr ough the hyp erplanes H k,α c ontaining x . Pr o of: W r ite W x for the stabilizer o f x in W e aff . The image of W x in t ⋊ W is conjugate to a subgroup of W , and so W x is finite. By assumption π 1 G has no non- trivial finite subgr oups. The exa ct sequence (4 .45) then sho ws that W x ⊂ W aff . The result then follo ws from Prop osition 4.44.  W r ite R ≥ 0 = [0 , ∞ ). Prop osition 4.4 7 . Supp ose that ( W, V ) is a finite r efle ction gr oup. Th e orbit sp ac e V /W is home omorphic to R n 1 × R n 2 ≥ 0 . The gr oup W is gener ate d by n 2 r efle ctions. In p articular, if W is non-t rivi al, then n 2 6 = 0 . LOOP GROUPS AND TWISTED K -THEOR Y I 49 Pr o of: This follows immediately from the Theor ems on pages 20 and 24 of [11].  Appendix A. Groupoids W e remind the reader that we are assuming througho ut this pap er that, unless otherwise s pecified, a ll s pa ces a r e lo cally contractible, para compact and completely regular. These assumptions implies the existence of partitions of unity [13] and lo cally con tractible slices thro ugh actions o f co mpact Lie g roups [2 7, 28]. A.1. Definitio n and Fi rst Prop erties. A gr oup oid is a category in which all morphisms are isomor phism. W e will consider gr oupoids in the category of top o- logical spaces. Thus a group oid X = ( X 0 , X 1 ) consists o f a space X 0 of ob jects, a space X 1 of mor phisms, and map “ iden tity map” X 0 → X 1 , a pa ir of ma ps “do - main” and “rang e” X 1 → X 0 , an a sso ciativ e co mposition law X 1 × X 0 X 1 → X 1 , and an “ inverse” map X 1 → X 1 . W r ite X n = X 1 × X 0 · · · × X 0 X 1 for the space o f n -tuples of comp oseable maps. Then the collection { X n } is a simplicial spa ce. The i th face map d i : X n → X n − 1 is giv en b y d i ( f 1 , . . . f n ) =      ( f 2 , . . . , f n ) i = 0 ( f 1 , . . . , f i ◦ f i +1 , . . . , f n ) 0 < i < n ( f 1 , . . . , f n − 1 ) i = n Even thoug h a group oid is a sp ecial kind o f simplicial space, we’ll re fer to the simplicial space as the nerve of X = ( X 0 , X 1 ) and write X • . Finally , we let | X | = a n X n × ∆ n / ∼ denote the geometric realization of X • . Example A.1 . (cf Segal [3 0]) Supp ose that G is a topolo g ical gro up acting on a space X . Then the pa ir ( G, X ) forms a gro upoid with s pace of ob jects X a nd in which a morphism from x to y is an element of g for which g · x = y . In this case X 0 = X a nd X 1 = G × X . The comp osition law is given by the multiplication in G . W e will wr ite X/ /G for this group oid. Example A.2 . (Segal [30]) Supp ose that X is a space and U = { U i } is a covering of X . The ner v e of the covering U is the nerve of a group oid. Indeed, let N U be the catego r y whose ob jects ar e pairs ( U i , x ) with U i ∈ U and x ∈ U i and in w hich a morphism from ( U i , x ) to ( U j , y ) is an elemen t w ∈ U i × X U j whose pro jection to U i is x and w ho se pro jection to U j is y . Then N U is a group oid. If U i and U j are open subsets of X then s uch a map exists if and o nly if x = y , in which ca se it is unique. W riting X 0 = ` U i , then X n = X 0 × X · · · × X X 0 , and the ner ve of this group oid is just the nerv e of the c overing U . Definition A.3. A map of groupo ids F : X → Y is an e quivalenc e if it is fully faithful a nd essentially surjective; tha t is, if every ob ject y ∈ Y 0 is isomorphic to one of the form F x , and if for ev ery a, b ∈ X 0 the map F : X ( a, b ) → Y ( F a, F b ) is a homeomorphism. 50 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN An equiv a lence of ordinary categories a utomatically admits a n inverse (up to natural isomorphism). Examples A.6 and A.8 b elow sho w that the same is not necessarily true of an equiv alence of topolo gical categor ies , or g roupo ids. Requiring the existence of a globally defined in verse is, on the o ther ha nd to o restrictive. Definition A.4. A lo c al e quivalenc e X → Y is an equiv alence of group oids with the additional prope rt y that each y ∈ Y 0 has a neigh bo rhoo d U admitting a lift in the diagram ˜ X 0 / /   X 0   Y 1 range / / domain   Y 0 U G G / / Y 0 in whic h the squar e is Car tesian. R emark A.5 . The term lo c al e quivalenc e der ives from t hinking of a gro upoid X as defining a shea f U 7→ X ( U ) on the categ ory of top ological spa c es. An equiv a - lence X → Y ha s a globally defined in verse if and only if for ev er y spa ce U the map X ( U ) → Y ( U ) is an equiv alence. As one easily c hecks, a map X → Y is a lo cal equiv alence if and only if it is an equiv alence on stalks. W e will say that t wo g roupo ids X a nd Y a s b eing we akly e quivalent , if there is a diagr am of lo cal equiv alences X ← Z → Y . Example A.6 . If U is an open cov ering o f a s pace X , then the map N U → X is a local equiv alence. Mor e genera lly , if U → V is a map of cov erings of X , th en N U → N V is a local equiv alence. Example A.7 . Given gr oupoids X and A , write X ( A ) for the g roupo id of maps A → X . Then a map X → Y is a loca l e q uiv alence if and only if for spaces S , the map lim − → U X ( U ) → lim − → U Y ( U ) is an equiv alence o f gr oupoids, wher e U ra nges ov er all cov erings of S . Stated more succinctly , a map o f group oids is a loc a l equiv alence if and only if the cor respo nding map of presheav es of group oids is a stalkwise equiv alence. Example A.8 . If P → X is a principal G -bundle ov er X , then P / /G → X is a local equiv alence. Example A.9 . If H ⊂ G is a subgroup, the map of groupo ids pt / /H → ( G/H ) / /G is a local equiv alence. LOOP GROUPS AND TWISTED K -THEOR Y I 51 The fiber pro duct o f functors P i   1 1 1 1 1 1 Q j         X is the group o id P × X Q whose ob jects consist of p ∈ P , q ∈ Q, x ∈ X and isomorphisms ip → x ← j q . The morphisms are th e evident co mm utative diagrams. T o give a functor S → P × X Q is to giv e a funct ors S p − → P , S q − → Q, S x − → X and natural isomorphisms i ◦ p → x ← j ◦ q . The gr oupoid P × X Q is usually called fib er pr o duct of P and Q ov er X , even though strictly sp eaking it is a kind of homotopy fib er pro duct a nd no t the c a te- gorical fiber pr oduct. W e will also say that the morphism P × X Q → Q is obtained from P → X b y change of base along j : Q → X . A na tural transfor ma tion T : j 1 → j 2 gives a natural isomorphism betw een the group oids obtained by c ha nge of base along j 1 and j 2 . One easily ch ecks that the class of loca l equiv alences is stable under comp osition and change o f base. Consequently , if P → X and Q → X a r e b oth lo cal equiv- alences, so is P × X Q → X . Using E xample A.7 o ne easily chec ks tha t if tw o of three maps in a co mposition are lo cal equiv a lences so is the third. Definition A.10. The 2-category Cov X is the category whose ob jects are lo cal equiv alences p : P → X and in which a 1 - morphism from p 1 : P 1 → X to p 2 : P 2 → X consis ts o f a functor F : P 1 → P 2 and a natural transforma tion T : p 1 → p 2 ◦ F making P 1 F / / p 1   4 4 4 4 4 4 P 2 p 2         T = ⇒ X commut e. A 2- morphism ( F 1 , T 1 ) → ( F 2 , T 2 ) is a natural transfor mation η : F 1 → F 2 for whic h T 2 = p 2 η ◦ T 1 . W e will denote b y Co v X the 1-category quotient of Cov X . The ob jects of Cov X are those of Cov X , and Cov X ( a, b ) is the set of isomorphism classes in Cov X ( a, b ). W e will see that Cov X and Cov X are not that differen t from each other. Lemma A.11. F or every a, b ∈ Cov X , the c ate gory Cov X ( a, b ) is a c o discr ete gr oup oid: ther e is a unique morphism b etwe en any two obje cts. 52 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN Pr o of: W rite a = ( F 1 , T 1 ) and b = ( F 2 , T 2 ) P 1 F 1 ,F 2 / / p 1   5 5 5 5 5 5 P 2 p 2         T 1 ,T 2 = ⇒ X . A morphism (natural transformation) η ∈ Co v X ( a, b ) asso ciates to x ∈ P 1 a map η x : F 1 x → F 2 x whose image p 2 η x : p 2 F 1 x → p 2 F 2 x is prescrib ed to fit into the diagram o f isomorphisms p 1 x T 1 x          T 2 x   ? ? ? ? ? ? ? p 2 F 1 x p 2 η x / / p 2 F 2 x. The map p 2 η x is therefore forced to b e ( T 2 x ) ◦ ( T 1 x ) − 1 , and so η x is uniquely determined since P 2 → X is an equiv alence.  Corollary A.12. The 1 -c ate gory quotient Cov X is a (c o-)dir e cte d class. Pr o of: Suppose that p i : P i → X , i = 1 , 2 are tw o ob jects of Cov X . The group oid P 12 = P 1 × X P 2 comes equipp ed with maps P 12 → P 1 and P 12 → P 2 . If f , g : P → Q ar e tw o morphisms in Cov X there is, b y Lemma A.1 1, a unique 2-morphism relating them, and s o in fa c t f = g in Cov X  A.2. F urther Prop erties of Group oids. W e now turn to several constr uctions which a r e in v aria n ts of lo cal equiv a lences. A group oid is a presen tation of a stack , and the inv a rian ts of lo cal equiv alences are in fact the inv ariants of the underlying stack. A.2.1. Point set t op olo gy of gr oup oids. The orbit sp ac e o r c o arse mo duli sp ac e of a group oid X is the space of isomorphism clas s es o f ob jects, top ologized as a quotient space of X 0 . W e denote the coarse moduli space of X by [ X ]. A t this lev e l of generality , the space [ X ] can be so mewhat pa tho lo gical, a nd without some further assumptions migh t not be in our cla ss o f lo cally cont ractible, para compact, and completely regular spaces. When X = S/ /G , then [ X ] is the or bit space S/G , a nd in that case we will revert to the more standard notation S/G . A loca l equiv alence Y → X gives a homeomorphism [ Y ] → [ X ]. F o r a subspace S ⊂ [ X ] w e denote X S the full sub-groupo id of X consisting of ob jects in the isomorphism class of S . There is a o ne to one c o rresp ondence betw een full sub-group oids A ⊂ X containing ev ery o b ject in their X -isomorphism class and subspaces [ A ] o f [ X ]. With this we transp ort many notions from the p oint set top ology of spaces to the context of gr oupoids. When S is closed (resp. op en) we will say that X S is a clos e d (resp. o pen) subgro upoid o f X . W e can sp eak o f the interior and closur e of a full subg roupo id. By an op en c overing of a group oid, we mean an op en co vering { S α } of [ X ], in which case the collection { X S α } forms a cov er ing of X by o p en sub-group oids. LOOP GROUPS AND TWISTED K -THEOR Y I 53 More g enerally , if f : S → [ X ] is a map, we can form a gr oupoid X S with ob jects the pairs ( s, x ) ∈ S × X 0 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 differen tly , the gr oupoid X S is the group o id who se nerve fits in to a pull-back squa re ( X S ) • − − − − → X •   y   y S − − − − → [ X ] . W e will say that X S is define d by pul lb ack from the map S → [ X ]. With this we can transp ort ma n y of the maneuvers of homo top y theory to the context of group oids. F o r instance if S = [ X ] × I , and f is the pro jection, then X S is the group oid X × I . One can then form mapping c ylinders and o ther similar constructions. F o r example, s uppose that [ X ] is paracompact, a nd written a s the union o f tw o sets S 0 , S 0 whose interiors co ver. W rite U i = X S i . The U i are (full) subgro upoids whose in teriors cov er X . Let N denote the g roupo id co nstructed fr o m the map cyl ( S 0 ← S 0 ∩ S 1 → S 1 ) → [ X ] . It is the double mapping c y linder o f U 0 ← U 0 ∩ U 1 → U 1 . F o llo wing Segal [3 0], a partition of unit y { φ 0 , φ 1 } sub ordinate to the covering { S 0 , S 1 } , defines a map [ X ] → cyl ( S 0 ← S 0 ∩ S 1 → S 1 ) . The composite [ X ] cyl ( S 0 ← S 0 ∩ S 1 → S 1 ) → [ X ] is the iden tity , and so we get functors X → N → X whose compo site is the identit y . On the other hand, the compo site cyl ( S 0 ← S 0 ∩ S 1 → S 1 ) [ X ] → cyl ( S 0 ← S 0 ∩ S 1 → S 1 ) is homoto pic t o the iden tit y , b y a homotop y (the ob vious line ar homotop y ) that preserves the map to [ X ] (it is a homotop y in the catego ry of spaces ov er [ X ] ). This homotopy then defines by pullback, a ho motop y N × ∆ 1 → N from the compo site N → X → N to the identit y map o f N , fixing the map to X . In this way X b ecomes a stro ng deforma tion retract of N , and N is decomp osed in a w ay esp ecially w e ll- suited for constr ucting sequences of May er-Vietoris type. A group oid X has pr op er diagonal if the map (A.13) X 1 (domain , range) − − − − − − − − − → X 0 × X 0 is proper , and [ X ] is Hausdorff. If (A.13) is pro per and X 0 is Hausdorff then X is prop er. If Y → X is a loca l equiv alence, then X has prop er diag onal if and only if Y has prop er diagonal. 54 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN A.2.2. L o c al and glob al qu otients. A gro upoid whic h is related by a chain of lo cal equiv alences to one of the form S/ /G , obtained from a group G acting on a space S , is s aid to be a glob al quotient . A lo c al quotient gr oup oid is a group oid X admitting a countable o pen cover { U α } with the property that each X U α is weakly equiv a len t to a gr oupoid of the form S/ /G with G a compact L ie gr o up, and S a Ha usdorff space. If Y → X is a lo cal equiv alence , then Y is a lo cal quotient group oid if and only if X is, so the prop ert y of being a lo cal quo tien t is in tr ins ic to the underlying stack. If X is a lo cal quotient group oid, then [ X ] is paracompact, lo cally contractible and completely regular. If X is a lo cal quotient g r oupoid with the prop ert y that there is at mo st one map b etw een any tw o ob jects (ie X 1 → X 0 × X 0 is a n inclusio n), the map X → [ X ] is a lo cal equiv alence, and so X is just a space. The following lemma is straightforw ard. Lemma A.14. Any gr oup oid c onst ructe d by pul lb ack fr om a lo c al quotient gr oup oid is a lo c al quotient gr oup oid. In p articular, any (ful l) sub gr oup oid of a lo c al quotient gr oup oid is a lo c al quotient gr oup oid, and the mapping cylinder of a map X S → X c onstructe d by pul lb ack along a map S → [ X ] to the orbit sp ac e of a lo c al quotient gr oup oid is a lo c al quotient gr oup oid. A.3. Fib er bundles o v er group oi ds and descen t. In this section we define the category of fiber bundles over a group oid, and s ho w that a lo cal equiv alence gives an equiv alence of categor ies of fib er bundles (Prop osition A.18). Th us the categor y of fiber bundles over a group oid is in trinsic to the underlyin g stack. A fib er bundle o ver a groupo id X = ( X 0 , X 1 ) consists of a fib er bundle P on X 0 together with identifications o f cer tain pullbacks to X n for v a r ious n . W e intro duce some conv enient notation for describing these pulled bac k bund les. Let’s denote a t ypical p oint of X n b y x 0 f 1 − → . . . f n − → x n . Given a bundle P → X 0 we’ll write P x i for the pullbac k of P along the map X n → X 0 ( x 0 → · · · → x n ) 7→ x i . Similarly , if P → X 1 is giv en, w e’ll write P f i for the pullbac k of P a long the map X n → X 1 ( x 0 f 1 − → . . . f n − → x n ) 7→ ( x i − 1 f i − → x i ) , and P f i ◦ f i +1 for the pullbac k along X n → X 1 ( x 0 f 1 − → . . . f n − → x n ) 7→ ( x i − 1 f i ◦ f i +1 − − − − − → x i +1 ) , etc. F o r small v a lues of n we’ll use sym b ols like ( a f − → b ) ∈ X 1 ( a f − → b g − → c ) ∈ X 2 . to denote t ypica l p oints. LOOP GROUPS AND TWISTED K -THEOR Y I 55 Definition A. 1 5. A fi b er bund le o n X consists o f a fiber bundle P → X 0 , together with an bundle isomorphism (A.16) t f : P a → P b on X 1 , for whic h t Id = Id, and satisfying the co cycle condition that P a t f / / t g ◦ f A A A A A A A A P b t g ~ ~ } } } } } } } P c commut es on X 2 . This wa y of describing a fib er bundle is conv enient when thinking of X as a category . The asso ciation a → P a is a fun ctor from X to spaces , that is contin uous in an appropriate sense. There is a more succinct w ay of describing a fiber bundle on a group oid. Namely , a fib er bundle on a group oid X = ( X 0 , X 1 ) is a group oid P = ( P 0 , P 1 ) and a functor P → X making P i → X i in to fib er bundles, and all of the structure maps in to maps of fiber bundles (i.e., pullbac ks squares). A functor F : Y → X b et ween groupo ids defines, in the evident way , a pullback functor F ∗ from the category of fib er bundles ov er X to the ca tegory of fib er bundles ov er Y . A natural transformation T : F → G defines a natural tr ansformation T ∗ : F ∗ → G ∗ . Example A.17 . Let U = { U i } be a cov ering of a spac e X . T o give a fiber bundle ov er N U is to give a fib er bundle P i on ea c h U i and the clutching (descent) data needed to a ssem ble the P i in to a fib er bundle ov er X . Indeed, pullback along the map N U → X gives an equiv alence b etw een the categor y of fib er bundles ov er X and the category of fib er bundles ov er N U . The following generaliza tion of Example A.17 will be refer red to as desc ent for fiber bundles ov er group oids. Prop osition A.18 . Supp ose that F : X → Y is a lo c al e quivalenc e. Then the pul lb ack functor F ∗ : { Fib er bund les on Y } → { Fib er bund les on X } is an e quivalenc e of c ate gories. Pr o of: Suppos e that P is a fib er bundle over Y , which we think of as a functor from Y to the ca teg ory of topo logical spaces. Since Y → X is a n equiv alence of categories , the functor F ∗ has a left adjoin t F ∗ , given by F ∗ P ( x ) = lim − → Y /x P, where Y /x is the catego ry of ob jects in y ∈ Y eq uipped with a mor phism F y → x . Since Y → X is a n equiv alence of group oids, there is a unique map b etw een any t wo ob jects of Y /x , and so F ∗ P ( x ) is iso mo rphic to P y for any y ∈ Y /x . F or ea c h x ∈ X , c ho ose a neighbor hoo d x ∈ U ⊂ X 0 , a map t : U → Y 0 , and a family of morphisms U → X 1 connecting F ◦ t to the inclusion U → X 0 . W e to polog ize [ x ∈ X F ∗ P x 56 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN b y requiring that the canonical map t ∗ P → F ∗ P | U be a homeomorphism. This gives F ∗ P the structure of a fib er bundle ov er X 0 . Naturality provides F ∗ P with the additional structur e required to ma k e it int o a fib er bundle over X . One easily checks that the pair ( F ∗ , F ∗ ) is an a djoin t equiv alence of the categ o ry of fib er bundles ov er X with the category of fibe r bundles ov er Y .  F o r a fib er bundle p : P → X write Γ( P ) for the space of sections Γ( P ) = Γ( X ; P ) = { s : X → P | p ◦ s = Id X } , topo logized a s a s ubspa ce of X P 0 0 × X P 1 1 . If f : Y → X is a lo cal equiv a lence, a nd P → X is a fiber bundle, then the eviden t map Γ( X ; P ) → Γ( Y ; f ∗ P ) is a homeomorphism. If P is a pointed fib er bundle, with s : X → P as a basep oin t, a nd A ⊂ X is a (full) subgr oupoid, write Γ( X, A ; P ) for the s pace of section x of P for which x | A = s . Now suppose that P → Q is a ma p of fibe r bundles ov er X , and { U α } is a cov er ing of X by op en sub-group oids. W rite P α → U α for the res triction of P to U α , and P α 1 ,...,α n for the restriction of P to U α 1 ∩ · · · ∩ U α n , and similarly fo r Q . Prop osition A.19. If for e ach non-empty finite c ol le ction { α 1 . . . α n } the map Γ( P α 1 ,...,α n ) → Γ( Q α 1 ,...,α n ) is a we ak homotopy e quivalenc e, then so is Γ( P ) → Γ( Q ) . Pr o of: This is a straig h tfor w a rd application of the tec hniques of Segal [30]. Let’s first consider the ca s e in whic h X is co vered by just t wo op en sub-group oids U and V . W e for m the “do uble mapping cylinder” C = U ∐ U ∩ V × [0 , 1 ] ∐ V / ∼ , and consider the functor g : C → X . A choice of par tition unity on [ X ] sub ordi- nate to the covering { [ U ] , [ V ] } gives a section of g making Γ ( X ; P ) → Γ( X ; Q ) a retract of Γ( C ; g ∗ P ) → Γ( C ; g ∗ Q ). It therefore suffices , in this case , to show that Γ( C ; g ∗ P ) → Γ( C ; g ∗ Q ) is a weak equiv alence. But Γ( C ; g ∗ P ) fits into a homoto p y pullback squar e Γ( C ; g ∗ P ) − − − − → Γ( U ; P )   y   y Γ( V ; g ∗ P ) − − − − → Γ( U ∩ V ; P ) , and similar ly for Γ( C ; g ∗ Q ) (to simplify the diagram we hav e not distinguished in notation b etw een P and its r estriction to U , V , and U ∩ V ). The result then follows fro m t he long exa ct (Mayer-Vietoris) seq uence of homotopy groups. An easy induction then gives that the map on spaces of sections of P → Q restricted to an y finite union U α 1 ∪ · · · ∪ U α n is a w eak eq uiv a lence. F or the case the co llection LOOP GROUPS AND TWISTED K -THEOR Y I 57 { U α } is count able (to whic h we are reduced when [ X ] is sec ond countable), or der the U α and write V n = U 1 ∪ · · · ∪ U n . F o rm the “ infinite mapping cylinder ” C = a V i × [ i, i + 1] / ∼ , and consider g : C → X . As before, a par tition of un ity on [ X ] subo rdinate to the cov ering [ V ] i defines a section of [ C ] → [ X ] and hence of C → X , making Γ( X ; P ) → Γ( X ; Q ) a retract of (A.20) Γ( C ; g ∗ P ) → Γ( C ; g ∗ Q ) . It therefor e suffices to s ho w that (A.20 ) is a weak equiv alence . But (A.20) is the homotopy inv er s e limit of the tow er (A.21) Γ( V n ; P ) → Γ( V n ; Q ) and so its homotopy groups (or sets, in the ca s e of π 0 ) are related to those of (A.21) b y a Milnor sequence, and the result follows. Alternatively , following Sega l [30], one can av oid the countabilit y hypothesis and the induction by using for C the nerve of the cov ering { U α } and the ho motop y sp e c tr al sequences of Bousfield- Kan and Bousfield [9, 8].  A.4. Hil b ert bundles. A Hilb ert bund le ov er a group oid X is a fib er bundle whose fiber s ha ve the structure of a separa ble Z / 2-gr aded Hilb ert space. R emark A.22 . There is a tricky issue in the p oint set top ology here. In defining Hilbert bundles as sp ecial kinds of fib er bundles, w e’r e implicitly using the compact op en top ology on U ( H ) and no t the norm top ology . This causes tr o uble when we form the asso ciated bundle of F redholm o perato rs ( § A.5), since w e cannot then use the norm top ology o n the space o f F r edholm op erators. This issue is raised and resolved by A tiyah-Segal [3], and we are following their discussion in this pap er. Definition A.23. A Hilbert bundle H is universal if for eac h Hilb ert bundle V there exists a unitary embedding V ⊂ H . The bundle H is said to hav e the absorption pr op erty if for an y V , there is a unitary equiv a lence H ⊕ V ≈ H . Lemma A.24. A universal Hilb ert bun d le has the absorption pr op erty. Pr o of: First note that if H is universal, then H ⊗ ℓ 2 ≈ H ⊕ H ⊕ · · · has the abs o rption prop ert y . Indeed, g iv en 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 ⊕ · · · . W e can then wr ite H ≈ H ⊗ ℓ 2 ⊕ V ≈ H ⊗ ℓ 2 , to conclude that H is a bsorbing.  Definition A.25. A Hilb ert bundle H ov er X is lo c al ly universal if for ev ery op en sub-group oid X U ⊂ X the restriction of H to X U is univ er sal. 58 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN Lemma A.26. If H and H ′ ar e universa l Hilb ert bund les on X , then ther e is a unitary e quivalenc e H ≈ H ′ .  R emark A.27 . Since the category o f Hilb ert bundles on X depends only on X up to lo cal equiv alence, if f : Y → X is a local equiv alence and H is a (lo cally) univ ersal Hilb ert bundle on X , then f ∗ H is a (lo cally) universal Hilb ert bundle on Y . Similar ly , if H ′ is a (lo cally) universal Hilb ert bundle on Y there is a (lo cally) univ ersal Hilbert bundle H on X , and a unitary equiv alence f ∗ H ≈ H ′ . W e now show that the existence of a lo cally universal Hilb ert bundle is a lo cal issue. Lemma A.28. Supp ose that X is a gr oup oid, and t hat { U i | i = 1 . . . ∞} is a c overing of X by op en sub-gr oup oids. If H is a Hilb ert bund le with the pr op erty that H i = H | U i is un iversa l, then H ⊗ ℓ 2 is un iversa l. Pr o of: Let V be a Hilber t-space bundle on X . Cho ose a partition of unit y { λ i } on [ X ] subordinate to the open cover [ U ] i . F o r eac h i c ho ose an em be dding r i : V | U i ֒ → H i . The map V → H ⊕ H ⊕ · · · = H ⊗ ℓ 2 with compo nen ts λ i r i is then an em b edding of V in H ⊗ ℓ 2 .  Corollary A. 29. In the situation of L emma A.28, if H | U i is lo c al ly un iversa l, then H ⊗ ℓ 2 is lo c al ly un iversal.  Lemma A.30. Supp ose that X is a gr oup oid, and t hat { U i | i = 1 . . . ∞} is a c overing of X by op en sub-gr oup oids. If H i is a lo c al ly universal Hilb ert bund le on { U i } , t hen ther e exists a Hilb ert bund le H on X with H | U i ≈ H i . Pr o of: This is an easy induction, using Lemma A.26.  Corollary A.31. S u pp ose that X is a gr oup oid, and that { U i | i = 1 . . . ∞} is a c overing of X by op en sub-gr oup oids. If H i is a lo c al ly universal Hilb ert bund le on { U i } , t hen ther e exists a lo c al ly universal Hilb ert bund le H on X .  Lemma A.32 . S upp ose that X = S/ / G is a glob al quotient of a sp ac e S by a c omp act Lie gr oup G . Then the e quivariant Hilb ert bund le S × L 2 ( G ) ⊗ C 1 ⊗ ℓ 2 is a lo c al ly universal Hilb ert bun d le on X . LOOP GROUPS AND TWISTED K -THEOR Y I 59 Here C 1 is the co mplex Clifford algebra o n one (o dd) gener a tor. It is there simply to mak e the odd comp onent of our Hilber t bundle large enough. Pr o of: Since the open (f ull) s ubg roupo ids of S/ /G cor respo nd to the G -stable op en subsets of S it suffices to s how that that L 2 ( G ) ⊗ C 1 ⊗ ℓ 2 is universal. Let V b e any Hilber t bundle on S/ / G , ie an equiv ariant Hilbert bundle on S . By Kuip e r ’s theorem, V is trivial as a (non-equiv ar ian t) Hilbert bundle on S . Cho ose an orthonormal homogeneo us basis { e i } , and let e i = h e i , − i : V → C 1 be the corresp onding pro jection opera tor. By the universal pro perty of L 2 ( G ), each e i lifts uniquely to an eq uiv a rian t map V → L 2 ( G ) ⊗ C 1 . T aking the sum of these maps gives an em b edding of V in L 2 ( G ) ⊗ C 1 ⊗ ℓ 2 .  Combinin g Lemma A.32 with Lemma A.30 giv e s : Corollary A. 3 3. If X is a lo c al quotient gr oup oid, then ther e exists a lo c al ly universal Hilb ert bund le on X .  Corollary A.34. Supp ose that X is a lo c al quotient gr oup oid, f : Y → X is a map c onstructe d by pul lb ack fr om [ Y ] → [ X ] . If H is lo c al ly universal on X , then f ∗ H is lo c al ly universal on Y . Pr o of: This is an easy consequence of Lemma A.32 and Corollary A.29.  Corollar y A.34 is needed is the pro of of e x cision in twi sted K -theory , and is the reason for our r estriction to the class of lo cal quotient gr o upoids. The following res ult is w ell-known, but we could not quite find a reference. Our pro of is taken from [25, Theor e m 1 .5] whic h gives the analogous result for equiv ariant embeddings of coun ta bly infinite dimensional inner product spa ces (and not Hilbert spaces). Of course the result a lso follo ws from Kuip er’s theorem, since the spa c e of em b eddings is U ( H ⊗ ℓ 2 ) /U ( V ⊥ ). But the contractibilit y of the space of embeddings is more e le mentary than the con tractibility of the unitary group, s o it seemed better to hav e proof that do esn’t mak e use of K uip er’s theorem. Lemma A.35. Supp ose that V and H ar e Hilb ert bund les over a gr oup oid X , and that ther e is a unitary emb e dding V ⊂ H ⊗ ℓ 2 . Then the sp ac e of emb e ddings V ֒ → H ⊗ ℓ 2 is c ontr actible. Pr o of: Let f : V ⊂ H ⊗ ℓ 2 be a fixed em b edding, and write H ⊗ ℓ 2 = H ⊕ H ⊕ · · · f = ( f 1 , f 2 , . . . ) . The co n tra ction is a concatenation o f tw o ho motopies. The first takes an embedding g = ( g 1 , g 2 , . . . ) to (0 , g 1 , 0 , g 2 , . . . ) and then the second is cos( π t/ 2) · (0 , g 1 , 0 , g 2 , . . . ) + sin( π t/ 2) · ( f 1 , 0 , f 2 , 0 . . . ) . 60 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN It is easier to write down the reverse of the first ho motop y . It, in turn, is the concatenation of an infinite s e quence o f 2-dimensional r otations (0 , g 1 , 0 , g 2 , 0 , g 3 , . . . ) 7→ ( g 1 , 0 , 0 , g 2 , 0 , g 3 , . . . ) 0 ≤ t ≤ 1 / 2 ( g 1 , 0 , 0 , g 2 , 0 , g 3 , . . . ) 7→ ( g 1 , g 2 , 0 , 0 , 0 , g 3 , . . . ) 1 / 2 ≤ t ≤ 3 / 4 · · · . One m ust c heck that the limit as t 7→ 1 is ( g 1 , g 2 , . . . ), a nd the that path is con tin- uous in the co mpact-open topo logy . Both facts are easy and left to the reader.  Lemma A.36. Supp ose that H is a lo c al ly universal Hilb ert bund le over a lo c al quotient gr oup oid X . Then the sp ac e of se ctions Γ ( X ; U ( H )) of t he asso ciate d bund le of unitary gr oups is we akly c ontr actible. Pr o of: This follows easily from Kuip er’s theorem (se e Ap p endix 3 of [3]), a nd Prop osition A.19.  W e co nclude this section with a useful cr iterion for a lo cal-quo tien t stack to b e equiv alent to a global quotient by a c o mpact L ie group. Prop osition A.37. The (lo c al ly) universa l Hilb ert bund le over a c omp act, lo c al- quotient gr oup oid, spli ts into a sum of finite-dimensional bund les iff the gr oup oid is e quivalent (in the sense of lo c al e quivalenc e) to one of the form X/ /G , with X c omp act, and G a c omp act gr oup. R emark A.38 . (i) This implies right aw ay that the extensions of gro upoids co rre- sp o nding to t wistings whose inv ariant in H 3 has infinite or der a r e not quotient stacks: indeed, an y 1-eigenbundl e for the central T is a pro jective bundle re pr esen- tative for the twisti ng, and hence m ust b e infinite-dimensional. (ii) There a re simple obstructions to a g roupo id b eing rela ted by a c hain of lo cal equiv alences to a global quotient b y a compact g roup; for instance, suc h quotient s admit contin uo us c hoices of Ad-inv ariant metrics on the Lie a lgebra stabilizers which are in teg r al on the co-weigh t lattices. The stack obtained by gluing the bo undaries of B ( T × T ) × [0 , 1] via the s hea ring automorphism of T × T do es not carry such metrics. The same is true for the quotien t s tac k A / / T ⋉ LT , where T is a tor us, and A is the space of connections on the triv ial T -bundle ov er the circle. In this cas e , the stack is fiber e d over T in B ( T × T )-stacks with the tautologica l shearing ho lonomies. Hence, the large r stacks A / / T ⋉ LG where G is a compact Lie g roup a nd A is th e space of connectio ns on the tr ivial G -bundle ov er the circle are not global quotien ts either. (iii) The r esult is curiously similar to T o taro’s characterisation of smo oth quotient stacks a s the Artin stac k s where coherent sheav es admit r esolutions b y vector bun- dles [32]. Pr o of. The ‘if ’ pa rt follo ws from our construction of the universal Hilbert bundle. F o r the ‘only if part,’ first note that a lo cal quotient group oid X is weakly equiv a len t to a gr oupoid of the form S/ / G , if and only if ther e is a principal G -bundle P → X with the prop erty that there is at mo st one map betw een an y tw o ob jects in P . In that case P is equiv alent to [ P ] ( § A.2.2 ), and X is w eakly equiv alent to [ P ] / /G . This latter co ndition holds if and o nly if for each x ∈ X 0 the map Aut ( x ) → G asso ciated to P is a mo nomorphism. Supp ose that H is the (lo cally) universal LOOP GROUPS AND TWISTED K -THEOR Y I 61 Hilbert bundle on X , and that w e can find an or tho g onal decomp osition H = ⊕ H α with each H α of dimension n α < ∞ . T ake P to be the pro duct of the frame bundles of the H α , and G to be the pro duct of the unitary groups U ( H α ) ≈ U ( n α ). T o chec k that Aut( x ) → G is a monomo rphism in this case it suffices to ch eck lo cally near x . The assertion is thus reduced to the case of a globa l q uo tien t by a compact group, where it follo ws from our explicit construction.  A.5. F redholm op erators and K -theory. W e will build our mo del of twisted K -theory using the “ sk ew-adjoint F redholm” mo del of A tiy ah-Singer [1]. In th is section w e recall this theory , and the mo difications describ ed in A tiyah-Segal [3] Let H be a Z / 2-g raded Hilbert bundle o ver a groupoid X . W e wish to asso- ciate to H a bundle of F redholm o pera to rs ov er X . As men tioned in Remark A.22, we ca nnot just use the norm topology on the space of F r edholm op erators here. W e hav e used the compact-o pen top ology on U ( H ), and the (conjuga tio n) action of U ( H ) in the compact-op en top ology o n F redholm in the nor m top ology is not contin uous. F ollowing A tiy ah-Sega l [3, Definition 3.2 ], w e make the following defi- nition. Definition A.39. ([3]) The space F r e d (0) ( H ) is the space of odd skew-adjoint F r edholm op erator s A , for which A 2 + 1 is co mpact, to polog ized as a subspace of B ( H ) × K ( H ), with B ( H ) given the compact-op en to polog y and K ( H ) the norm topo logy . Let C n = T { C n } / ( z 2 + q ( z ) = 0) denote the complex Clifford alg ebra asso ciated to the quadratic form q ( z ) = P z 2 i . W e wr ite ǫ i for the i th standard basis elemen t of C n , r egarded as an element of C n . F o llo wing Atiy ah-Singer [1], for an o perato r A ∈ F red (() C n ⊗ H ), with n o dd, let w ( A ) = ( ǫ 1 . . . ǫ n A n ≡ − 1 mo d 4 i − 1 ǫ 1 . . . ǫ n A n ≡ 1 mo d 4 . The opera tor A is then even and self-a djoin t. Definition A.40. ([3]) The space F red ( n ) ( H ) is the subspace o f F red (0) ( C n ⊗ H ) consisting of o dd op erators A whic h commute (in the gr aded sense) with the action of C n , and fo r which the essen tial sp ectrum of w ( A ), in case n is odd, contains b oth po sitiv e and negative e ig en v a lues. A tiyah and Segal [3] show that the “identit y” map from F red ( n ) ( ℓ 2 ) in the norm topo logy to F red ( n ) ( ℓ 2 ) in the a bov e top ology is a weak homotopy equiv a lence. It then follows from Atiy ah-Singer [1, Theorem B(k)] that the map F r ed ( n ) ( ℓ 2 ) → Ω ′ F r ed ( n − 1) ( C 1 ⊗ ℓ 2 ) A 7→ ǫ k cos( π t ) + A sin( π t ) is weak homotopy equiv alence, wher e we are making the eviden t ident ification C n ≈ C n − 1 ⊗ C 1 , and Ω ′ denotes the space of paths from ǫ k to − ǫ k . Comb ining these leads to the follo wing simple consequence. Prop osition A.41. If X is a lo c al quotient gr oup oid, and H a Z / 2 -gr ade d, lo c al ly universal Hilb ert bund le over X , the map Γ( X ; F red ( n +1) ( H )) → Ω ′ Γ( X ; F red ( n ) ( H )) is a we ak homotopy e quivalenc e. 62 DANIEL S . FREED, MICHAEL J. HOPKINS, AND CONST ANTIN TELEM AN Pr o of: By Prop osition A.19, the question is lo cal in X , so we may a ssume X = S/ /G , with G a compact Lie gr oup. By o ur ass umption on the existence o f lo cally contractible slices, we may reduce to the cas e in whic h S is equiv aria n tly contractible to a fixed p oint s ∈ S . Fina lly , since the question is homoto py inv a rian t in X , w e reduce to the cas e S = pt . W e a re therefore reduced to showing that if H is a univ ers al G -Hilbert space, then the map of G -fixed p oints (A.42) F red ( n ) ( H ) G → Ω ′ F r ed ( n +1) ( H ) G is a weak equiv a le nce . F or each irr educible repre s en tatio n V of G , let H V denote the V -isotypical co mponent of H . Then (A.42) is the pro duct ov er the irreducible representations V of G , of (A.43) F red ( n ) ( H V ) G → Ω ′ F r ed ( n +1) ( H V ) G Since H is universal, the Hilb ert space H V is isomorphic to V ⊗ ℓ 2 , and the map F r ed ( n ) ( ℓ 2 ) → F red ( n ) ( V ⊗ ℓ 2 ) G T 7→ Id ⊗ T is a homeomo rphism. The P ropo sition is thus reduced to the r esult of Atiy ah-Singer quoted above.  W e now asse mble the spaces Γ( X ; F r ed ( n ) ( H )) into a sp ectrum in the sense o f algebraic top ology . T o do this requires sp ecifying ba sepoints in F red ( n ) ( H ). Since our o pera to rs ar e o dd, w e can’t take the iden tit y map a s a bas epoint and a different choice m ust b e made. There ar e so me technical difficulties that arise in trying to sp e c ify consistent choices and we hav e just c hosen to b e unspecific on this p oint. The difficulties don’t amount to a s erious pro blem since any in vertible op erator ca n be ta k en a s a basep oin t, and the space o f inv ertible op erators is contractible. The reader is referred to [2 3] for further discussion. W e will use the sym b ol ǫ to refer to a chosen basep oin t in F red ( n ) ( H ), as well as to the constan t section with v a lue ǫ in Γ( X ; F r ed ( n ) ( H )). Prop osition A.41 giv es a homotopy equiv alence (A.44) Γ( X ; F r e d ( n +1) ( W )) → ΩΓ( X ; F r e d ( n ) ( W )) As described in [1], the fact that C 2 is a matrix algebra g iv es a homeomorphism (A.45) Γ( X F red ( m ) ( W )) ≈ Γ( X F red ( m +2) ( W )) . W e the sp ectrum K ( X ) b y taking K ( X ) n = ( Γ( X, F r ed (0) ( W )) n even Γ( X, F r ed (1) ( W )) n o dd with structure map K ( X ) n → Ω K ( X ) n +1 to b e the ma p (A.44) when n is o dd, and the composite of (A.45) a nd (A.44) when n is ev en. 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Hi gh Energy Phys. (1998) , no. 12, P aper 19, 41 pp. (electronic). MR 2000e:811 51 Dep ar tment of Ma thema tics, University of Texas, Aus tin, TX E-mail addr ess : dafr@math.ut exas.edu Dep ar tment of Ma them a tics, Massachusetts Institute of Techn ology, Cam bridge, MA 02139-4307 E-mail addr ess : mjh@math.mit .edu Cambridge, Cambridge E-mail addr ess : c.teleman