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Equivariant K-theory, groupoids and proper actions Lopez, Jose Maria Cantarero 2009

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Equivariant K-theory, groupoids and proper actions by Jose Maria Cantarero Lopez,  B.Sc., Universidad de Malaga, 2004  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in The Faculty of Graduate Studies (Mathematics)  The University Of British Columbia (Vancouver) April 2009  ©  Jose Maria Cantarero Lopez, 2009  Abstract Equivariant K-theory for actions of groupoids is defined and shown to be a cohomology theory on the category of finite equivariant CW-complexes. Under some conditions, these theories are representable. We use this fact to define twisted equivariant K-theory for actions of groupoids. A classification of possible twistings is given. We also prove a completion theorem for twisted and untwisted equivariant K-theory. Finally, some applications to proper actions of Lie groups are discussed.  11  Table of Contents Abstract  ii  Table of Contents  iii  Acknowledgements  .  .  .  v  1  Introduction  1  2  Groupoids  6  2.1  Basic facts  6  2.2  Equivalent groupoids  2.3  Groupoid actions  9 11  2.4  Equivariant CW-complexes  12  2.5  Equivariant cohomology theories  13  2.6  Fiber bundles  18  2.7  Vector bundles  21  2.8  Hilbert bundles  23  Equivariant K-theory  25  3  4  3.1  Extendable K-theory  25  3.2  The Mayer-Vietoris sequence  26  3.3 3.4  The Bott periodicity  31  Equivariant representable K-theory  34  Twisted equivariant K-theory  43  4.1  Twisted extendable K-theory  43  4.2  The orbit category  48  111  Acknowledgements I would like to express my utmost gratitude to my supervisor Alejandro Adem in the first place, for he has guided me through my Ph.D. program patiently, introduced me into the research world and given me many oppor tunities to advance professionally and personally. I owe special thanks to Antonio Viruel for helping me with the choice of school and advisor for my graduate studies. I am very grateful to some of my colleagues who were specially helpful with my research at some point, namely, Bernardo Uribe, Nitya Kitchloo, Jose Manuel Gomez and Ali Duman. Thanks to the mem bers of my supervisory committee, Denis Sjerve and Dale Rolfsen, and the rest of the examining committee for their time. On a personal level, I would like to thank my family and friends for their support and encouragement. My girlfriend Dacil Garcia deserves my special appreciation for her love and positive influence, which have been fundamental during this particularly stressful time. Finally, I would like to thank everybody who was important to the successful realization of this thesis and the completion of my Ph.D. program, and apologize that I could not mention personally all of them.  V  Chapter 1  Introduction Symmetries have always played a very important role in mathematics. In algebraic topology, these are usually realized by actions of groups on topo logical spaces. A groupoid is a generalization of a group with the difference that multiplication is not globally defined. Actions of groupoids give rise to more general symmetries. Groupoids can also be seen as a generaliza tions of topological spaces. A particular kind of groupoids, orbifolds, have been extensively studied lately in algebraic topology, algebraic geometry and physics. When studying orbifolds, it is convenient to consider two orbifolds to be the same not when they are isomorphic, but when they satisfy a weaker condition, called Morita equivalence. In fact, a similar concept, that of weak equivalence of groupoids, helps us identify when two actions are equivalent in a sense. The recent theorem of Freed, Hopkins and Teleman [17, 18, 19] relates the complex equivariant twisted K-theory of a simply-connected compact Lie group acting on itself by conjugation to the Verlinde algebra. This result links information about the conjugation action with the action of the ioop group on its universal space for proper actions. If we use the language of groupoids, the two associated groupoids are Morita equivalent. The invari ance of orbifold K-theory under Morita equivalence [1] also seems to suggest that the language of groupoids is an appropriate framework to work with proper actions. We introduce all the necessary background on groupoids in chapter 2, as well as new constructions that will allow us to construct equivariant K-theory for g.roupoid actions. The complex representation ring of a compact Lie group G can be iden 1  Chapter 1. Introduction tified with the G-equivariant complex K-theory of a point. Equivariant complex K-theory is defined via equivariant complex bundles, but this pro cedure does not give a cohomology theory for proper actions of non-compact Lie groups in general, as shown in [35]. Phillips constructed an equivari ant cohomology theory for any second countable locally compact group G on the category of proper locally compact G-spaces. This is done using infinite-dimensional complex G-Hilbert bundles. Sometimes it is enough to use finite-dimensional vector bundles, for example in the case of discrete groups [28]. In this paper we will construct complex equivariant K-theory for actions of a Lie groupoid by using extendable complex equivariant bun dles, defined in section 2.7. These bundles are finite-dimensional, but are required to satisfy an additional condition which will make sure that we have a Mayer-Vietoris sequence. The Grothendieck construction then gives a cohomology theory on the category of 9-spaces. For any 9-space X, K(X) is a module over K(Go) and the latter can be identified with Kb(9) when 9 is an orbifold. But this theory does not satisfy Bott periodicity in general. In fact, it may fail to agree with classical equivariant K-theory when the action on the space is equivalent to the action of a compact Lie group. In order to solve this problem we introduce 9-cells, which are 9-spaces whose 9-action is equivalent to the action of a compact Lie group on a finite complex. The condition of Bredon-compatibility makes sure that 9-equivariant K-theory agrees with classical equivariant K-theory on the 9-cells. This condition also implies Bott periodicity for finite 9-CW-pairs. The following theorem is proved in chapter 3: Theorem 1.0.1. If  9 is a Bredon-compatible Lie groupoid, the groups  K(X, A) define a 7L/2-graded multiplicative cohomology theory on the cat  egory of finite 9-CW-pairs. Atiyah and Segal twist equivariant K-theory for actions of a compact Lie group G using G-stable projective bundles [6]. Since stable projective bundles and sections behave well under weak equivalences, it seems natural 2  Chapter 1. Introduction  to use 9-stable projective bundles, which can be defined in an similar way, to twist 9-equivariant K-theory. G-equivariant K-theory can be represented by a space of Fredhoim op erators on a G-stable Hubert space. This is used to construct twisted Ktheory in [6]. For a G-stable projective bundle, we can consider a suitable bundle of Fredholm operators associated to it and define twisted K-theory with the sections of this bundle [6]. For actions of groupoids, we have a representability theorem if we consider not all maps into such a space of Fredholm operators, but only those which are extendable. The definition of extendable sections and the space Fred’ (H) can be found in section 3.4. Theorem 1.0.2. Let H be a stable representation of a Bredon-compatible  finite Lie groupoid 9. Then:  (X) 9 K  =  t [X,Fred’(H)]  Choosing all sections of the Fredholm bundle corresponds to choosing all vector bundles in the untwisted case. To make these new theories an exten sion of untwisted K-theory, we need to consider extendable sections. Then we can define twisted 9-equivariant K-theory as the group of extendable homotopy classes of extendable sections of a suitable Fredholm bundle, that is, homotopy classes where the homotopies run over extendable sections. Extending it to all degrees as in [6], we obtain a cohomology theory: Theorem 1.0.3. If 9 is a Bredon-compatible finite Lie groupoid, the groups K(X) define a Z/2-graded cohomology theory on the category of finite  9-CW-complexes with 9-stable projective bundles, which is a module over untwisted 9-equivariant K-theory.  The category of 9-orbits behaves similarly to the corresponding category for a compact Lie group. In particular, we are able to use some of these properties to prove an analogue of Elmendorf’s construction [15]. This con struction is the key to the classification of 9-stable projective bundles. In fact, we show that isomorphism classes of 9-stable projective bundles over X 3  Chapter 1. Introduction are classified by H (X). This is done by constructing a particular model for the space that represents H(—) which admits a natural 9-stable projective bundle on it. Chapter 4 contains all the definitions and results concerning twisted K-theory. In chapter 5, we prove corresponding completion theorems for twisted and untwisted K-theory. We introduce a universal 9 space E9 as the limit of a sequence of free 9-spaces E’9 as in the case of compact Lie groups. The quotient of E9 by the 9-action is B9, the classifying space of 9. We can then form the fibered product X XT E9 over G 0 and prove a generalization of the completion theorem of Atiyah and Segal [5] when 9 is finite, that is, when G 0 is a finite 9-CW-complex: Theorem 1.0.4. Let 9 be a Bredon-compatible, finite Lie groupoid and X  a finite 9-CW-complex. Then we have an isomorphism of pro-rings {K(X)/IK(X)}  {K(X x’,. E9/9)}  The recent completion theorem for twisted equivariant K-theory for ac tions of compact Lie groups in [26] provides the necessary results to use induction over cells. In the twisted case, however, the completion theo rem will relate the completion of twisted 9-equivariant K-theory of X with respect to I to the twisted 9-equivariant K-theory of X x’,- E9. Theorem 1.0.5. Let 9 be a Bredon-compatible finite Lie groupoid, X a  finite 9-CW-complex and P a 9-stable projective bundle on X. Then we have an isomorphism of K(Go)-modules: PXE9KTh(X x  E9)  Some applications are discussed in chapter 6. When S is a Lie group, not necessarily compact, we can define twisted equivariant K-theory for proper actions of S using the groupoid S >i ES, where ES is the universal space for proper actions of S. These groupoids provide particular instances where these theorems apply and can be used to study the proper actions of these 4  Chapter 1. Introduction  particular groups. The case of compact Lie groups and finite groups were studied by Atiyah and Segal [5]. Discrete groups are dealt with in [28]. Al most compact groups and matrix groups are studied in [34]. Proper actions of pro-discrete groups are shown to be Bredon-compatible in [38]. The re sults here provide a way to define twisted K-theory for such actions as well as completion theorems.  5  Chapter 2  Groupoids 2.1  Basic facts  In this section we review some basic facts about groupoids. All this material can be found in [1] and [32]. 0 of objects Definition 2.1.1. A topological groupoid 9 consists of a space G and a space G 1 of arrows, together with five continuous structure maps, listed below.  • The source map s : G 1  —*  1 its source 0 assigns to each arrow g E G G  s(g).  G its target 0 assigns to each arrow g E 1 G t(g). For two objects x, y Go, one writes g x —+ y to indicate that 1 is an arrow with s(g) = x and t(g) = y. g EG  • The target map t : G 1  —*  t(g), one can form their composi • If g and h are arrows with s(h) tion hg, with s(hg) = s(g) and t(hg) = t(h). The composition map hg, is thus defined on the m: G , defined by m(h,g) 1 1 X 1 —p G ,t G 3 fibered product 1 1 x,t G G  =  {(h,g)  e  1 1 x G G  I  s(h)  =  t(g)}  and is required to be associative.  • The unit map u : G 0 —* G 1 which is a two-sided unit for the composi tion. This means that su(x) = x = tu(x), and that gzt(x) = g = u(y)g forallx,yeGoandg:x—*y.  6  Chapter 2. Groupoids • An inverse map i : G 1  , written i(g) 1 G  =  g’. Here, if g x then g’ : y —* x is a two-sided inverse for the composition, which means that g’g = u(x) and gg’ = —*  —  Definition 2.1.2. A Lie groupoid is a topological groupoid 9 for which 1 are smooth manifolds, and such that the structure maps are 0 and G G smooth. Furthermore, s and t are required to be submersions so that the domain G 1 of m is a smooth manifold. 1 x,t G Example 2.1.3. Suppose a Lie group K acts smoothly on a manifold M.  1 = KxM, OnedefinesaLiegroupoidKNMby(K>M)o = Mand(KxM) with s the projection and t the action. Composition is defined from the  multiplication in the group K. This groupoid is called the action groupoid.  9 be a Lie groupoid. For a point x e Go, the set of all arrows from x to itself is a Lie group, denoted by 9 and called the Definition 2.1.4. Let  isotropy group at x. The set ts (x) of targets of arrows out of x is called 0 consisting of all the orbits in 9 is the orbit of x. The quotient 191 of G called the orbit space. Conversely, we call 9 a groupoid presentation of 191. 0 is G x G 0 a proper map. Note that in a proper Lie groupoid, every isotropy group is compact. Definition 2.1.5. A Lie groupoid  Definition 2.1.6. Let  1 9 is proper if (s, t) : C  —  9 and FC be Lie groupoids. A strict homomorphism  1 Go and : H commute with all the structure maps for the two groupoids. J-C  —*  0 9 consists of two smooth maps g: H  —>  —  1 that G  Given a Lie groupoid 9, we can associate an important topological con struction to it, namely its classifying space B9. Moreover, this construction is well-behaved under Morita (weak) equivalence. For ri.> 1, let G be the iterated fibered product G  =  ,s(g) 1 {(gi, ...,g,) I g E G  =  ),i 1 t(g+  =  1, ...,n  —  1}  , these G have the structure of a sim 0 Together with the objects G plicial manifold called the nerve of 9. Here we are really just thinking of 7  Chapter 2. Groupoids  9 as a category. Following the usual convention, we define face operators d : G  —k  1 for i =O,...,n, given by G_  j’ ...,  g)  =  (9,...,9n)  ifiO  (ga,  if i  ...,  I  =  otherwise  for 0 <i < n when n> 1. Similarly, we define do(g) when n = 1.  g) d ( s(g) and 1  t(g)  For such a simplicial space, we can glue the disjoint union of the G x /M —+ z be the as follows, where L is the topological n-simplex. Let ö : as the i-th face. We define the classifying linear embedding of L into space of 9 (the geometric realization of its nerve) as the identification space B9  =  JJ(G x  )/(d(g),x)  (g,ö(x))  This is usually called the fat realization of the nerve, meaning that we have chosen to leave out identifications involving degeneracies. The two definitions will produce homotopy equivalent spaces provided that the topo logical category has sufficiently nice properties. Another nice property of the fat realization is that if every G has the homotopy type of a CW-complex, then the fat realization will also have the homotopy type of a CW-complex. Definition 2.1.7. A smooth left Haar system for a Lie groupoid 9 is a family {A’ a E Go}, where each A is a positive, regular Borel measure on the manifold t’ (a) such that: t(V) x W, and if G satisfying V • If (V,’) is an open chart of 1 Aw is the Lebesgue measure on Rk restricted to W, then for each o is equivalent to )w, and the map a e t(V), the measure (a,w) —* d(?o’i/.’a)/dA(w) belongs to C°°(t(V) xW) and is strictly positive.  8  Chapter 2. Groupoids • For any x  , we have C°°(G ) 1  1 and f C  f  f(xz)d)(z)  t’ (s(x))  =  f  f(y)dt(x)(y)  t’ (t(x))  Proposition 2.1.8. Every Lie groupoid admits a smooth left Haar system.  LI  Proof. The proof can be found in [33j.  2.2  Equivalent groupoids  Definition 2.2.1. A strict homomorphism is called an equivalence if:  : 3-C  —*  9 between Lie groupoids  • The map t7rj : C 1  0 H  —+  Go  is a surjective submersion, where the fibered product of manifolds 1 G 0 is defined as {(g,y) I g e Gi,y e Ho,s(g) = H • The square 1 H (s,t)  (s,t)  xG 0 G  xH 0 H is a fibered product of manifolds.  0 can be connected The first condition implies that every object x e G is by an arrow g : (y) —* x to an object in the image of q , that is, 5 essentially surjective as a functor. The second condition implies that induces a diffeomorphism Hi(y,z)  —‘  Q(y),b(z)) 1 G  9  Chapter 2. Groupoids 1 to the space of all arrows from the space of all arrows y z in H . In particular / is full and faithful as a functor. 1 (y) —* (z) in C —  A strict homomorphism ql : J-C —* 9 induces continuous maps —* and Bg B3-C —* B9. Moreover, if q is an equivalence, is a homeomorphism and Bb is a homotopy equivalence. This follows from the fact that an equivalence induces an equivalence of categories. Definition 2.2.2. A local equivalence 3-C  —*  9 is an equivalence with the  additional property that each go e G 0 has a neighbourhood U admitting a lift to Ho in the diagram Io  1 G  U in which the square is a pullback square. Definition 2.2.3. Two Lie groupoids 9 and 9’ are Morita equivalent if there exists a third groupoid FC and two equivalences  9  —  :3-f:  —  Definition 2.2.4. Two Lie groupoids  9’  9 and 9’ are weakly equivalent if  there exists a third groupoid 3C and two local equivalences  9  <—  }( —9’  10  Chapter 2. Groupoids  Groupoid actions  2.3  Definition 2.3.1. Let 9 be a groupoid. A (right) 9-space is a manifold  E equipped with an action by 9. Such an action is given by two maps 1 — E. The latter E —* C 0 (called the anchor map) and i : E xG 0 G map is defined on pairs (e,g) with Tr(e) = t(g) and written u(e,g) = e.g. They must satisfy ir(e  .g)  =  s(g), e u(7r(e))  =  e and (e .g) h  =  e (gh).  Example 2.3.2. Let M be a 9-space. We can construct the action groupoid  00 G M which has space of objects M and morphisms M x . This 1 for action notion a group earlier groupoid action groupoid generalizes the of 3-C  =  9  >i  and the structure maps are formally the same as in that case. Definition 2.3.3. Let 9 be a groupoid and let X, Y be 9-spaces. A map  Y is 9-equivariant if it commutes with the anchor maps and satisfies f(x g) = f(x) g whenever one of the sides is defined.  f  : X  —*  .  • G 0 is a final object in the category of 9-spaces with the action given by e g = s(g) and projection given by the identity. • If  X  X xY  and =  Y  {(x,y)  9-spaces,  are  I  r(x)  fibered  the  product  over  , 0 C  ir(y)} becomes a 9-space with coordinate-  =  0 wise action. In particular X x’,- C  =  X.  • Similarly, if X is a 9-space and Y is any other space, X x Y is a 9-space with trivial action on the second factor. In fact, XxY = Xx(YxGo). Let I denote the unit interval [0, 1]. Given a 9-space X with anchor map lrx, we give X x I the structure of a 9-space with anchor map 7rxxJ(x, )) = lrx(x)  and action (x, A) g  =  (x  .  g,  A) when  lrxxJ(x,  A)  =  7rx(x)  t(g).  Y be two 9equivariant maps between two 9-spaces X and Y. We say that f and g are 9-homotopic if there is a 9-map H X x I —* Y such that H(x, 0) = f(x)  Definition 2.3.4. Let 9 be a groupoid and let f,g  X  —*  and H(x, 1) = g(x) for all x e X. We call H a 9-homotopy between g. g. This is an equivalence relation and we denote it by f  f  and  11  Chapter 2.  Groupoids  Definition 2.3.5. We say two 9-spaces X and Y are 9-homotopy equivalent  if there are 9- maps gf g 1 x•  2.4  f  : X  —÷  Y and g Y  —>  X such that fg  ly and  Equivariant CW-complexes  Definition 2.4.1. An n-dimensional 9-cell is a space of the form D x U  where U is a 9-space such that 9 x U is weakly equivalent to an action groupoid corresponding to a proper action of a compact Lie group C on a finite G-CW-complex. Definition 2.4.2. A 9-CW-complex X is a 9-space together with an  9-  invariant filtration  X 1 ø=X_ ç 0 c ...cxc...cuX=X n>O such that X is obtained from X 1 for each n 0 by attaching equivariant n-dimensional cells, that is, there exists a 9-pushout II U x S 1  “iEI  q  iEI,  II UxD  U.iEIr  iEI  and X carries the colimit topology with respect to this filtration. Definition 2.4.3. A 9-CW-complex X is finite if it is made up out of a  finite number of cells. A 9-CW-complex X is finite-dimensional if there is a positive integer N such that X = XN. Definition 2.4.4. A 9-CW-pair (X, A) is a pair of 9-CW-complexes. Definition 2.4.5. We say a groupoid  0 is a finite 9-CW9 is finite if G  complex. 12  Chapter 2. Groupoids  2.5  Equivariant cohomology theories  Definition 2.5.1. Let S be a Lie group and R a commutative ring.  A proper S-cohomology theory H with values in R-modules is a collection of covariant functors H from the category of proper S-CW-pairs to the cate gory of R-modules indexed by n e Z together with natural transformations 1 (X, A) for n e Z such that the following axioms ö(X, A) : H(A) —* H are satisfied: • S-homotopy equivariance. If fo, fi : (X, A) —* (Y, B) are S-homotopic maps of proper S-CW-pairs, then H(f ) = H(f 0 ) for n Z. 1 • Long exact sequence of a pair. Given a pair (X, A) of proper S-CW complexes, there is a long exact sequence H(X, A) where i : A  —÷  X and  j:X  H(j)  —‘  H(X)  H(i)  H(A)  (X, A) are the inclusions.  • Excision. Let (X, A) be a proper S-CW-pair and let f : A —* B be an S-map. Equip (XUf B, B) with the induced structure of a S-CW-pair. Then, the canonical map (F, f) : (X, A) —* (X Uf B, B) induces an isomorphism H(F,f) : Hg(X,A) • Disjoint union axiom. Let {X complexes. Denote by j : X Then the map  i —b  -  H(XUfB,B)  I} be a family of proper S-CW HiEI X the canonical inclusion.  fl H (ji) : H (II x) II H (Xi) IEI  IEI  iEI  is an isomorphism. Definition 2.5.2. Fix a groupoid  9 and a commutative ring R. A 9-  cohomology theory H with values in R-modules is a collection of covari 13  Chapter 2.  Groupoids  ant functors H from the category of 9-CW-pairs to the category of R modules indexed by integers n E Z together with natural transformations o(X, A) : H(A) —* Hr (X, A) for n E Z such that the following axioms 1 are satisfied: • 9-homotopy equivariance. If fo, fi : (X, A) —* (Y, B) are 9-homotopic ) for n E Z. 1 maps of 9-CW-pairs, then H(fo) = H(f • Long exact sequence of a pair. Given a pair (X, A) of 9-CW-complexes, there is a long exact sequence H(X,A)  where i : A  —*  X and  j  : X  H”(j)  —*  H(X)  H’(i)  (X, A) are the inclusions.  • Excision. Let (X, A) be a 9-CW-pair and let f : A —* B be a 9map. Equip (X Uf B, B) with the induced structure of a 9-CW-pair. Then, the canonical map (F, f) : (X, A) —+ (X Uf B, B) induces an isomorphism H(F,f) : H(X,A)  H(XUfB,B)  • Disjoint union axiom. Let {X i E I} be a family of 9-CW-complexes. Denote by j : X —* JJ 1 X the canonical inclusion. Then the map flH(j) : H(flX) iEI  iI  flH(X) iEI  is an isomorphism. Let C be the category whose objects are pairs (X, F) where X is a proper S-CW-complex and P is an S-projective bundle on X. A morphism  14  Chapter 2. (X, P)  —*  (Y,  Q)  Groupoids  is a diagram F  where F is a map of S-proj ective bundles and  f  is an S-map.  Definition 2.5.3. Let S be a Lie group and R a commutative ring. A proper S-cohomology theory H on the category of proper S-spaces with S-projective bundles with values in R-modules is a collection of covari ant functors H, n E Z, from the category C to the category of R modules that take (X, F) to H(X) together with boundary homomor (X) for ii Z for any pushout diagram 1 ‘H phisms d : PIAH(A) A  2 x where PA  =  )*(F and such that the following axioms are satisfied: (i ) 2  • 5-homotopy equivariance. If fo and fi are S-homotopic maps X — Y of S-CW-complexes and P is a 5-projective bundle on Y, then we have a commutative diagram for all n E Z. H(Y)  ) 0 H(f  15  Chapter 2. Groupoids  • Mayer-Vietoris sequence. For any pushout square of proper S-CW complexes A  2 x  j2  and any S-projective bundle P on X, let Pk = j(P) for k ). Then there is a natural exact sequence 2 PA = (12)* (F —f  where j  ‘H(X) =  12  Pn(j)  P1H_n(X)P2H—n(X)  and i  =  PkHfl(ik)  for k  =  1,2  =  1,2 and  d’ PA H (A) —>  1,2.  • Disjoint union axiom. Let {X i E I} be a family of proper S CW-complexes and P an S-projective bundle on X. Denote by ji: X.j —* fl 1 X the canonical inclusion. Then the map flH(j) : Hic’1iIP1(JJX) iEI  iEI  flPnK iEI  is an isomorphism. Let C 9 be the category whose objects are pairs (X, F) where X is a 9A morphism CW-complex and P is a 9-projective bundle on X. (X, F) —* (Y, Q) is a diagram F  where F is a map of 9-proj ective bundles and f is a 9-map. 16  Chapter 2. Groupoids Definition 2.5.4. Fix a groupoid  9 and a commutative ring R. A 9-  cohomology theory H on the category of 9-spaces with 9-projective bun dles with values in R-modules is a collection of covariant functors  e  Z, from the category C 9 to the category of R-modules that take (X, P) to F’H(X) indexed by ii e Z together with boundary homomorphisms n(A) 9 for n E Z for any pushout diagram d : PIAH n  A  2 x where PA  =  )* (F 2 (j ) and such that the following axioms are satisfied: 2  • 9-homotopy equivariance. If fo and f 1 are 9-homotopic maps X —* Y of 9-OW-complexes and P is a 9-projective bundle on Y, then we have a commutative diagram for all n E Z. “H(Y)  ) 0 H(f  fo()H(x)  * 1 f  ()H(X)  • Mayer-Vietoris sequence. For any pushout square of 9-OW-complexes 1 .X  A  jl  2 x  32  and any 9-projective bundle P on X, let Pk  =  j(P) for k  =  1,2 and 17  Chapter 2. Groupoids PA  =  )*(P (i ) 2 . Then there is a natural exact sequence  d’  where  P  ‘H(X)  j  =  H (X  H(jk) and i  =  ) 2 )$P2H_n(X  PkHn(ik)  for k  =  2  PA H (A)  1,2.  • Disjoint union axiom. Let {X I i e I} be a family of 9-CW-complexes and P a 9-projective bundle on X. Denote by j : X —* HEI X the canonical inclusion. Then the map  llH(j)  n(llxi) 9 HiEIPIH  .  jI  iEI  fli(X) iEI  is an isomorphism.  2.6  Fiber bundles  Definition 2.6.1. Let X be a 9-space. A 9-fiber bundle on X is a fiber bundle ir : P  —*  X for which P is a 9-space and r is a 9-map.  Proposition 2.6.2. Suppose that F: 9 the pullback functor {J-C  —  Fiber bundles on Ho}  —*  —*  J-f is a local equivalence. Then  {9  —  Fiber bundles on Go}  is an equivalence of categories Proof. Suppose that P is a 9-fiber bundle over G . Since F is an equivalence 0 F* has a left adjoint F, given by of categories, the functor FP(x)  =  lim P  —*x 0 G  0 equipped with a morphism x are the elements of G Fy — x. Since F is an equivalence of groupoids, there is a unique map between any two objects of G 0 —* x, and so FP(x) is isomorphic to P, for , a 0 , choose a neighbourhood x e U C H 0 any y E G 0 — x. For each x E H  where G 0  —*  18  Chapter 2. Groupoids map t: U  —  inclusion U  0 and a family of morphisms U C  —  —*  1 connecting F o t to the H  . We topologize 0 H U FP  0 xEH  by requiring that the canonical map t*P  4  F*PIu  be a homeomorphism. This gives FP the structure of a fiber bundle over . Naturality provides F P with the additional structure required to make 0 H it an J{-fiber bundle. One easily checks that the pair (F, F*) is an adjoint 0 with the category of equivalence of the category of J-C-fiber bundles over H . 0 9-fiber bundles over G Proposition 2.6.3. Suppose that F :  9  3f is a local equivalence.  —f  Then the pullback functor induces a homeomorphism from the space of 3-C0 to the space of 9-equivariant equivariant sections of an 3-C-fiber bundle on H sections of the pullback 9-fiber bundle on Go. Proof. Assume we have a local equivalence 9 — 3-C. Given a section v of , we can consider the section F*(v) G —* F*(P) 0 a fibre bundle P —# H defined by F*(v)(x) = (x,v(F(x))). And on the other hand, given a section 0 —* F(Q) w of a fibre bundle Q —* Go, we can consider the section F(w) : H 1 0 is such that there is h e H defined by F(t)(x) = (x,w(y)) where y e C E that satisfies F(y) = s(h) and t(h) = x. Definition 2.6.4. Let P be a 9-space and T  =  P/9. Note that T is not  T is a principal 9-bundle if it admits local sections and if the map 1 G xG 0 P — P XT P induced by the action and homeomorphism. projection maps is a a 9-space. We say that P  —>  Definition 2.6.5. A principal 9-bundle E  —*  B is universal if every prin  19  Chapter 2. Groupoids cipal 9-bundle P  —*  X over a paracompact base admits a 9-bundle map  P  X unique up to homotopy. For any group G we can construct the universal C-space EG in the sense of Milnor [31] or Milgram [301. We have analogous constructions for a groupoid. The first construction is the analogue to the universal space of Milgram and it is also described in [20]. Given the groupoid 9, construct the transla tion groupoid = 9 G , which has G 1 1 as its space of objects and only one arrow between two elements if they have the same source or none otherwise, 3G i.e., the space of arrows is G 1 x . The nerve of this category is given by: 1 kt. x 8 G 5 x 1 G 3 N9k=G1 1 There is a natural action of 9 on N9k with ir given by the source map and h, ..., fk+lh). With this action we have N9k/9 1 Gk, (fi, ..., fk+1) Ii = (f N9 and therefore B9/9 B9. It is clear that B9 Go and 9 so N/9 acts freely on B9. .  The second construction imitates Milnor’s universal G-space. We con struct the 9-spaces: Th E 9  =  5 * 1 G  ..  1 G 3 *  =  {)jgj I s(gi)  =  = ...  1}  =  with the 9-action given by the anchor map 7r(> 1 )jgj) = s(gi), action map (D ).‘jgjg and the subspace topology from 1 )jgj) . g = * .‘. *G 1 G . 1 20  Chapter 2. Groupoids  Now define E9 to be the direct limit of the sequence of 9-spaces 9. Th E Note 9 acts freely on E’9 for all n and thus on E9. It can be checked that E9/9=B9. Both E9 and B9 define universal 9-principal bundles. Therefore E9/9 and B9/9 = B9 are homotopy equivalent. From this point on we will use E9 as our universal 9-space and identify E9/9 with B9. Definition 2.6.6. Given a 9-space X, we define the Borel construction X  =  (X x , E9)/9. 7  In the case of group actions, this construction is also known in the liter ature as the homotopy orbit space. G x M, where G is a topological group and M is a G-space. Then, we have E9 = M x EH and B9 = MH.  Remark 2.6.7. Let  9  =  Remark 2.6.8. Let M be a 9-space. Consider the groupoid X , E9 and B3-C = M 7 . 9 In this case, E3-C = M x  2.7  =  9  xi  M.  Vector bundles  Definition 2.7.1. A complex vector bundle over an orbifold groupoid  9 is a  9-space E for which r : E G is a complex vector bundle, and the action 0 of 9 on E is fibrewise linear. Namely, any arrow g x y induces a linear isomorphism g’ : E E. In particular, E is a linear representation of —*  —  —÷  the stabilizer 9. We will only consider complex vector bundles, so will omit the word complex from now on. Definition 2.7.2. Let  9 be a groupoid. A 9-vector bundle on a 9-space X  is a vector bundle p: E —* X such that E is a 9-space with fibrewise linear action and p is a 9-equivariant map.  21  Chapter 2. Groupoids X a 9-vector bundle. We say V is extendable if there is a 9-vector bundle W — G 0 such that V is a direct summand of lr*W. Definition 2.7.3. Let X be a 9-space and V  —*  • Direct sum of 9-extendable vector bundles induces an operation on the set of isomorphism classes of 9-extendable vector bundles on making this set a monoid. We can also tensor 9-extendable vector bundles. • The pullback of a 9-extendable vector bundle by a 9-equivariant map is a 9-extendable vector bundle. • All 9-vector bundles on C 0 are extendable. Note that 9-extendable vector bundles on G 0 are equivalent to orbifold vector bundles on 9. Example 2.7.4. Let 9 = H x M. Then, a 9-space is a H-space X with a H-equivariant map to M. In this case, 9-equivariant vector bundles on X correspond to H-vector bundles on X. Example 2.7.5. Let M be a 9-space. Consider the groupoid 7C  =  9  >  M.  An 3{-space is a 9-space X with a 9-equivariant map to M. As in the previous example, 3-C-equivariant vector bundles on X are just 9-equivariant vector bundles on X. Proposition 2.7.6. All 9-vector bundles on a free 9-space X are extend-  able. Proof. It suffices to prove that vector bundles on X/9 pull back to extend 0 induces a map able 9-vector bundles. The anchor map ri : X —* G — 1r2 : X/9 191. These maps fit into a commutative diagram: 7tl X  P1  X/9  191 22  Chapter 2. Groupoids Given a vector bundle V on X/9, there is a vector bundle W on that irW = V A for some vector bundle A on X/9. Consider W’  9 =  such  pW.  We have =  2.8  irpW =prW =p(VA) =pVEPpA  Hubert bundles 9  be a Lie groupoid and X a 9-space. A 9-Hubert bundle on X is a 9-space E with an equivariant map p : E —f X which is also a locally trivial Hubert bundle with a continuous linear 9-action. Definition 2.8.1. Let  Definition 2.8.2. A universal 9-Hilbert bundle on X is a 9-Hilbert bundle  E such that for each Hilbert bundle V on X there exists a 9-equivariant unitary embedding V c E. Definition 2.8.3. A locally universal 9-Hilbert bundle on X is a 9-Hilbert  bundle E such that there is a 9-equivariant countable open cover {U} of X such that Eu. is a universal 9-Hilbert bundle on U. Definition 2.8.4. A local quotient groupoid is a groupoid  9  0 such that G  admits a 9-equivariant countable open cover {U} with the property that 9 xi U is weakly equivalent to an action groupoid corresponding to the proper action of a compact Lie group G on a finite G-CW-complex.  Corollary 2.8.5. A finite Lie groupoid is a local quotient groupoid. Proposition 2.8.6. If  9 is a local quotient groupoid, then there exists a  locally universal 9-Hubert bundle on 0 G that is unique up to unitary equiv alence. Proof. See [17]  LI  Corollary 2.8.7. If 9 is a finite Lie groupoid, then there exists a locally universal 9-Hilbert bundle on G 0 that is unique up to unitary equivalence. We denote it by U(9). 23  Chapter 2. Groupoids Definition 2.8.8. Let 9 be a Lie groupoid with a locally universal 9-Hubert bundle U(9) on G 0 and X a 9-space. A 9-stable Hilbert bundle on X is a 9-Hilbert bundle E X  =  ir(U(9)) Go we call E a stable representation of 9. —*  X such that E  Proposition 2.8.9. Suppose that F:  E. In the case when  3C is a local equivalence. Then the pullback functor induces an equivalence of categories, namely, from the category of locally universal 3-C-Hilbert bundles on H 0 to the category of lo cally universal 9-Hubert bundles on G . 0  9  —*  Proof. See [17] Corollary 2.8.10. Suppose that F: 9 the pullback functor  {Stable representations of J-C}  LI —*  —*  Jf is a local equivalence. Then  {Stable representations of 9}  is an equivalence of categories. Definition 2.8.11. Let 9 be a Lie groupoid and X a 9-space. A 9projective bundle on X is a 9-space P with a 9-equivariant map p: P — X such that there exists an equivariant open covering {U} of X for which . Moreover, we shall 0 U.j x,- P(E) for some 9-Hilbert bundle E on G P ® ir*lP(U(9)) for some locally call P a 9-stable projective bundle if P universal 9-Hilbert bundle U(9) on G . 0 =  24  Chapter 3  Equivariant K-theory 3.1  Extendable K-theory  Definition 3.1.1. Let X be a 9-space. Vectg(X)  =  {isomorphism classes of extendable 9-vector bundles on X} (X) 9 K  =  K(Vectg(X))  where K(A) is the Grothendieck group of a monoid A. We call K (X) the 9 extendable 9-equivariant K-theory of X.  G  M, then Kg(M)  =  Remark 3.1.3. Let M be a 9-space and 3-C  =  Remark 3.1.2. If 9  =  >i  KG(M).  9  >  M, then K(M) does  not necessarily coincide with Kg(M), as we will see later on. We can now define the extendable K-groups as in [28]: Th K (X) K(X, A)  =  =  Ker[Kg(X x ) Th S Ker[K(X UA X)  Kg(X)] K(X)]  where i : X — X x S is the inclusion given by fixing a point in S and 32 : X —* X UA X is one of the maps from X to the pushout. We equip X x sn with a 9-action by taking as the anchor map the composition of the projection onto the first coordinate and the anchor map for X. Then let the groupoid act trivially on the sphere. The anchor map for X UA X is the unique map to G 0 making the pushout diagram commutative. The action is induced by the action of 9 on X.  25  Chapter 3. Equivariant K-theory The following lemma follows easily from the definitions: Lemma 3.1.4. Let (X, A) be a 9-pair. Suppose that X disjoint union of open 9-invariant s’ubspaces X and set A  = =  the An X. Then  H X,, 1  there is a natural isomorphism Th K (X,A)  —  fJK(x,A) jEl  From now on  9  will be a Lie groupoid.  Corollary 3.1.5. If fo, between 9-pairs, then  f  =  f f  : (X,A)  —*  : (Y,B) Th K  (Y,B) are 9-homotopic 9-maps  —  K(X,A)  for all n> 0. E  Proof It follows from the existence of a Haar system.  3.2  The Mayer-Vietoris sequence  ) 2 : 1 ,(X Xo) —* (X,X 2 U X . 1 0 X = Ixo, and assume that X 2 be 9-extendable vector bundles, 2 —* X P2 : E 2U E . 1 strong map covering o, and set E = E is a 9-extendable vector bundle over X. Lemma 3.2.1. Let  be a map of 9-spaces, set —* X 1 and Let p : —* E : 2 be a let Then p = P1 Up : E —b X  1 is locally X is locally trivial. Since E trivial, so is EIx_x 0 _ 1 Ex x So it remains to find a neighbourhood of 2 . 0 1 of X 2 over which E is locally trivial. Choose a closed neighbourhood W X Proof We have to show that p : E  —*  . By the 0 W —p X in X 1 for which there is a strong deformation retraction r : 1 homotopy invariance for nonequivariant vector bundles over paracompact  1 wi —* : E spaces, r is covered by a strong map of vector bundles 2U 0 W . Then extends, via the pushout, to 1 . Set W = X 1 which extends i a strong map of vector bundles EIw is locally trivial.  —.  2 which extends E  2  and hence Elw  26  Chapter 3. Equivariant K-theory : X —* Y be a 9-equivariant map and let E’ —> X be a 9-extendable vector bundle. Then, there is a 9-extendable vector bundle E —* Y such that E’ is a summand of q*E.  Lemma 3.2.2. Let  Y —G . Now, 1rq: X —* G 0 . Since E’ is extendable, 0 bundle on such that E’ is a direct summand of there is a 9-vector V 0 G ()*V Let E = 7r*V. E is a 9-vector bundle on Y and it is the pullback Proof. Consider  ir :  of an extendable 9-vector bundle, hence it is extendable. And we have that E E’ is a direct summand of (7r)*V = *E Lemma 3.2.3. Let  A  2 x  32  be a pushout square of 9-spaces. Then there is a natural exact sequence, infinite to the left dZ’ K(X)  —÷  K’(A)  ) 1 K(X  K(X) -? K(A)  K(X) --? K(X ) 1  ) -? K(A) 2 K(X  (3.1)  Proof. We first show that the sequence Kg(X)  3132  (X 9 K ) )e2 1 Kg(X  12  Kg(A)  (3.2)  is exact; and hence the long sequence in the statement of the theorem is exact T K ) 1 (X ) K(X h for all n. Clearly the composite is zero. So fix an ele at 2  Ker(i—i). By the previous lemma, we can add an element of the form ([jE’}, [jE’j) for some 9-vector bundle E’ —* X, and arrange ] for some pair of 9-vector bundles Ek —* Xk. 2 that a 1 = [E ] and a2 = [E 1 ment  cr2) E  27  Chapter 3. Equivariant K-theory Then iE 1 and iE 2 are stably isomorphic, and after adding the restrictions of another bundle over X, we can arrange that iE 1  . Lemma 3.2.1 2 iE  now applies to show that there is a 9-vector bundle E over X such that jE  Ej fork  1,2, and hence that  =  ) 2 (ai,c  =  J, [E 1 ([E ]) 2  e Im(j ej).  Assume now that A is a retract of X . We claim that in this case, 1 (X) 9 Ker[K  (X 9 K ) 2 ]  (Xi) 9 Ker[K  K(A)]  is an isomorphism. It is surjective by the exactness of (3.2). element [E]  EIxi  =  —  (3.3) So fix an  e Ker(j e j). To simplify the notation, we write  [E’]  Let Pi : X 1 —* A be a retraction, and let 2 be its extension to X. By the previous lemma, we can ar X  jE, EIA  =  ijE,  .  .  .  p X —* range that EIxk retraction p : X  E’Ixk for k = 1,2. Applying the same lemma to the —* X , we obtain a 9-vector bundle F’ —> X 2 2 such that p*F/. Stabilizing again, we can assume that E’ p*F! E’ is a sumniand of and hence that F’ phisms bk  EIxk  —*  1 E’1x and E’x 2 p(F’tA) p(E’IA). Fix isomor E’ IXk covering the identity on X. The automorphism  (‘cbIA) o (1IAY’ of E’IA pulls back, under P1, to an automorphism q 5 of b by o we can arrange that hA = “1)2 IA. Then E’Ixj. By replacing 1 i U “1)2 is an isomorphism from E to E’, and this proves the exactness.  Now for each  ii  1,  K(A)  =  (A x ) 9 Ker[K Th S  —*  Kg(A)]  i*  (X UAxpt (A 9 Ker{K  X  Sn))  ((Xi x D) UAxsn-1 (X 9 Ker[K 2 x D))  Kg(X)] Kg(X)]  1 x pt U A x D) is a strong deformation retract of Xj. x the last step since (X 2 x D”) and define D. Denote Y = (X 1 x D”) UAxs,--1 (X K(A)  —*  K(X) to be the homomorphism which makes the  28  Chapter 3. Equivariant K-theory following diagram commute:  (_,pt)* K(A)  o  incl*  d o  (X) 9 K  Kg(Y)  0  Id  (X x S 9 K (X) 9 ) (_,pt): K 1  K’(X)  0  We have already shown that the long sequence (3.1) is exact at (X x D) H(X ) K(X 1 K(X ) for all n. Denote Z = 1 2 2 x D) and 1 X D) UAxpt (X 2 x D). To see exactness at K+l(X) and W = (X 1, apply the exactness of (3.3) to the following split  Th for any n K (A)  inclusion of pushout squares:  1 fix x 2  —  1 fiX (X ) x S 2  x fix 2  I  I  X  X  x  x  Z  X x S 1  x  The upper pair of squares induces a split surjection of exact sequence whose kernel yields the exactness of (3.1) at K+(X). And since  (W) 9 Ker[K (Z) 9 Ker[K  —*  (X 9 K 1  -*  Kg(X)1  )] 2 fix  ) 1 K(X  Th K ) 2 (X  by (3.3), the lower pair of squares induces a split surjection of exact se (A). E quences whose kernel yields the exactness of (3.1) at Lemma 3.2.4. Let q : (x,A)  —  (Y,B) be a map of finite 9-CW-pairs  29  Chapter 3. Equivariant K-theory such that Y  B UIA X. Then K(Y,B)  is an isomorphism for all n  —*  K(X,A)  0.  Proof. The square x  X UA X  —  Y UB  is a pushout, and X is a retract of X UA X. So its Mayer-Vietoris sequence splits into short exact sequences  Th UB Y) K 0—> (Y and so K(Y, B)  —*  K(X UA X)  K(Y)  —>  K(X) —*0  Th A). K (X,  Lemma 3.2.5. Let (X, A) be a finite 9-CW-pair. Then the following se  quence, extending infinitely far to the left, is natural and exact: K(X,A)  L  K(X)  K(X,A)  K’(X,A)  K(X)  K(A)  Proof. This follows immediately from the Mayer-Vietoris sequence for the square A  X  Z1XX  30  Chapter 3. Equivariant K-theory  3.3  The Bott periodicity  We now consider products on K(X) and on K(X, A). We follow [28]. (X) into a com 9 Tensor products of 9-extendable vector bundles makes K f* (Y) —* K 9 K (X) are ring homo 9 mutative ring, and all induced maps morphisms. For each n, m 0, (X) m 9 K  (X x S) m Ker[K  = Ker[K (X x S x ) 9 tm S  —*  —*  Kg(X x S)  (X)] m K  =  Kg(X x )] tm S  where the first isomorphism follows from the usual Mayer-Vietoris sequences, hence (X x 9 K  5fl)  (X 9 ®K  Sm)  mu1top)  (X 9 K  n  5)  restricts to a homomorphism (X) m K(X) ® K  —  (X) m K  By applying the above definition with n = 0 or m = 0, the multiplica tive identity for K (X) is seen to be an identity for K(X). Associativity of 9 the graded product is clear and graded commutativity follows upon showing that composition with a degree —1 map 5fl tm induces multiplication S by —1 on (X). Th This product makes K(X) into a graded ring. Clearly, K K(Y) —* K(X) is a ring homomorphism for any 9-map f X —p Y. This makes K(X) into a K(Go)-algebra, since C 0 is a final object in the category of 9-spaces. We will now construct a Bott homomorphism. Recall that we have ) —* K(pt)] Z, and is generated by the Bott element 2 ) = Ker[K(5 2 K(S ), where H is the canonical 2 B e K(S ), the element [52 x C] [H] E k(S 2 complex line bundle over = CP . For any 9-space X, there is an obvious 1 —  31  Chapter 3. Equivariant K-theory pairing Ker[K(X x S ) 2  Th ® k(S) K (X)  K(X)]  induced by (external) tensor product of bundles. Evaluation at the Bott element now defines a homomorphism b  =  b(X) : K(X)  —*  which by construction is natural in X. And this extends to a homomorphism b  =  b(X, A) : K(X, A)  defined for any 9-pair (X, A) and all n  —  (X, A) 2 K  0.  Definition 3.3.1. A groupoid 9 is Bredon-compatible if given any 9-cell U, all 9-vector bundles on U are extendable. Note that if U is a 9-cell and 9 is Bredon-compatible, then Ku(U) = K(U). Example  3.3.2. An example of a Bredon-compatible groupoid  is  G >i M, where G is a compact Lie group and M is a finite G-CW complex. A 9-cell U is a finite G-space with an equivariant map to M and 9-vector bundles on U are just G-vector bundles. By [40], for any G-vector bundle A on U, there is another G-vector bundle B such that A B is a trivial bundle, that is, the pullback of a G-vector bundle V over a point. Consider the unique map from M to a point. The pullback of V over this map is a G-vector bundle on M. If we pull it back to U we recover A B  9  =  and therefore 9 is Bredon-compatible. Corollary 3.3.3. If 9 is Bredon-compatible and U is a 9-cell, then K(U) K(M) for some compact Lie group G and a finite G-CW-complex M. Theorem 3.3.4. If 9 is Bredon-compatible, the Bott homomorphism b  =  b(X,A) K(X,A)  —>  (X,A) 2 K 32  Chapter 3. Equivariant K-theory is a natural isomorphism for any finite 9-CW-pair (X, A) and all n  0.  ) where U x D m m is a 9-cell. Proof. Assume first that X = Y U (U x D ) 1 Assume inductively that b(Y) is an isomorphism. Since K(U x S”— ) m ), m is isomorphic to K(M x Sm_l) and K(U x D K(M x D ) are isomorphisms by m the Bott homomorphisms b(U x Sm_l) and b(U x D the equivariant Bott periodicity theorem for actions of compact Lie groups. The Bott map is natural and compatible with the boundary operators in the Mayer-Vietoris sequence for Y, X, U x sm_i and U x D and so b(X) is an isomorphism by the 5-lemma. The proof that b(X, A) is an isomorphism LI follows immediately from the definitions of the relative groups. Based on the Bott isomorphism we just proved, we can now redefine for all n E Z K(x, A) For  any K(A)  finite —p  —  9-CW-pair  f  (X, A) 9 K°  if ri is even  K’(X, A)  if n is odd  (X,A),  define  (X,A) to be 6 : K’(A) 1 K  the —f  boundary  operator  (X,A) if n is odd, 9 K°  and to be the composite K(A)  -L K (X,A) 1 (A) L K 2  if n is even. We can collect all the information we have so far about 9-equivariant K-theory in the following theorem: Theorem 3.3.5. If  9 is a Bredon-compatible Lie groupoid, the groups  K(X, A) define a 7Z/2-graded multiplicative cohomology theory on the cat  egory of finite 9-CW-pairs. Note that for a general Lie groupoid 9, K(—) is a multiplicative coho mology theory on the category of 9-spaces, but it is not clear whether we have Bott periodicity.  33  Chapter 3. Equivariant K-theory  3.4  Equivariant representable K-theory  Let H be a stable representation of 9. Consider the associated bundle of Fredhoim operators Fred(H) on G , and the subbundle Fred’(H) of 0 operators A for which the action g — gAg 1 is continuous. See [16] for more details on the correct topology for this space.  f  Definition 3.4.1. We say that a 9-equivariant map  : X  —*  Fred’ (H) is  extendable if there is another 9-equivariant map g : X —> Fred’ (H) such that gf = vlrx for some section v of Fred’(H) — Go, where 7rx : X —> C 0 is the anchor map. Definition 3.4.2. We say that a homotopy H : X x I  —,  Fred’(H) of  9-equivariant maps is extendable if each FI is an extendable 9-equivariant map X Fred’ (H). —,  Definition 3.4.3. Let X be a 9-space, H a stable representation of  n  9 and  0. Define the 9-equivariant representable K-theory groups of X to be RK(X)  =  [X, ?7Fred’(H)] t  where this notation denotes the extendable homotopy classes of extendable 9-maps. For 9-pairs (X,A), define RK(X, A)  where j2 : X  —*  =  Ker[RK(X UA X)  --  RK(X)]  X UA X is one of the maps from X to the pushout.  We could have defined the representable K-groups as in [28]: RK(X)  =  (X x S) 9 Ker[RK  —--*  (X)] 9 RK  t and i : X —f X x S is the inclusion where RK (X) = [X, Fred’(H)] 9 given by fixing a point in 5Th• Both definitions are clearly equivalent.  We will now prove this defines a cohomology theory on the category of 34  Chapter 3. Equivariant K-theory 9-spaces. Since the definition is given by homotopy classes of maps, the next corollary follows from the definition. Corollary 3.4.4. If fo, fi : (X, A)  —*  (Y, B) are 9-homotopic 9-maps  between 9-pairs, then  f for all n  =  f  : RK (Y, B) Th  —*  RK(X, A)  0.  The following lemma follows easily from the definitions: Lemma 3.4.5. Let (X, A) be a 9-pair. Suppose that X  =  II X,, the disjoint iEI  union of open 9-invariant subspaces X and set A a natural isomorphism (X, A) Th RK  —  =  An X. Then there is  (X, A) Th H RK iEI  Lemma 3.4.6. Let q : X  —*  Y be a 9-equivariant map, H be a stable  representation of 9 and let s : X —* Fred’(H) be a 9-extendable map. Then, there is a 9-extendable map t : Y —* Fred’ (H) such that s’s = tq’i for some 9-extendable map s’ : X —* Fred’ (H). Proof. Since s is extendable, there is a 9-extendable map s’ : X —* Fred’ (H) —f Fred’(H). such that s’s = v = ‘Irx = v7ryb for some section v : Choose t = vlry. This is a 9-extendable map for it is the pullback of a 9-extendable map and we have s”s = tb. Lemma 3.4.7. Let  A  hi  2 x  35  Chapter 3. Equivariant K-theory be a pushout square of 9-spaces, H a stable representation of 9 and i 1 a cofi bration. Let 5 k : Xk —* Fred’ (H) be 9-extendable maps for k = 1,2 such that s i and s2i2 are 9-extendable homotopic maps from A to Fred’(H). 1 Then, there is a 9-extendable map t : X —* Fred’(H) such that tjk is 9-extendable homotopic to 5 k for k = 1,2. Proof. Let F : A x I 0 F  Fred’(H) be a 9-extendable homotopy with  —*  i and F 1 s 1 = s2i2. There is v2 : G 0 —* Fred’ (H) such that 2 —* Fred’ (H). By the previous lemma, = vir2 for some S’2 : X there is a 9-extendable homotopy F : C 0 x I — Fred’(H) such that F’F = F o (irA x id) for some F’ : A x I —* Fred’ (H). We can also make it satisfy F ’ = s’ 1 i2 by multiplying by a convenient constant homotopy for 2 =  . 0 C Now, since A x I  1 x I is a 9-equivariant cofibration, there are X : X 1 x I —‘ Fred’(H) that extend F and F’ respectively. Therefore C, G’ 1 x I. In fact, by the previous G’G must be an extension of Fo (irA x id) to X 1 x id). Therefore C is lemma, we can choose G and G’ so that G’G = F(ir a 9-extendable homotopy. Let C 0  =  s  and C 1  =  The extendable 9-homotopy classes of these  .  1 = s2i2. So we can easily extend G =F i 1 and tj this to a map t : X —* Fred’ (H) such that tj 1 = 2 = 2• Therefore tj is 9-extendable homotopic to 8 k for k = 1, 2. In fact, it is given by: two maps are equal, and s i 1  =  .  t(x)  =  f  s(x)  if x  =  (x j ) 1  52(X2)  if x  =  j2(x2)  We only need to prove that t is extendable. Let  have s i 1  =  i 1 C’  =  ’ 1 F  =  t’(x)  ii’  =  G,  s’i  =  F. We  . Consider i 2 s’  —  f1  ’(x s ) 1  if x  =  (x j ) 1  52(X2)  if x  =  j2(x2)  36  Chapter 3. Equivariant K-theory  Let F 1  =  v1. Then we have v(x)  =  f 1.  =  . Now consider the map: r 1 v  vi(7r1(xi)) V(7t(X))  This map is well defined. If lrl(xl)  =  if x if x  =  (xj) 1 r  =  1r2(X2)  7r2(x2),  then x  and vi(lri(xi)) = vJ7rAa = Fl7rAa = F1’lrAa = 2 (x Fiir ) a routine check that t’t = vrx, thus t is extendable  = =  iia  and x2  =  v2(7r2(x2)).  i2a,  It is Li  Lemma 3.4.8. Let il  A  j1 32  2 x  1 a cofibration. Then there is a natural be a pushout square of 9-spaces and i exact sequence, infinite to the left 1 RK”(X) dZ4  — RK’(A)  —*  ) —? RK 2 (X Th RK (A) Th  ) 1 RK(X  ) 1 RK(X) 2? RK(X  RK(X) —? RK(A)  Proof. It is a consequence of the two previous lemmas, the results in [8] and LI the proof of lemma 3.8 in [28].  For any stable representation H of 9 there is a 9-map Fred’(H), which therefore induces a Bott map 2 iFred’(H) —* + (X). By the definition of the relative groups, 2 b(X) : RK(X) —* RK (X, A). We will 2 we also have Bott maps b(X, A) : RK(X, A) —* RK prove that these maps are isomorphisms for finite 9-CW-complexes. Lemma 3.4.9. Suppose that F : have an isomorphism:  9  —*  F* : RK(H ) 0  3f is a local equivalence. Then we  —*  RK(Go) 37  Chapter 3. Equivariant K-theory ) 0 Proof RK(H =  =  , l’Fred’(H)] 0 [H t  Extendable i-C-homotopy classes of extendable sections of Fred’(H) Extendable 9-homotopy classes of extendable sections of F*(lnFred(H))  =  Extendable 9-homotopy classes of extendable sections of 1ViFred’(F*H)  =  t [Go, crFredI(F*H)] U  =RK(Go)  Corollary 3.4.10. If 9 and J-C are weakly equivalent, we have an isomor ) RK(Go) 0 phism RK(H Corollary 3.4.11. If 9 is a Bredon-compatible finite Lie groupoid and U K(M) for some compact Lie group G and a is a 9-cell, then RK(U)  finite G-CW-complex M. Proof Since U is a G-cell, we know that 9 x U is weakly equivalent to G x M for some compact Lie group C and a finite G-CW-complex M. Therefore, by the previous corollary: RKu(U)  RKM(M)  Let H be a locally universal representation of 9. We want to see that 7r(H) = U x H is a locally universal 9 i U-Hilbert bundle. Notice that if U is a 9-cell, so is any open 9-subspace of U. Therefore it is enough to prove the previous assertion with universal Hilbert bundles. So assume H is a universal 9-Hilbert bundle. Now let V be a 9 x U-vector bundle on U. This a 9-vector bundle on U, and since 9 is Bredon-compatible, there is a 9-vector bundle W on G 0 such that ir(W) = V V’ for some other 9-vector bundle V’ on U. Since H is universal, there is a unitary 9-embedding W —* H and so 38  Chapter 3. Equivariant K-theory ir(W) ‘—* ‘4(H) = U x H. Since V is a direct summand of ir(W), we have a unitary 9 x U-embedding V U x H. Thus, if E is a locally universal Hubert representation of C, then E x M is a locally universal G i M-Hilbert bundle. RKu(U)  =  [U,QFred’(U x H)]  = [U, U x VFred’(H)ju = (9  >  =  =  ,- rFred’(H) over U 7 U)-extendable sections of U x  = 9-extendable sections of U x- QFred’(H) over U t = [U, c2”Fred’(H)]  RK3M(M)  =  =  =  RK(U)  =  [M, Fred’(M Th x E)]M 2  = [M,M x Fred’(E)jM  =  =  = (C x M)-extendable sections of M x Fred’(E) over M = (G x M)-sections of M x iFred’(E) over M = [Mj2Fred’(E)]G  =  =  =  K(M)  Therefore, RK (U) Th  K(M)  LI  Theorem 3.4.12. If 9 is a Bredon-compatible finite Lie groupoid, the Bott homomorphism b  =  b(X,A) : RK (X,A) Th  —p  (X,A) 2 RK’  39  Chapter 3. Equivariant K-theory is a natural isomorphism for any finite 9-CW-pair (X, A) and all n  0.  ) where U x D m m is a 9-cell. Proof. Assume first that X = Y u (U x D isomorphism. Since (U>< Th RK ) is 1 S— Assume inductively that b(Y) is an ) RK(M x D m ), m isomorphic to RK(M x Sm_i) and RK(U x D ) are isomorphisms by m the Bott homomorphisms b(U x Sm_l) and b(U x D the equivariant Bott periodicity theorem for actions of compact Lie groups. The Bott map is natural and compatible with the boundary operators in the Mayer-Vietoris sequence for Y, X, U x Sm—i and U x D and so b(X) is an isomorphism by the 5-lemma. The proof that b(X, A) is an isomorphism follows immediately from the definitions of the relative groups. Based on the Bott isomorphism we just proved, we can now redefine for all n E Z  RK(X, A)  =  1..  RK° ( 9 X, A) RK’(X,A)  For any finite 9-CW-pair (X,A),  if n is even ifnisodd  define the boundary operator  RK(A) —* RK’(X,A) to be 6 : K’(A) odd, and to be the composite RK(A)  -L  (A) 2 RK  —*  (X,A) if n is 9 K°  RK’(X, A)  if n is even. We can collect all the information we have about 9-equivariant repre sentable K-theory in the following theorem: Theorem 3.4.13. If 9 is a Bredon-compatible finite Lie groupoid, the groups RK(X, A) define a 7/72-graded multiplicative cohomology theory on the cat  egory of finite 9-CW-pairs. Note that for a general finite Lie groupoid 9, RK(—) is a multiplicative cohomology theory on the category of 9-spaces, but it is not clear whether we have Bott periodicity.  40  Chapter 3. Equivariant K-theory Corollary 3.4.14. Let  a 9-cell. Then K(U)  9 be a Bredon-compatible finite Lie groupoid and U RK(U)  Proof. If U is a 9-cell, then 9 U is weakly equivalent to G >i M for some compact Lie group G and a finite G-CW-complex M. By corollary K(M). By corollary 3.4.11, we also have 3.3.3, we also have K(U) RK(U)_KtJ(M). Theorem 3.4.15. Let  9 be a Bredon-compatible finite Lie groupoid and X  a finite 9-CW-complex. Then K(X)  RK(X)  Proof. Assume first that X = Y U, (U x D ) where U x D m m is a 9-cell. Assume inductively that we have an isomorphism K(Y) --* RK(Y). We know that K(U x S— ) is isomorphic to RK(U x Sm_l) and 1 ) by the previous corollary. In fact, since m K(U x D ) m RK(U x D these last two isomorphisms follow from choosing a weak equivalence from the same 9-cell to the action of a compact Lie group on a finite equivariant CW-complex, these isomorphisms are natural with respect to the Mayer Vietoris sequences for RK and K. Let us denote the corresponding groups ), A m Th = K(Y) K(U x D ), m by RA = RK(Y)eRK(U x D Th = RK(U x Sm-l) and B = K(U x S”RB ) , then: 1 B’  A’’  1 RA  —  RB’  —  K(X)  —  I  RK(X)  and so the result follows by the 5-lemma.  A  —  RA  B  -  RB LI  In other words, we have just proved that the cohomology theory K(—) is representable by extendable maps. Corollary 3.4.16. Let  9 be a Bredon-compatible finite Lie groupoid, X a  41  Chapter 3. Equi variant K-theory finite 9-CW-complex and H a stable representation of 9, then: K(X)  =  f  [X,Fred’(H)] t t [X, f2Fred’(H)]  if n is even if n is odd  We would like to make two observations: 1. Note that all constructions and results in this sections remain true if we relax the condition of  9 being finite to 9 having a locally universal  Hilbert representation U(9). 2. This cohomology theory should not be confused with the 9-equivariant representable K-theory defined in [16]. In their paper, they define 9-equivariant representable K-theory of X as the KK-groups asso ciated to Co(X) and show that this is actually representable (by all 9-equivariant continuous maps) by a corresponding Fredholm bundle. Note that in our case only a special class of maps are considered to have a correspondence with extendable vector bundles, but the Fredholm bundles in both cases are equivalent.  42  Chapter 4  Twisted equivariant K-theory 4.1  Twisted extendable K-theory  Let X be a 9-space and P —* X a 9-projective bundle on X. We can then construct the bundle End(P) on X whose fibre at x is the vector space End(H) of endomorphisms of a Hubert space H such that P = lP(H). Similarly, we can replace End(H) by Fred(H), the space of Fredhoim operators from JI to H, and define in this way a bundle Fred(P) — X. Now consider the subbundle Fred’(P) of Fredholm operators A such that 1 for which the expression makes g —* gAg’ is continuous for all g E G sense. Definition 4.1.1. We say that a 9-equivariant section s of Fred’ (F) —‘ X is extendable if there is another 9-equivariant section t such that ts = vrrx for some section v of Fred’(P) — Go. Definition 4.1.2. We say that a homotopy H  X x I  —  Fred’ (F) of  9-  equivariant sections is extendable if each H is an extendable 9-equivariant section of Fred’(P) —* X. Definition 4.1.3. Let P be a 9-stable projective bundle and X a 9-space. We define the 9-equivariant twisted extendable K-theory of X with twist ing P to be the group of extendable homotopy classes of extendable 9(X) 9 equivariant sections of Fred’(P) and we denote it by F’K In order to define the rest of the twisted extendable K-groups, we need to introduce the fibrewise iterated loop-space 12Fred’(P), which is a 9-bundle on X whose fibre at x is rFred’(H). 43  Chapter 4.  Twisted equivariant K-theory  Definition 4.1.4. The extendable homotopy classes of sections of this bun  dle will be denoted by K(X). The groups K(X) are functorially associated to the pair (X, F) and so an isomorphism P  —  isomorphism K(X) Corollary 4.1.5. If  P’ of 9-stable projective bundles on X induces an —*  ‘‘K(X) for all n  0  9 is a Bredon-compatible finite Lie groupoid and P  is a trivial 9-stable projective bundle on a finite 9-CW-complex X, then K(X)  K(X).  Proof. It follows from the representability of 9-equivariant K-theory, that E  is, corollary 3.4.16.  Corollary 4.1.6. Let P be a 9-stable projective bundle on Y. If the maps  Jo, fi X —f Y are 9-homotopic 9-maps between 9-spaces, then f(P) is isomorphic to f (P) and we have a commutative diagram:  f (P) for all n  (X)  0.  Lemma 4.1.7. Let  A 2 i 2 x  ji 32  be a pushout square of 9-spaces and P a 9-stable projective bundle on X. . Then there is a natural )*(P 2 (i Let Pk = j(P) for k = 1,2 and PA = ) 44  Chapter 4.  Twisted equivariant K-theory  exact sequence, infinite to the left d’  K(X)  PA K’ (A)  P-n  ‘K(X,)  —*  K(X)  J2  (X  )  PA K (A)  ‘K(X,)  Proof. The proof is essentially the same as that of lemma 3.4.8.  PAO)  E  If we use a mod 2 graded version Fred(°) (1) of the bundle of Fredhoim operators associated to a projective bundle P, as in [6], we have a multipli cation: (X) 9 (X) 0 F”K 9 ‘K°  (X) 9 ‘K°  A 0 1 + 1 0 A’ defined on the spaces of degree 1 self-adjoint Fredhoim operators. This extends the multiplication in untwisted 9-equivariant K-theory and makes “K(X) into a K(X) coming from the map (A, A’)  —*  module. Just like in the case of representable K-theory, for any 9-stable Hilbert bundle there is a 9-map  Fred’ (H)  —  2 Fred’ (H). Therefore, for any 1  9-stable projective bundle P on X there is a Bott map: b(X,P) :  _n(X) 9 PK  .S  P-n-2  We do not know if this map is an isomorphism in general. Now we will prove that b(X, F) is an isomorphism when X is a finite 9-CW-complex using a similar argument to the one used for untwisted 9-equivariant Ktheory.  3C be a local equivalence, and P a H . Then F induces an isomorphism stable projective bundle on 0  Proposition 4.1.8. Let F : ) 0 K(H  —*  9  —k  F*(P)K*(Go)  Proof. First of all, F*(P) is a 9-stable projective bundle by corollary 2.8.10 and proposition 2.6.2. Since these groups are defined using sections, the result follows from proposition 2.6.3. 45  Chapter 4. Twisted equivariant K-theory Corollary 4.1.9. If  9 and J-C are weakly equivalent and P is a Jf -stable  , then ‘K(Ho) 0 projective bundle on H Corollary 4.1.10. If  F*(P)K(GO)  9 is a Bredon-compatible finite Lie groupoid, U is a  9-cell and P is a 9-stable projective bundle on U, then K(U) QK(M) for some compact Lie group G, some finite G-CW-complex M and some 0-stable projective bundle Q on M. Proof. Since U is a G-cell, we know that 9 U is weakly equivalent to 0 > M for some compact Lie group G and a finite G-CW-complex M. Therefore, by the previous corollary:  for some G >i M-stable projective bundle Q on M. In the proof of corol lary 3.4.11, we saw that if H is a locally universal representation of 9, then U xH is a locally universal 9 i U-Hilbert bundle. And also that if E is a lo cally universal Hilbert representation of 0, then E x M is a locally universal G x1 M-Hilbert bundle. It follows that if P is a 9 x U-stable projective bundle on U, then P is a 9-stable projective bundle on U. Similarly, if Q is a G M stable projective bundle on M, then Q is a 0-stable projective bundle on M. Ku(U)  =  (9  =  U)-extendable sections of fl’Fred’(P) over U  = 9-extendable sections of c2’Fred’(P) over U  =  =  = PK*(U)  QKM(M)  =  = (0 x M)-extendable sections of 2Fred’(Q) over M  =  46  Chapter 4. Twisted equivariant K-theory =  G-sections of fFred’(Q) over M  =  = QK,(M) Therefore, 1 ’K(U)  Theorem 4.1.11. If  QK,(M)  9 is a Bredon-compatible finite Lie groupoid, the Bott  homomorphism b  =  b(X,P) : F’K(X)  is an isomorphism for any finite 9-CW-complex X, all 9-stable projective bundles on X and all n 0. m is a 9-cell. Let ) where U x D m Proof. Assume that X = Y U, (U x D P be a 9-stable projective bundle. Assume inductively that b(Y, I) is )Kn(U x D m ) m QK(M x D ) and m an isomorphism. Since PI(UxD _n(U x S 9 PRUxSm_l)K ), the Bott homo 1 ) QI(MXSm_l)KG_n(M < S— 1 morphisms b(U x Sm_i,PI(UXSm_1)) and b(U x Dm,PI(UXDm)) are isomor phisms by the Bott periodicity theorem in twisted equivariant K-theory for actions of compact Lie groups [6]. The Bott map is natural and compati ble with the boundary operators in the Mayer-Vietoris sequence for Y, X, m and so b(X, P) is an isomorphism by the 5-lemma. U x 3m—i and U x D Based on the Bott isomorphism we just proved, we can now redefine for all n Z ‘K(X) if n is even K(X) = P() if n is odd  f  We can collect all the information we have so far about twisted 9equivariant K-theory in the following theorem: Theorem 4.1.12. If 9 is a Bredon-compatible finite Lie groupoid, the groups  K(X) define a Z/2-graded cohomology theory on the category of finite  47  Chapter 4. Twisted equivariant K-theory 9-CW-complexes with 9-stable projective bundles, which is a module over untwisted 9-equivariant K-theory. Note that for a general Lie groupoid 9, K(—) is a cohomology theory on the category of 9-spaces, but it is not clear whether we have Bott periodicity. All constructions and results in this section are true if we relax the condition of 9 finite to 9 admitting a locally universal 9-representation.  4.2  The orbit category  Recall from section 2.4 that a 9-cell is a 9-space U for which the groupoid 9 x U is weakly equivalent to the action of a compact Lie group G on a finite G-CW-complex. The orbit category Og is a topological category with discrete object space formed by the 9-cells. The morphisms are the 9-maps, with a topology such  (U, V) x U —p V are continuous for all 99 that the evaluation maps Hom -space we shall mean a continuous contravariant functor 9 cells U, V. By an 0 from 09 to the category of topological spaces. Definition 4.2.1. Let X be a 9-space. The fixed point set system of X, (U,X) and given 9 written X, is an Og-space defined by X(U) = Map  e  U  —*  V, X()(f)  =  fG. We also denote X’  =  Mapg(U,X).  0 space T such that each space -space is an 9 9 Definition 4.2.2. A CW-0 T(U) is a CW-complex and each structure map T(U) —+ T(V) is cellular.  -spaces —* 9-spaces and a 9 Theorem 4.2.3. There is a functor C : 0 -space T and 9 : 4C —* Id such that for each 0 natural transformation : (CT)U —* T(U) is a homotopy equivalence, in fact a strong each U, deformation retraction. Proof. We first construct the 9-space CT. Let OT denote the topological category whose objects are triples (U, s, y) where U is a 9-cell, 48  Chapter 4. Twisted equivariant K-theory s E J(U) U and y E T(U). Let us consider the nerve of this cate gory as a topological simplicial space. This is the bar complex B(T, Og, J), where J: Og —* Top is the covariant functor which forgets the 9-action. Then B(T,0g,J) consist of (n + 2)-tuples (y,fl,f2,. ,f,s) where the f.j : U U, and 1 are composable arrows in 09, s E J(U) U_ ). The boundary maps are given by: 0 y E T(U .  .  —  o(y,f1,f2,. .,f,s) .  (y,fl,f2,...,fn,s) m 8  i(y,fl,f2,... ,f,s) 8  =  =  =  (f(y),f2,f3,..  .,f,s)  (y,fl,f2,...,fn_1,(fn)*(s))  (y,fl,f2,.. ,fi—1,fifi+1,fi+2,.  ..  ,f,S)  Degeneracies are the insertion of identity maps in the appropriate spots.  The groupoid  9 acts simplicially on B (T, Og, J) and consequently the geo  metric realization B(T, 09, J) is a 9-space. We define CT We now require the homotopy equivalence 9-cell U, natural in U. We have: (CT)U  =  ,J)U 9 B(T,0  =  : (CT)U  =  —f  B(T, 09, J). T(U) for each  ,Hom B(T,0 ( 9 U,—))  The second equality follows from the fact that  9 acts on the last coordi  nate only. Now it is a general property of the bar construction that for any topological category C, contravariant functor F: C — Top and object A of C, there is a natural map B(F,C,Homc(A,—))  —k  F(A)  which is a strong deformation retraction. This map is induced by a simplicial map i  :B(F,C,Homc(A,—)) —*F(A)  where F(A) is the simplicial space all of whose components are F(A) and  49  Chapter 4. Twisted equivariant K-theory all whose face and degeneracy maps are the identity. In our case, by the formula:  is given  n(y,f1,f2,...,fn,f)=(f1o...ofnof)*(y)  Now f is an element of Horn 09 (U, tJ). The proof that is a strong de formation retraction is a standard simplicial argument contained in chapter 12 of [29]. E  4.3  The classification of projective bundles  (X) for the group of isomorphism 9 Now we follow [6] closely. Let us write Pic classes of complex 9-line bundles on X (or equivalently, of principal S 1 bundles on X with 9-action), and Projg(X) for the group of isomorphism classes of 9-stable projective bundles. Applying the Borel construction to line bundles and projective bundles gives us homomorphisms: Picg(X)  —*  Pic(Xg)  (X) 9 Proj  —*  Proj(X)  H(X; Z) H(X; Z)  which we shall show are bijective. Definition 4.3.1. A topological abelian 9-module is a 9-space such that each of the fibres of its anchor map is a topological abelian group and the action is linear. Example 4.3.2. Given any topological abelian group B, G 0 x B is a 9module with the anchor map given by projection on the first coordinate. When there is no danger of confusion we will denote it by B. Let us introduce groups H(X; A) defined for any abelian 9-module A. These are the hypercohomology groups of a simplicial space x whose real ization is the space X . Whenever a Lie groupoid 9 acts on a space X we 9 can define the action groupoid whose space of objects is X and space of morphisms is G 1 x’,- X. Let x be the nerve of this groupoid regarded as a 50  Chapter 4. Twisted equivariant K-theory simplicial space, that is,  =  G x ,- X, where G is the space of composable  p-tuples of arrows in 9. For any simplicial space with an action of a Lie groupoid 9 and any topo logical abelian 9-module A we can define the hypercohomology IHI*(x; sh(A)) with coefficients in the sheaf of continuous equivariant A-valued functions. It is the cohomology of a double complex C, where, for each p cochain complex C calculates H*(p; sh(A)).  Definition 4.3.3. H(X;A) These groups are the 7 X; sh(A)). E’ = H(G x  =  0, the  IHI*(x;sh(A))  abutment  of a spectral sequence with  Lemma 4.3.4. If 9 is a Bredon-compatible Lie groupoid and X is a finite  9-CW-complex, Hr’(X;Z)  H(X;S’) for anyp>0.  Proof. If we compare the spectral sequences for the 9-CW-structure of X and H(—;S’), we with respect to the cohomology theories notice that we have an isomorphism in each cell by the similar result in U  [6].  Proposition 4.3.5. Let 9 be a Bredon-compatible finite Lie groupoid and  X a 9-space. Then we have:  1. H(Go; Z)  Hom(9, Go x S’)  0 x S’), the group of equivalence classes of cen 2. H(Go; Z) Ext(9, G FC —* 9 — 1. 1 tral extensions 1 —* G 0 x S —,  3. H(X;Z) .  H(X; Z)  Picg(X) Projg(X)  H(pt;sh(Sl)) = 0 ;sh(G x S H(G 1. We have E = 0 Proof. )) 1 when X = G 0 and in the previous lemma we have seen that  51  Chapter 4. Twisted equivariant K-theory E = Hc.c.(9; S ) is the cohomology of 9 defined by continuous Eilenberg 1 Maclane cochains. So H(Go;Z)  =  ) 1 H(Go;S  H(9;Si)  Hom(9,Go x S ) 1  2. In this case the spectral sequence gives us an exact sequence  )--*E” 1 O—*E° —*H(Go;S  —  that is,  0  —*  H(9;Si)  —*  H(Go;S’)  —>  Pic(9)  H(9;Si)  H’(9;sh(Go x S’)) = Pic(9), and E 1 is the subgroup of primitive elements, that is, of circle bundles J-C on 9 such that m*3f are the pr ® prC, where prl,pr2, m : G 1 xG 1 —* 0 G for E ’ 1  =  obvious maps. Equivalently, Pic(9)prjm consists of circle bundles H on 9 equipped with bundle maps i7i: H 1 xG 1 —‘ H 1 covering the 0 H multiplication in 9. It is easy to see that the composite  Ext(9,Go x S’)  —p  H(Go;Go x S’)  —‘  Pic(9)  takes an extension to its class as a circle bundle. On the other hand  (9; 31) is just the group of equivalence classes of extensions x 1 — 3C —* 9 which as circle bundles admit a continuous section, so its image in Ext (9, Go x S’) is precisely the kernel of this composite. It remains only to show that the image of Ext(9, G 0xS ) in Pic(9),,.jm 1 is the kernel of Pic(9)prim —* H ). This map associates to a 1 (9; S 3 bundle J{ with a bundle map ñi as above precisely the obstruction to changing  by a bundle map 1 G xG 1 0 G ciative product on i-C. ni  —  S’ to make it an asso  52  Chapter 4.  Twisted equivariant K-theory  3. The spectral sequence gives 0—> E°  -  H(X;S’)  —*  —*  Now E? ’ is the subgroup of circle bundles S —* X 0 2 1 = Pic(X), and E which admit a bundle map ñi G 1 xG 0 S —* S covering the 9-action on X. As before, ñi can be made into a 9-action on S if and only if an obstruction in (9; Map(X, Go x 5’)) vanishes. Finally, the kernel of Picg(X) —* Pic(X) is the group of 9-actions on X x 5’, and this is just E° = H’(9; Map(X, G 0 x 5’)).  4. First we shall prove that the map Projg(X)  —*  H(X; Z) is injective.  Consider the filtration Projg(X)  Proj’  Proj°  Here Proj(’) consists of the stable projective bundles which are trivial when the 9-action is forgotten, that is, those that can be described by cocycles c : C, xG 0 X —* PU(C) that satisfy the condition Q(g2,glx)c(g,,x) Proj(°)  =  (g2g,,x).  consists of those projective bundles for which c lifts to  00 X —* U(J-C) satisfying the equality a map C, x c(g2,g,x)cx(g,,x) = c(g2,g,,x)a(g2g,,x) for some c: G 2 xX —->5’ We shall compare the filtration of Projg(X) with the filtration H(X;S’) D H’ 1 D  defined by the spectral sequence. By definition H(’) is the kernel of H(X; 5,) —> E,° 2 = H (X; sh(Si)) = Proj(X), and the composite 2 9 —-> H(X; S,) —* Proj(X) is clearly the map which forgets the Proj 9-action. Thus Proj g(X)/Proj(’) maps injectively to H(X, S’)/H(’) Now let us consider the map Proj(’)  —>  H(’). The subgroup H(°) is 53  Chapter 4. the kernel of H(’)  —*  Twisted equivariant K-theory E’, while Er”  =  Pic(Gi  0 xG  X). We readily  check that an element of Proj(’) defined by the cocycle o maps to the element of Pic(Gi 0 xG X) which is the pullback of the circle bundle U(ff) — PU(FC), and can conclude that c maps to zero in E’ if and only if it defines an element of P’roj(°). Thus Proj(’)/Proj(°) injects into H(’)/H(°). Finally, assigning to an element c of Proj(°) the class H(9;Map(X,Go x S’)) of the cocycle c, we see that if vanishes, then the projective bundle comes from a 9-Hilbert this class in E°  =  bundle, which is necessarily trivial, as we have already explained. So Proj(°) injects into H°).  Now we will construct a universal 9-space C(P) with a natural 9-stable projective bundle on it and show that the composite map  [X,C(P)]g  —,  (X) 9 Proj  —*  H(X;Z)  is an isomorphism.  Let U be a 9-CW-cell and consider the spaces  P(U)  =  JJ  BPU(H)U  HEExt(U,S’)  where we represent an element of Ext(U, .9’) by a Hilbert bundle H with a stable projective representation of 9 U inducing the extension. Note that even though there may be different choices of H, they all determine the same class in Hu(U; Z), and so this space P(U) is well-defined.  In fact, P is an Og-space. Now, we can use theorem 4.2.3 to construct P(U) for every 9-cell the 9-space C(P). This space satisfies C(P)U U. Also, it carries a tautological 9-stable projective bundle, and so we have a 9-map C(P) —* Map(E9, BPU(H)) into the space that repre 54  Chapter 4. Twisted equivariant K-theory sents the functor X  H (X, Z). This map induces an isomorphism  —*  9 [X,C(P)j  —*  H(X,Z)  U x Si, where U is a 9-cell. In fact, since finite G-CW-complexes are built out of spaces of the form G/H x S, where H is a closed subgroup of G, it is enough to check this for the cases X = U x S, where 9 U is weakly equiv alent to G xi G/H. But this reduces to proving the isomorphism 7r(P(U)) HS_i(B(9 i U),Z), which follows from the diagram: it is enough to check the cases X  ir(P(U))  (PH)  —  =  _z(B(9 3 H  U),Z)  (BH,Z) 3 H  where the map in the bottom row is an isomorphism by the results in [6].  E  55  Chapter 5  The completion theorem 5.1  The completion map  Lemma 5.1.1. Let B9  =  Th Any product of n elements in K* (B”9, G E 9/9. ) 0  is zero. Proof. Consider the subsets Uj = {Ajgj + + Th and let E I # O} in 9 = (1,79. These sets are open in B9. They are also homotopy equivalent : 0 to G ...  u  h  1 G  gJ 1 [EA  g  gj UjxI ([Ag], t)  gj  =  g  .Uj —  {tAg + (1  —  t)g]  These maps are 9-equivariant, so they define maps f : (I —* . Wehavefh=lG 3 h:Go—UandH:UxI-—-*U andHisa 0 homotopy between l. and hf. We can see G 0 inside all the Uj’s.  56  Chapter 5. The completion theorem If n  =  2, consider the following commutative diagram:  9, U 2 9, 1 2 K(B ) 2 U ® K(B )  9, U 2 K(B 1 U U)  ,U 9 2 K(B ) 1 9 ®K(B ,Go)  9,U 2 K(B ) 1  ,Go) 9,Go)®K(B 2 K(B 9  I  =  0  9,Go) 2 K(B  Similarly, consider the diagram: ® K(B’9, U)  9, (12) ® Th K(B9, U ) ® K(B 1  K(B9, U ) ® K(B9, Go) ® 1  K(B9, Go) ® K(B9, Go) ®  ...  ..  —  K(B”9, U 1 U  .  U (J)  =  K(B”9, U ) 1  ) 0 0 K(B9, G  ® K(B9, Go)  ..  .-  9, G 7 K(B ) 0 E  0 is a finite 9-OW-complex, then Lemma 5.1.2. If 9 acts freely on X and G (X) 9 K  K(X/9).  Proof. Given x e X, there exists a sufficiently small neighbourhood Uir(x) acts on U = of ir(x) such that 9() acts on U,r(). Then ir(x) is a compact Lie group and so there is a local slice for that action at 9 57  0  Chapter 5. The completion theorem x. We have an augmentation map K(G ) 0  K(Go) given by forgetting  the 9-action. Let Ig be the kernel of this map. The composite homomorphism ) 0 K(G  —*  K(E9)  K(B9)  K*(Go)  is the augmentation map, whose kernel is Ig. )/I. 0 K(Go) —* K(E’9) factors through K(G  Therefore, the map  For any 9-space X, K(X) is a module over K(Go) and by naturality the homomorphism K(X)  —  K(X  K(X)/IK(X) Conjecture 5.1.3. Let  XIT  E9) factorizes through  —*  K(X  X7  E9)  9 be a Lie groupoid and X a 9-space  .  Then we  have an isomorphism of pro-rings {K(X)/IK(X)} If a groupoid  {K*(X x E9/9)}  9 satisfies this conjecture for X  =  0 we will say that G  9  satisfies the completion theorem.  5.2  The completion theorem  Lemma 5.2.1. Let  9  =  G  i  X, where G is a compact Lie group and X is a  G-CW-complex such that K(X) is finite over R(G). Then 9 satisfies the completion theorem. Proof. Let 1 x be the kernel of K(X) —* K*(X). We would like to prove {K(X x EG)}. By the that there is an isomorphism {K(X)/Ifl completion we have an isomorphism Atiyah-Segal theorem, {K(X)/IK(X)} {K(X x G)}. Th So it suffices to prove that the E 58  Chapter 5. The completion theorem  IG-adic topology and the Ix-adic topology are the same in K(X). Since K,(X) is a module over R(G), we have IGK(X) C Let K be the kernel of o : K(X) —* K(X x ERG). As a corol lary to the Atiyah-Segal completion theorem, we know that the sequence of ideals {K} defines the IG-adic topology on K(X). In particular, there is m e N such that Km C IGK(X). Note that K 1 =1 x• Consider the com K*(X). Since position K,(X) —* K(X x G) tm —* K(X x E’G) E X x EG is the union of m open sets which are G-homotopy equivalent to G)m = 0. Thus the first 1 X x E’G = X x G, we have K(X x E G,X x E m map factors through I, thus I C Km. Hence I C IGK(X). LI Lemma 5.2.2. If 9 and 3-C are locally eqztivalent, then 9 satisfies the com pletion theorem if and only if H does.  Proof. If 9 and 3-C are locally equivalent by a local equivalence 3-C — 9, then ) --* K(Ho). The following diagram 0 we have an isomorphism f : K(G is commutative: K(Go)  K(Ho)  K*(Go)  K*(Ho)  and Therefore have f(Ig) C I. Let g be the inverse of f, x e But since Since ,6(x) = 0, we have c(y) = 0. y = g(x). = K*(Go) Zek*(Go) and so a(y) = (n,(a)). This c(y) = (n,a) ). 0 implies in particular n, = 0, that is, ag(I) C k*(G  )m = 0. Then, cg(I) c k*(Go)m = 0 and 0 Let m e N such that k*(G so g(I) C 19. Thus I = fg(I) C f(19) and the topologies induced by Ig and ‘3-C on K(Ho) are the same. Therefore we have an isomorphism of pro-rings {K*(Go)/Ifl {K*(Ho)/Ij.  59  Chapter 5. The completion theorem The local equivalence also induces a homotopy equivalence between B9 and B3-C. If we consider the associated filtrations to each of these spaces, 9} and {BFC}, the homotopy equivalence must take B9 to some Th {B Bi-C and BJ-C to some BTh+1c’9. Hence we have an isomorphism of prorings {K*(Bfl9)}  {K*(BJC)}. The lemma follows then by looking at the  diagram:  {K*(Go)/Ifl  {K*(Bfl9)}  {K*(Ho)/I}  {K*(BJ{)}  E  From the previous lemma, we obtain the following theorem: Theorem 5.2.3. If  9 and J-C are weakly equivalent, then 9 satisfies the  completion theorem if and only if J-f does. In particular, we have this corollary:  9 is Bredon-compatible and U is a 9-cell, K(U) is a finitely generated abelian group and the groupoid 9 x U satisfies the comple Corollary 5.2.4. If  tion theorem. This corollary tells us that the completion theorem is true for 9-cells. Now we move on to prove this for finite 9-CW-complexes. Let X be a finite 9-CW-complex and consider the spectral sequences for the maps  f:X  —*  X/9 and Ii : X x E9  =  —*  X/9 in 9-equivariant K-theory:  K(X)  fJ K(f’U)  iEI  =  U) 1 fi K(h  ==  ) 9 K(X  icIp  Now, since h (U) 1  =  f’(U) x,- E9 there is a map of spectral sequences 60  Chapter 5. The completion theorem E  —*  E induced by the projections f (U) x,,. E9 1  —*  f’(U).  The following lemma will be useful in what follows. Lemma 5.2.5. Fix any commutative Noetherian ring A, and any ideal I C —* M —* M” of finitely generated AA. Then for any exact sequence  modules, the sequence: {M’/PM’}  —>  {M/PM}  —>  M”} Th {M”/I  of pro-groups is exact. Proof. See [28], section 4.  U  The spectral sequence E is a spectral sequence of K(Go)-modu1es. From this point we assume that 9 is finite. Note that this implies that K(Go) is a Noetherian ring. All elements in these spectral sequences are finitely gen erated over K(Go). By the previous lemma, the functor taking a module M to the pro-group {M/IM} is exact and so we can form the following spectral sequence of pro-rings: =  {Jlj K(L )/IK(L’U)} U 1  =  jaip  Similarly consider the maps h : X x another spectral sequence of pro-rings: =  {fl K(lç’U)}  ,‘  E9  —*  X/9. They give us  {K(X x. E’9)}  jaIp  We have a map of spectral sequences çb : F  —*  F.  If 9 is Bredon-compatible, the groupoids 9 x f’(U) satisfy the comple tion theorem for all i. Since we are taking quotient by the ideal 19 and not by ‘9xf—’(u) , we need to check that both topologies coincide. We consider the long exact sequence in equivariant and non-equivariant K-theory for the 61  Chapter 5. The completion theorem pair (Cu, V) where V is any f’(U) and Cv is the mapping cylinder of the 0 and map 7t : V —> G . Note that Cv is 9-homotopy equivalent to C 0  VCC. G 0 ( 9 K )  (Cu,U) 9 K  I  K(Cu,U)  —‘  I  K(Go)  (U) 9 K  —  I  K(U)  K(Cu,U)  I  —p-  K’(Cu,U)  (U) C 1 K 9 N Let Iv I9)f-1(u.). It is clear that 1 v Now let m such that K(Cv,V) m = 0 and n e N such that Kg(Cv,V)” = 0. Then m c 1 i (V) and so the topologies coincide. K 9 is an isomorphism when restricted to any particular el F’ and therefore, it is an isomorphism of spectral sequences. In ement 3 particular, we have {K (X)/IK”(X)} {K7(X x E9)} This proves  Theorem 5.2.6. Let  9 be a Bredon-compatible Lie finite groupoid and X a  finite 9-CW-complex. Then we have an isomorphism of pro-rings {K(X)/IK(X)}  {K*(X  x,.  E9/9)}  5.2.7. Under the same circumstances, the homomorphism K*(X ) 9 -adic completion of 9 induces an isomorphism of the 1 K(X) K*(Xg). K(X) with Corollary —‘  5.3  The twisted completion theorem  For any 9-stable projective bundle P on a 9-space X, consider the 9-stable projective bundle P ,,E9 on X x , E9. The following diagram commutes:  62  Chapter 5.  The completion theorem  9 Th Px,E  Xx,,.E’9  Therefore we have a map: K(X)  PXrE’9K*(X  r 7 X  E’9)  ‘K(X) is a module over K(Go) and PxirE9K*(X x E9) is a mod  ule over K(E9). In fact we have a commutative diagram: K(Go)  I  K(X)  ‘  —.  K(E9)  I  PXE’9K*(X  x 9) Th E  From the setup for untwisted K-theory in section 5.1, we know that the last map factors through I and therefore, by naturality we have a map: PXrE’9*(  , E9) 7 x  We can also look at these maps as a map of pro-K(Go)-modules: {‘K(X)/I’K(X)}  J(*()( {PXtE’ 9 2  x E’9)}  Taking limits we obtain a map of K(Go)-modules: *(X PXE K 9  Conjecture 5.3.1. Let  x E9)  9 be a finite Lie groupoid, X a 9-space and P a 9-  stable projective bundle. Then we have an isomorphism of K(Go)-modules: PXTrE9K*(X  x E9)  63  Chapter 5. The completion theorem  0 and all 9-stable pro If a groupoid 9 satisfies this conjecture for X = G jective bundles on G , we will say 9 satisfies the twisted completion theorem. 0 In what follows, we will use the following result from [261, stated here in a simpler form which suffices for our purpose: Theorem 5.3.2. Let X be a finite G-CW-complex, where C is a compact  Lie group. Then K(X) is finitely generated over R(G) and the projection EG x X —* X induces an isomorphism: PxEG K*  (EG x X)  for any G-stable projective bundle P on X. Lemma 5.3.3. Let 9 = G x X, where G is a compact Lie group and X is a compact G-space such that K(X) is finite over R(G). Then 9 satisfies  the twisted completion theorem. Proof. It follows from lemma 5.2.1 and the completion theorem for twisted equivariant K-theory for actions of compact Lie groups, that is, theorem LI  5.3.2.  Lemma 5.3.4. If 9 and Jf are locally equivalent, then 9 satisfies the twisted completion theorem if and only if FC does. . Then, F*(P) is 90 Proof. Let P be a J-f-stable projective bundle on H stable. If 9 and 2f are locally equivalent by a local equivalence F : J-f —* 9,  then we have an isomorphism f : F*PK(GO) diagram is commutative:  I  ‘K(Ho). The following  ) 0 K(H  ) 0 K(G  F*PK*(G)  —-*  -  ) 0 K(H  64  Chapter 5.  The completion theorem  By lemma 5.2.2, the topologies induced by 19 and Ij- are the same and therefore we have an isomorphism of pro-rings: {F*PK* (Go)/IF*PK(Go)  }  } )/I7K(H 0 {K(H )  The local equivalence also induces a local equivalence between the groupoids Th to 9. E Th Therefore we have a E 9 x E9 and J-C x E3-f and it takes 3-C {F* (P xrE’K) {PxE K(E3-f) }, K (E9) } homomorphism of pro-rings It is also the case that which is an isomorphism in the limit. F*(P x EJ-C) = F*(P) x E9 and so we have a commutative diagram: {F  (Go)/Ir*PK(Go)}  -  } )/IK(H 0 {K(H )  CK*(Efl3{)} 3 {PXrE’  {F*(P)XE’9K*(Eflg)}  The lemma follows then by looking at the diagram: F* p z-*  (f-I  A  ) 9 F*(P)xE9K*(E  =  P  T/*  fizz-  I  A  -CK*(E) 3 PXIE  E  From the previous lemma, we obtain the following theorem: Theorem 5.3.5. If 9 and 3-C are weakly equivalent, then 9 satisfies the  twisted completion theorem if and only if 3-f does. Now from this theorem, lemma 5.3.3 and theorem 5.3.2, we obtain this corollary: Corollary 5.3.6. If 9 is a Bredon-compatible finite Lie groupoid, U is a  ’K(U) is a finitely gen 9-cell and P is a stable 9-projective bundle on U, 1 erated abelian group and the groupoid 9 i U satisfies the completion theorem. 65  Chapter 5. The completion theorem  This corollary tells us that the twisted completion theorem is true for 9-cells. Now we move on to prove this for finite 9-CW-complexes. Let X be a finite 9-CW-complex and P a 9-stable projective bundle on X. Consider the spectral sequence for the maps f : X —* X/9 in twisted 9-equivariant K-theory with twisting given by the restrictions of F:  =  fl  f*(P.)q(f_l  )  Pjçp+qy  iEI  The spectral sequence E is a spectral sequence of K (Go)-modules. As sume 9 is a finite groupoid so that K(Go) is a Noetherian ring. All elements in these spectral sequences are finitely generated over K (Go). The functor taking a K(Go)-module M to the K(Go)-module M is exact [28] and so we can form the following spectral sequence of K(Go)-modules.  =  u 1 fi Qiq(f-  P+x  iEIp  where Q = f*(P). Similarly consider the map h : X x E9 gives us another spectral sequence of K(Go)-modules:  =  jq(h_lu 9 QixE ) II 1  ==  —  X/9. It  7 E9) K(X x 9  iEIp  since h*(P) = Q x E9. We have h’(U) = f’(U) x E9 so there is a map of spectral sequences F —* F induced by the projections onto the first coordinate f’(U) x E9 —* f’(U). ) satisfy the twisted 1 If 9 is Bredon-compatible, the groupoids 9 f (U completion theorem for all i. From the previous section, we know that the ) are the 1 1 on K(fU topologies determined by the groupoid 9 and 9 x U same and therefore they are the same on  This proves q is an isomorphism when restricted to any particular el 66  Chapter 5. The completion theorem ement  and therefore, it is an isomorphism of spectral sequences. In  A 1 particular, we have PK-I-(X)  Theorem 5.3.7. Let  PxE9KP+(X x E9)  9 be a Bredon-compatible finite Lie groupoid, X a  finite 9-CW-complex and P a 9-stable projective bundle on X. have an isomorphism of K(Go)-modules: PXIE9n(  Then we  x E9)  67  Chapter 6  Proper actions of Lie groups Throughout this whole chapter S will be a Lie group, but not necessar ily compact. To study proper actions of S, we can consider the groupoid 9 = S x ES, where ES is the universal space for proper actions of S as defined in [27]. This space is a proper S-CW-complex such that ESG is contractible for all compact Lie subgroups G of S. The existence of ES is shown in [27]. It is also shown there that every proper S-CW-complex has an S-map to S and this map is unique up to 5-homotopy. Some immediate consequences follow: • Proper S-CW-complexes are 9-CW-complexes. • Extendable 9-bundles on a proper S-CW-complex X are extendable 5bundles for any S-map X —* S, since all of them are S-homotopic. ES x ES and this space is S-homotopy equivalent to ES, so B9 is homotopy equivalent to BS.  • E9  =  • 9 is finite if and only if ES is a finite proper S-CW-complex. • Extendable 9-sections on a proper S-CW-complex X are extendable 5sections for any S-map X —* ES, since all of them are 5-homotopic. • If H is a locally universal S-Hilbert representation, then ES x H is a locally universal 9-Hilbert bundle. • Stable 9-projective bundles on X are stable 5-projective bimdles on  x. By abuse of language, we say that proper actions of S are Bredon compatible if the corresponding groupoid 9 = S ES is Bredon-compatible. 68  Chapter 6. Proper actions of Lie groups For such actions, the following results follow from the corresponding results for groupoids. We will denote K(X)  =  K(X) and K(X)  =  Theorem 6.0.8. If S is a Lie group with Bredon-compatible proper actions,  the groups K(X, A) define a Z/2-graded multiplicative proper cohomology theory on the category of finite S-CW-pairs. Theorem 6.0.9. If S is a Lie group with Bredon-compatible proper ac  tions and a finite model for ES, the groups RK(X, A) define a 7Z/2-graded multiplicative proper cohomology theory on the category of finite proper 5CW-pairs. Corollary 6.0.10. Let S be a Lie group with Bredon-compatible proper  actions and a finite model for ES, X a finite S-CW-complex and H a stable representation of 5, then:  —  f1  [X,Fred’(H)] t [X, 2Fred’(H)1 t  if n is even if n is odd  Theorem 6.0.11. IfS is a Lie group with Bredon-compatible proper actions  and a finite model for ES, the groups ‘K(X) define a 7/72-graded proper cohomology theory on the category of finite proper S-CW-complexes with S-stable projective bundles, which is a module over untwisted 5-equivariant K-theory. Theorem 6.0.12. Let S be a Lie group with Bredon-compatible proper ac  tions and a finite model for ES and X a finite S-CW-complex. Then we have an isomorphism of pro-rings {K(X)/IK(X)}  {K*(X x, E”S/S)}  Theorem 6.0.13. Let S be a Lie group with Bredon-compatible proper ac  tions and a finite model for ES, X a finite S-CW-complex and P a S-stable projective bundle on X. Then we have an isomorphism of K(ES)-modules: PXrESKfl(X Xlr  ES) 69  Chapter 6. Proper actions of Lie groups All actions of finite groups and compact Lie groups are shown to be Bredon-compatible in [40]. Equivariant K-theory for these actions was in troduced in [40]. It is a well-known fact that for these actions are Bredon compatible [40]. The completion theorem in untwisted K-theory was proven in [5] and for twisted K-theory, it was recently proven in [26]. K-theory for proper actions of discrete groups was constructed in [28]. In this paper, actions of discrete groups are shown to be Bredon-compatible and a completion theorem is proven under some conditions. Twisted Ktheory for proper actions of discrete groups for some particular twistings was defined in [14], but no completion theorem existed up to date in the literature. In general, vector bundles may not be enough to construct an interest ing equivariant cohomology theory for proper actions of second countable locally compact groups [35], but they suffice for two important families, al most compact groups and matrix groups [34]. Almost compact groups, that is, second countable locally compact groups whose group of connected components are compact, always have a maximal compact subgroup. Any space with a proper action of one of these groups is the induction of a space with an action of that compact subgroup and so the study of proper actions of almost compact groups are reduced to studying compact Lie group actions. This is carried on in [34], and so these action groupoids are Bredon-compatible and we have a completion theorem. In fact, this is also proved in [34], by showing that the completion maps are compatible with the reduction map to the maximal compact subgroup. With different techniques it is proven that proper actions of matrix groups, that is, closed subgroups of GL(n, R), are Bredon-compatible and so a com pletion theorem follows. A particular instance of this case are proper actions of abelian Lie groups. Using the associated groupoids, we now can define twisted K-theory for actions of these groups and a completion theorem.  70  Chapter 6. Proper actions of Lie groups Proper actions of totally disconnected groups that are projective limits of discrete groups are shown to be Bredon-compatible in [38], where the cor responding K-groups are introduced. Since that theory coincides with the one constructed here, we now have a completion theorem for such actions. The previous constructions give a way of defining twisted K-theory and the corresponding results yield a completion theorem. One particular example is given by the group SL (). 2 In general 9 need not be a Bredon-compatible groupoid. When 9 = S x ES is a Bredon-compatible groupoid we must have Vectg(S/G) = Vect (pt) 0 for a compact subgroup G of S. Let S be a Kac-Moody group and T its maximal torus. Note that S is not a Lie group, but our constructions could be generalized to the context of topological groupoids. There is an S-map S/T —* S which is unique, up to homotopy. Given an S-vector bundle V on ES, the pullback to S/T is given by a finitedimensional representation of T invariant under the Weyl group. This rep resentation gives rise to a finite-dimensional representation of S. But this representation must be trivial. In particular, extendable S-vector bundles on S/T only come from trivial representations of T. In order to deal with these groups, it is more convenient to use dom inant K-theory, which was developed in [22]. Kac-Moody groups possess an important class of representations called dominant representations. A dominant representation of a Kac-Moody group in a Hilbert space is one that decomposes into a sum of highest weight representations. Equivariant K-theory for proper actions of Kac-Moody groups is defined as the repre sentable equivariant cohomology theory modeled on the space of Fredholm operators on a Hilbert space which is a maximal dominant representation of the group. It is expected that twisted dominant K-theory can be defined in the same way using a corresponding Fredholm bundle over a projective bundle which is stable with respect to a suitable Hilbert space of dominant representations. 71  Chapter 6. Proper actions of Lie groups  An example of a Kac- Moody group which is relevant to the work of Freed,  T > LG ([22], section 8) associated to the loop group LG of a simply connected simple compact Lie group G, where T is a circle acting by rotation of the loops. The clas sifying spaces for proper actions of LG and K(A) are the same. Let us call that space X(A), following the notation in [22]. The K(A)-equivariant dominant K-theory of X (A) can be identified with the Verlinde algebra of G, which is generated by projective representations of LG. The based loop group flG does not have any nontrivial compact subgroups and X(A)/2G is homeomorphic to G via the holonomy map [36]. This map carries the Hopkins and Teleman [17, 18, 19] is the group K(A)  =  action of LG on X(A) to the conjugation action of G on itself. Further calculations in section 5 of [22] imply that the K(A)-equivariant dominant K-theory of X(A) is isomorphic to the twisted G-equivariant K-theory of C with the conjugation action. The question arises whether it is possible to generalize this to general ized Kac-Moody algebras, introduced by R. Borcherds in [7]. These alge bras have the potential to define topological groups (Borcherds groups) by amalgamation using the same tools as in [25], and also have highest weight representations, which would give rise to the dominant representations of the group. We could use these representations to construct an equivariant cohomology theory for proper actions of these groups. A completion map is also possible for dominant K-theory [23], although in this case the topology could not induced by an ideal of the base ring in general. This would be something worth studying in the case of Borcherds groups, to obtain some knowledge about the homotopy type of their classifying spaces.  72  Bibliography [1] A. Adem, J. Leida, and Y. Ruan. Orbifolds and stringy topology, volume 171 of Cambridge Tracts in Mathematics. 2007. [2] M. Atiyah. Characters and cohomology of finite groups. Inst. Hautes  Etudes Sci. Publ. Math., 9:23—64, 1961. [3] M. Atiyah. K-theory. W.A. Benjamin. 1967.  [4] M. Atiyah and F. Hirzebruch. Vector bundles and homogeneous spaces. Proc. Sympos. Pure Math., 3:7—38, 1961. [5] M. Atiyah and G. Segal. Equivariant K-theory and completion. J. Duff. Geom., 3:1—18, 1969. [6] M. Atiyah and G. Segal. Twisted K-theory. Ukr. Mat. Visn., 1:287— 330, 2004. [7] RE. Borcherds. A characterization of generalized Kac-Moody algebras.  J. Algebra, 174:1073—1079, 1995. [8] G.E. Bredon. Equivariant cohomology theories, volume 34 of Lecture Notes in Mathematics. 1967. [9] E.H. Brown. Cohomology theories. Ann. of Math., 75:467—484, 1962. [10] J.L. Brylinski and B. Zhang. Equivariant K-theory of simply-connected Lie groups. K-Theory, 20:23—36, 2000. [11] J. Cantarero. Equivariant K-theory, groupoids and proper actions. arXiv:0803. 3244, 2008.  73  Bibliography [121 J. Cantarero. Twisted equivariant K-theory, groupoids and proper ac tions. arXiv:0902.0659, 2008. [131 P. Donovan and M. Karoubi. Graded Brauer groups and K-theory with local coefficients. Inst. Hautes Etudes Sci. Pubi. Math., pages 5—25, 1970. [14] C. Dwyer. Twisted equivariant K-theory for proper actions of discrete groups. 2005. [15] A. D. Elmendorf. Systems of fixed point sets. Trans. Amer. Math. Soc., 277:275—284, 1983. [16] H. Emerson and R. Meyer.  Equivariant representable K-theory.  arXiv:0710. 11Ov1, 2007. [17] D. Freed, M. Hopkins, and C. Teleman. Twisted K-theory and Loop Group Representations. arXiv:math/O31 2155, 2003. [18] D. Freed, M. Hopkins, and C. Teleman. Loop groups and twisted Ktheory II. arXiv:math/0511282, 2005. [19] D. Freed, M. Hopkins, and C. Teleman. Twisted equivariant K-theory with complex coefficients. J. Topol., 1:16—44, 2008. [20] D. Gepner and A. Henriques.  Homotopy theory of orbispaces.  arXiv:math.AT/0701 916, 2007. [21] S. Jackowski. Families of subgroups and completions. J. Pure Appl. Algebra, 37:167—179, 1985. [22] N. Kitchloo. Dominant K-theory and integrable highest weight repre sentations of K&-Moody groups. arXiv:math/0710.0167v1, 2007. [23] N. Kitchloo. Private communication. 2008. [24] N. Kitchloo and J. Morava. Thom Prospectra for Loopgroup represen tations. arXiv:math/O4O.4 5 1, 2004.  74  Bibliography [25] S. Kumar. Kac-Moody groups, their flag varieties and representation  theory, volume 204 of Progress in Mathematics. 2002. [26] A. Lahtinen. The Atiyah-Segal completion theorem in twisted K-theory.  arXiv:0809. lPf/3, 2008. [27] W. Luck. Survey on classifying spaces for families of subgroups. Infinite  groups: geometric, combinatorial and dynamical aspects, Progr. Math., 248:269—322, 2005. [28] W. Luck and B. Oliver. The completion theorem in K-theory for proper actions of a discrete group. Topology, 40:585—616, 2001.  [29] J.P. May. Classifying spaces and fibrations, volume 155 of Mem. Amer. Math. Soc. 1975. [30] R.J. Milgram. The bar construction and abelian H-spaces. Illinois J. Math. 11, 11:242—250, 1967.  [311 J. Milnor. Construction of universal bundles, I. Ann. of Math., 63:272— 284, 1956.  [32] I. Moerdijk and J. Mrcun. Introduction to foliations and Lie groupoids. Cambridge University Press. 2003. [33] Alan L.T. Paterson. Groupoids, inverse semigroups, and their operator algebras, volume 170 of Progress in mathematics. 1998. [34] NC. Phillips. Equivariant K-theory for proper actions II: Some cases in which finite dimensional bundles suffice. Index theory of elliptic  operators, foliations and operator algebras, Contem. Math., 70:205—227, 1988. [35] N.C. Phillips. Equivariant K-theory for proper actions, volume 178 of Pitman research notes in mathematics. 1989. [36] A. Pressley and G. Segal. Loop groups. Oxford Science Publications. 1988. 75  Bibliography [37] J. Rosenberg. Continuous-trace algebras from the bundle theoretic point of view. J. Austral. Math. Soc. Ser. A, 47:368—381, 1989. [38] J. Sauer. K-theory for proper smooth actions of totally disconnected groups. High-dimensional manifold topology, World Sci. Pubi., pages 427—448, 2003. [39] G. Segal. Classifying spaces and spectral sequences. Inst. Hautes Etudes Sci. Publ. Math., 34:105—112, 1968. [40] G. Segal. Equivariant K-theory. Inst. Hautes Etudes Sci. Pubi. Math., 34:129—151, 1968. [41] G. Segal. The representation ring of a compact Lie group. Inst. Hautes Etudes Sci. Publ. Math., 34:113—128, 1968.  76  

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