AN EQUIVARIANT BIVARIANT CHERN CHARACTER

Transcription

1 The Pennsylvania State University The Graduate School Department of Mathematics AN EQUIVARIANT BIVARIANT CHERN CHARACTER A Thesis in Mathematics by Jeff Raven c 2004 Jeff Raven Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2004

2 We approve the thesis of Jeff Raven. Date of Signature Paul Baum Thesis Adviser Evan Pugh Professor of Mathematics Chair of Committee Nigel Higson Distinguished Professor of Mathematics Head, Department of Mathematics Victor Nistor Professor of Mathematics Pablo Laguna Professor of Physics, Astronomy & Astrophysics

3 Abstract Using notions from homological algebra and sheaf theory Baum and Schneider defined a bivariant equivariant cohomology theory which shares many of the properties of equivariant KK-theory; indeed, these two theories have so much in common that when the group under consideration is profinite they are rationally isomorphic. This, combined with other similar results, led Baum and Schneider to conjecture that the same should be true for any totally disconnected group. We verify the conjecture for a large class of such groups, namely the countable discrete groups. iii

4 Contents Acknowledgments vi Chapter 1. Introduction I Equivariant KK-Theory 5 Chapter 2. Proper Actions and Equivariant K-Theory Proper Actions Equivariant Vector Bundles Equivariant K-Theory Equivariant Spin c Structures Equivariant Bott Elements Chapter 3. Kasparov s Equivariant KK-Theory C -Algebras Hilbert modules Kasparov s KK-Theory Vector Bundles and KK G (X, X) Dirac Operators and KK G (M, pt) Principal Induction and the Thom Isomorphism Chapter 4. Topological Equivariant KK-Theory Topological Cycles and Bordism Vector Bundle Modification Topological KK-Theory A Technical Lemma Normal Bordism The Eilenberg-Steenrod Axioms Induction and the Dimension Axiom II Equivariant Bivariant Homology 60 Chapter 5. Homological Algebra Abelian Categories Complexes and Homotopy Triangulated Categories Localization of Categories iv

5 5.5 The Derived Category Derived Functors Example : Modules Chapter 6. Equivariant Sheaf Theory Sheaves Direct and Inverse Images Borel Cohomology Proper Supports Equivariant Bivariant Cohomology III An Equivariant Bivariant Chern Character 99 Chapter 7. An Equivariant Bivariant Chern Character The K-theory Chern Character The Chern Character of a G-Spin c Manifold The Bivariant Chern Character Concluding Remarks Appendices 114 Appendix A.Straightening the Angle Appendix B. Smoothing Spin c Structures Bibliography v

6 Acknowledgments Many thanks to my adviser Paul Baum for our numerous discussions and his enthusiastic support. My thanks also go out to the other members of my committee for their feedback on my thesis. Finally I would also like to thank my wife, Marjorie Raven, for her assistance in proofreading the final draft. vi

7 Chapter 1 Introduction K-theory first appeared on the mathematical landscape in the early sixties, when it played a central role in the proofs of Riemann-Roch and the Atiyah-Singer index theorem. As the natural home for the symbols of elliptic operators on a manifold M, the use of the K-theory group K 0 (T M) allowed for a much more elegant proof of the index theorem than earlier bordism-based arguments. At the time, it was realized that this process of associating classes in K 0 (T M) to elliptic operators on M should be a manifestation of Poincaré duality, so that elliptic operators on M should themselves provide elements in the K-homology of M; unfortunately at the time the only means available for defining K-homology was via the Bott spectrum, a viewpoint unsuited to analytic objects. By generalizing the notion of elliptic operator, Atiyah [Ati70] was ultimately able to provide a notion of analytic K-cycles, but he was unfortunately unable to find a suitable equivalence relation which would recover K-homology. It wasn t until a decade later, with the work of Brown, Douglas, and Fillmore [BDF73] on extensions of C -algebras, that an analytic picture of K-homology emerged. This was quickly followed by a more general theory of extensions due to Kasparov [Kas80b], which provided a natural framework for working with not only K-theory and K-homology, but maps between them as well. The Kasparov groups KK (A, B) form a bivariant theory on C -algebras, contravariant in the first variable and covariant in the second. Of particular importance is the existence of a composition product KK i (A, B) KK j (B, C) KK i+j (A, C) which serves to generalize the cup and cap products. One can obtain a theory for spaces simply by letting KK (X, Y ) = KK (C 0 (X), C 0 (Y )), where C 0 (X) denotes the continuous functions on X vanishing at infinity. The fact that C 0 is contravariant means that KK (X, Y ) is a homology theory in the first variable and a cohomology in the second; a bit of work shows that setting one of the spaces to be a point allows one to recover K-theory and K-homology. The corresponding equivariant groups have played an important part throughout the development of these theories. The original K-theoretic proof of the index theorem relied in part on equivariant K-theory, while equivariant K-homology lies at the heart of the Baum-Connes and Novikov conjectures. As in the non-equivariant case, the equivariant Kasparov groups KK G (A, B) serve to unify these many constructions. Baum and Connes [BC98] were among the first to consider the problem of defining a Chern 1

8 character in equivariant K-theory, focusing on the case when G is discrete; using a markedly different approach, Lück and Oliver [LO01] later extended these results to a much larger class of spaces. Smooth actions of Lie groups were the next to be analyzed, first by Baum, Brylinksi and MacPherson [BBM85], who looked at smooth S 1 -actions, but later by many others including work by Block and Getzler [BG94] which approached the problem using equivariant cyclic homology. Meanwhile on the homological side, for discrete groups Lück [Lüc02] produced a Chern character isomorphism from an equivariant homology group to rational equivariant K-homology. However, until recently very little consideration had been given to the problem of defining a Chern character isomorphism ch G : KK G (X, Y ) C ĤH G (X, Y ; C), where ĤH G denotes an appropriate bivariant equivariant theory. Combining earlier work of Nistor [Nis91] on bivariant Chern characters with more recent work by Voigt [Voi03] on bivariant equivariant periodic cyclic homology provides one means of approaching the problem (at least for certain spaces), but one might hope that a more general Chern character could be obtained using topological techniques. Matters are substantially simplified by working with a finite group over the complex numbers. Classical representation theory then tells us that R(G) C = [γ] G//G so that every R(G) C-module decomposes as a sum of local terms. Applying this to the case of KK G (X, Y ) C, Baum and Schneider [BS02] have shown that the summand corresponding to the conjugacy class [g] can be identified with the bivariant equivariant cohomology HH Z(γ)(X γ, Y γ ), and thus KK G (X, Y ) C = HH Z(γ)(X γ, Y γ ). [γ] G//G A limiting argument then allowed them to extend this result to profinite groups, though in this case the target is more complicated. This, combined with some of the other results mentioned above, led them to conjecture that a similar result should hold for all totally disconnected groups. Theorem. For any countable discrete group G, finite proper G-CW complex X and G-space Y there is a natural Chern character isomorphism KK G C, (X, Y ) C ĤH G (X, Y ). In the end, the only real obstacle to proving such a result lies in defining a sufficiently natural Chern character; once this is done, the fact that it is an isomorphism follows more or less by abstract nonsense. However, to do this one needs a more concrete model for KK G (X, Y ) than is provided by Kasparov s definition; it is at this point that an approach first used by Baum and Douglas [BD82] to describe K (X) = KK (X, pt) becomes quite useful. In this framework, one considers triples (M, ξ, f) consisting of a Spin c manifold M, a class ξ K(M), and a continuous map f : M X. There is a natural notion of bordism on such 2

9 cycles, but it alone is not sufficient to produce K (X). One must also make use of vector bundle modification given a smooth Spin c bundle E M one can construct a new cycle (M, ξ, f) E := (S(E 1), β E π ξ, f π). Two triples are then equivalent if they can be connected by a series of bordisms and vector bundle modifications. That the resulting group is isomorphic to K (X) would quickly follow from the fact that it forms a homology theory, but it is not entirely obvious that this is the case. In particular, it is unclear whether one has a six-term exact sequence associated to a CW-pair (X, A) the essential difficulty being that the equivalence relation does not provide a concise condition for when a triple is equivalent to zero. There are a variety of arguments that establish the existence of the six-term exact sequence, yet they inevitably involve the introduction of an auxiliary group involving framed bordism. However, a more direct (though similar) approach is available. One can define a relation of normal bordism on triples, in which two triples (M i, ξ i, f i ) i=0,1 are equivalent if and only if there exist Spin c normal bundles ν i for each M i such that the triples (M i, ξ i, f i ) νi are bordant. It turns out that this produces the same equivalence relation as that of Baum and Douglas, but is explicit enough that the usual arguments for the long exact sequence in bordism can be carried out with only slight modification. These ideas extend naturally to the study of KK (X, Y ); in fact, so long as G is compact they even apply to KK G (X, Y ). However, while the same result is ultimately true when G is a countable discrete group and X is a finite proper G-CW complex, the argument in this case is more subtle. The problem is that normal bordism relies heavily on the fact that every Spin c manifold has a Spin c normal bundle. In the equivariant case one must instead work with G-Spin c normal bundles, and unfortunately these need not exist when G is non-compact. To deal with this, one must shift to the category G-Top Z of G-spaces over a proper G-space Z. In this context one defines a normal bundle for a G-Spin c manifold M Z to be a G-Spin c bundle ν such that TM ν is isomorphic to the pullback of a G-Spin c bundle on Z. When G is compact and Z is a point this reduces to the usual notion of normal bundle, but when G is non-compact the extra G-Spin c bundles provided by Z can help one sidestep the difficulties mentioned above. In particular, a theorem of Lück and Oliver allows one to conclude that when G is discrete and Z is a finite proper G-CW complex every G-Spin c manifold M Z will have a normal bundle. This in turn allows normal bordism to be used to define a homology theory tkk G Z (X, Y ) on G-Top Z. A little more work shows that this group is actually independent of the choice of Z and, in fact, is isomorphic to KK G (X, Y ) whenever X is a finite proper G-CW complex. Before finally defining the Chern character there is still one last matter to attend to the target of the map, the group denoted ĤH G (X, Y ). As when G was finite, the target arises as a sum of local contributions, one for each finite order conjugacy class of G. Specifically, ĤH G(X, Y ) := [γ] G tor//g 3 HH Z(γ) (Xγ, Y γ ),

10 where G tor G denotes the elements of finite order in G, G tor //G its conjugacy classes and HH Z(γ) (Xγ, Y γ ) is the bivariant Z(γ)-equivariant cohomology of the γ-fixed sets. Thus defined, ĤH G (X, Y ; C) naturally enjoys many of the same properties as KK G (X, Y ); in particular, it has long exact sequences in each variable, and there is an associative product ĤH G (X, Y ) ĤH G (Y, Z) ĤH G (X, Z). Moreover any proper G-equivariant map f : Y X naturally yields a corresponding class [f] ĤH 0 G (X, Y ). With all of that said, defining the Chern character ĉh G : KK G (X, Y ) C ĤH G (X, Y ) for a discrete group G now simply comes down to describing for each triple (M, ξ, f) corresponding classes ĉh G (M) ĉh G (ξ) and using the composition product to construct ĤH G (M, pt) ĤH G (M Y, M Y ) [f] [π M ] ĉh G(ξ) (ĉh G(M) 1 Y ) ĤH G (X, Y ). The content of this thesis falls quite naturally into three parts. In the first part we begin by reviewing the necessary material regarding proper actions of discrete groups and their equivariant K-theory; we then move on to review the properties of Kasparov s equivariant bivariant theory. We conclude by using the notion of topological KK G -cycles described above to define the tkk G - groups, and prove that they are isomorphic to Kasparov s analytic groups in those cases of interest. The next part of the thesis sees the development of the homological algebra and equivariant sheaf theory needed to define the bivariant equivariant cohomology groups and establish its properties. Finally, in the third part we define the bivariant equivariant cohomology classes needed to construct the equivariant bivariant Chern character and show that it gives a well-defined isomorphism. We conclude by discussing some extensions of this result, as well as describing an alternate approach to the problem which might hold promise for (non-discrete) totally disconnected groups. Conventions Unless specifically noted, all spaces are assumed to be compactly generated, paracompact and Hausdorff. Note that in particular this implies that all spaces are assumed to be normal. 4

11 Part I Equivariant KK-Theory 5

12 Chapter 2 Proper Actions and Equivariant K-Theory 2.1 Proper Actions Let G be a countable discrete group. Definition A G-space consists of a topological space X along with a continuous action of G on X. Given a closed G-invariant subspace A X we will refer to (X, A) as a G-pair. If X and Y are two G-spaces, a G-map from X to Y is a continuous G-equivariant map X Y ; a G-map between G-pairs (X, A) and (Y, B) is a G-map X Y that carries A into B. In general the action on a G-space X can be rather pathological, in that there is nothing that ensures a reasonable quotient topology on X/G. The prototypical example of this is the action of Z on S 1 = [0, 1]/{0, 1} generated by an irrational translation T(x) = x + α. Indeed, while the quotient S 1 /Z is quite large, consisting of uncountably many orbits, the topology induced by the quotient map contains only the trivial open sets. Ultimately the problem with this example is that although the action is free, it is not proper there are no Z-invariant neighborhoods of the form U = Z V. Definition ([BCH94]). A G-space X is proper if for every point p X there exists a G-invariant open neighborhood U, a finite subgroup H G, and a G-map φ : U G/H. There are various other notions of proper action in the literature, but most are equivalent given some minor assumptions about the spaces involved. The advantage of the definition given here is two-fold. First it allows for very concise proofs of the inheritance properties for proper actions, and second it emphasizes a view of proper spaces which is central to many arguments namely, that every proper G-space is locally obtained from a finite group action. Indeed, given (U, H, φ) as in the definition above and letting V = φ 1 ([H]), one finds that U is G- homeomorphic to the induced space G H V. This phenomenon will often allow us to reduce global questions about proper G-actions to local ones regarding finite group actions, for which many more techniques are available. Lemma Let X and Y be G-spaces, f : X Y a G-map. Then if Y is a proper G-space so is X. In particular, any subspace of a proper G-space is a proper G-space. Proof. Choose a point x X and consider the image f(x) Y. Since Y is a proper G-space, we can find a triple (U, H, φ) for f(x); one can then easily check that (f 1 (U), H, φ f) gives a triple for x X, and it follows that X is a proper G-space. 6

13 Lemma Let X be a proper G-space and G G a subgroup. Then X is a proper G -space under the restricted action. Proof. Consider any point x X and let (U, H, φ) be the triple guaranteed by the properness of the G-action. Let V denote the component of U containing x; without loss of generality we may assume that H fixes V. Since (G H)/H = G /(G H) the restriction of φ to G V yields a G -map φ : G V G /(G H). Together this gives a triple (G V, G H, φ ) for the G -action around x, and thus X is a proper G -space. One of the more significant consequences of properness is that the action eventually separates various sets. To allow for a more concise discussion of this phenomena we will need the following extra bit of notation. Definition Let X be a G-space with subsets A and B. Define G(A, B) to be the subset of G given by {g G : A (g B) }. Proposition A G-space X is proper if and only if for every pair of points x, y X there exist neighborhoods W x and W y of x and y respectively such that G(W x, W y ) is finite. Proof. First suppose that X is proper and we wish to prove the finiteness claim. Choose two points x, y X and let (U, H, φ) be the triple for x guaranteed by the properness of the G-action. Let V denote the component of U containing x; without loss of generality we may assume that H fixes V and U = G H V. There are essentially two cases to consider. The simplest is when y = γ x. In this case we simply let W x = V and W y = γ V it is then easy to check that #G(W x, W y ) = #H. So let us instead assume that y / G x. Note that the properness of the action implies that orbits are closed in X, and thus U\G y is open. Furthermore since X is normal we can find an open G-invariant neighborhood W x G x with W x U\G y. Let W y = X\W x ; by construction W y is a G-invariant open set containing y such that G(W x, W y ) =. Now for the converse. For this it suffices to consider the case of x = y; the finiteness condition then guarantees an open neighborhood W x with G(W x, W x ) finite. In particular this implies that the stabilizer G x G(W x, W x ) is finite, and thus to prove properness we need only find a G x - invariant neighborhood x V W x such that U = G V = G Gx V. Since X is normal we can assume that W x G x = {x}. Let Z be the union of the translates γ W x for γ G(W x, W x )\G x. This is a closed set which doesn t contain the point x, and hence V = W x \Z is an open neighborhood of x. Note that since G x is finite we can also choose V to be a G x -invariant open neighborhood. It is now straightforward to verify that G V = G Gx V, and thus the action is proper. Note that the proof of the proposition has the following corollary. Corollary Every proper G-space has Hausdorff quotient. Proposition Let X be a G-space. If X is proper, then for every pair of compact subsets K, L X the set G(K, L) is finite. The converse holds whenever X is locally compact. 7

14 Proof. Suppose that X is proper and we are given compact sets K and L. By the previous proposition for each (x, y) K L we can find open neighborhoods W x and W y such that G(W x, W y ) is finite. Together the open sets W x W y form a cover of K L, and hence by compactness there is a finite subcover {W x(i) W y(i) }. But then G(K, L) i G(W x(i), W y(i) ), and this last set is a finite union of finite sets, hence finite. Now suppose the finiteness condition is satisfied and we wish to prove that the action is proper. Consider two points x, y X. When X is locally compact there exist precompact neighborhoods W x and W y for x and y respectively. The finiteness condition then implies that the set G(W x, W y ) is finite, and thus G(W x, W y ) is finite. The previous proposition then shows that X is proper. Corollary Let X be a locally compact and proper G-space. Then averaging over G gives a well-defined -linear map from C c (X) to C b (X) G. A G-space X is cocompact if the quotient space X/G is compact. Note that when the G-action is proper this implies that X is both locally compact and σ-compact. Lemma Let X be a proper cocompact G-space. Then there exists f C c (X), f 0, such that γ G γ f = 1. Proof. Let π : X X/G denote the quotient map. Through the combination of compactness and properness we can find a finite open cover U = {U i } of X/G such that π 1 (U i ) = G Hi V i and each V i is precompact. Now let {φ i } be a partition of unity for U; since π(v i ) = U i we can lift each φ i to obtain a function φ i supported only on V i X. It follows that the function g = φ i i is compactly supported and g = γ G γ g is everywhere non-zero. Now set f = g/g. Given a G-space X, we will spend a good deal of our time working with various fixed-point subsets. For any subgroup S G let N(S) denote the normalizer of S in G; note that the centralizer Z(S) sits within N(S) as a normal subgroup. Then the action of G on X restricts to give actions of N(S) and Z(S) on the space X S of S-fixed points. Proposition Let X be a proper G-space. Then for any subgroup S G the fixed-point subspace X S is a proper N(S)-space. Moreover, if X is a cocompact G-space then X S is a cocompact N(S)-space. Similar statements hold with N(S) replaced by Z(S). Proof. It follows immediately from Lemmas and that X S is both a proper N(S)-space and a proper Z(S)-space, so it only remains to prove cocompactness. Note that since every proper cocompact G-space is a finite union of closed induced subspaces it suffices to prove the claim for spaces of the form X = G H K, where H G is a finite subgroup and K is a compact H-space. Moreover, since X S /N(S) is a quotient of X S /Z(S) we need only consider the Z(S)-action. Consider the projection π : X = G H K G/H. By assumption the preimage of each point is compact; thus the same is true for π : X S /Z(S) (G/H) S /Z(S). The desired result would 8

15 then follow so long as the set (G/H) S /Z(S) were finite, and this is the content of the following lemma. Lemma Let H, S G be two finite subgroups of G. Then the set (G/H) S /Z(S) is finite. Proof. A simple calculation shows that (G/H) S /Z(S) = Z(S)\{γ G : γ 1 Sγ H}/H. But this set maps injectively into Hom(S, H)/H, where Hom(S, H) denotes the set of group homomorphisms from S to H. Since S and H are finite groups this last set is finite. The concept of a CW-complex plays a vital rôle throughout much of algebraic topology, allowing one to prove results by inductively climbing up the skeleta. For the same reason it will be useful to have an equivariant notion of CW-complex. Definition A G-space X is a G-CW complex if it can be constructed as the limit of an ascending chain of closed G-invariant subsets = X 1 X 0 X 1 X n X where X n is constructed from X n 1 as a pushout of the form n S n 1 X n 1 n D n X n for some discrete G-space n. Remark A G-CW complex is proper if and only if each of the G-spaces n are proper. The quotient of a G-CW complex is in a natural way a CW-complex. Our primary interest will be with finite G-CW complexes those G-CW complexes whose quotients are finite CWcomplexes. Note that in the world of CW-complexes being finite is equivalent to being compact, and hence a G-CW complex is finite if and only if it is cocompact. Unfortunately there will be occasional moments when we will need to work with spaces with slightly more structure than a G-CW-complex. Definition Let V be a G-set. A G-simplicial complex consists of a family K of subsets of V, called simplices, such that {v} K for each v V, if s K and s s, then s K, and if s K and γ G, then γ s K. As a extension of simplicial complexes every G-simplicial complex K possesses a geometric realization K ; thanks to the conditions placed on the simplices of K this realization comes with a natural G-action. A G-simplicial complex is proper (i.e. has a proper geometric realization) if and only if the G-set V is proper, and is cocompact if and only if the G-set V is cofinite. 9

16 Note that a G-simplicial complex is generally not a G-CW complex the problem is that an element which preserves a simplex may nonetheless act non-trivially on its vertices. However this is easily remedied by replacing the G-simplicial complex with its barycentric subdivision, which does carry a natural G-CW structure. Though we shall have no use for it, the following proposition shows that the reverse is also true, at least up to G-homotopy; the proof mirrors that of the non-equivariant case (see Theorem 2C.5 of [Hat02]). Proposition Every G-CW-complex X is G-homotopy equivalent to a G-simplicial complex, which can be chosen to be of the same dimension, proper if X is proper, and cocompact if X is cocompact. As noted earlier we will often be working with the fixed-point subspaces of proper G-spaces; when those spaces are G-CW or G-simplicial complexes we have the following elaboration on Proposition Proposition Let X be a proper G-CW complex. Then for any subgroup H G the fixed-point subspace X H is a proper N(H)-CW complex. Moreover, if X is a cocompact G- CW complex then X H is a cocompact N(H)-CW complex. Similar statements hold with N(H) replaced by Z(H), and for G-simplicial complexes in place of G-CW complexes. Finally, despite our fondest wishes we will often be forced to work with arbitrary proper cocompact G-spaces. In these cases the following result will often prove to be quite useful. Lemma Every proper cocompact G-space has a G-map to a proper cocompact G-simplicial complex, and hence to a finite proper G-CW complex. Remark In fact, analogous to the non-equivariant case every proper cocompact G-space can be realized as the inverse limit of a system of finite proper G-CW complexes. However, for our purposes a single map to a G-CW complex will suffice. Proof. Let f C c (X) be the function guaranteed by Lemma , and let U = f 1 (t > 0). The properties of f guarantee that the collection U = {γ U} γ G is a locally finite cover of X. Let N(U) denote the nerve of U; it is easily verified that N(U) is a proper cocompact G-simplicial complex. Finally, the assignment x (f(γ 1 x)) γ G defines a G-map from X to N(U). 2.2 Equivariant Vector Bundles Definition A complex (resp. real) G-vector bundle on a G-space X consists of a complex (resp. real) vector bundle E X along with a continuous action of G on E given by bundle maps which cover the action on X. The set of all isomorphism classes of complex (resp. real) G-vector bundles over X will be denoted by Vect C G (X) (resp. VectR G (X)). 10

17 Lemma Let Z be an H-space for some subgroup H G. Then there are natural isomorphisms ind G H : VectC H (Z) = Vect C G (G H Z) ind G H : Vect R H(Z) = Vect R G(G H Z) given by sending an H-vector bundle E Z to the induced vector bundle ind G H (E) = G H E. Unfortunately, without further assumptions this is more or less the limit of what one can say about the G-vector bundles on a G-space X; the same pathologies which can conspire to make X/G so poorly behaved also limit our ability to use local arguments to understand G-vector bundles. However, when the space X is proper and cocompact most non-equivariant results can be made to carry over in one form or another. Lemma Let (X, A) be a proper cocompact G-pair, and let E be a G-vector bundle on X. Then any G-invariant section s : A E A can be extended to a G-invariant section on X. Proof. Since X is proper and cocompact, by Lemma there exists a positive f C c (X) such that γ G γ f = 1. Then s 0 = fs is a compactly supported section of E A, and we can apply the corresponding non-equivariant extension theorem to obtain a compactly supported section ŝ 0 on X. Finally, setting ŝ = γ G γ ŝ 0 yields the desired invariant extension, since ŝ A = γ ŝ 0 A = fs = s. γ G γ Gγ Given the previous lemma, arguments identical to the non-equivariant case lead to the following results (see [Ati67]). Lemma Let (X, A) be a proper cocompact G-pair, and let E and F be G-vector bundles over X. Then any G-equivariant bundle map φ : E A F A extends to a G-equivariant bundle map φ : E F. Moreover, if φ is an isomorphism (resp. monomorphism) then there exists a G- invariant open set U containing A such that φ U is also an isomorphism (resp. monomorphism). Lemma Let X and Y be proper cocompact G-spaces, f t : X [0, 1] Y a G-homotopy and E a G-vector bundle on Y. Then f0 E = f1 E. Lemma If f : X Y is a G-homotopy equivalence between proper cocompact G-spaces then the induced maps f : Vect C G (Y ) VectC G (X) f : Vect R G (Y ) VectR G (X) are bijective. It will often be essential to have some form of inner product on our G-vector bundles. To this end we define a Hermitian (resp. Euclidean) G-vector bundle to be a G-vector bundle with a G-invariant Hermitian (resp. Euclidean) structure. The following lemma guarantees that such beasts do in fact exist. 11

18 Lemma Let X be a proper G-space, and E be a complex (resp. real) G-vector bundle on X. Then E admits a G-invariant Hermitian (resp. Euclidean) structure. Proof. We shall only worry about the complex case; the argument in the real case is identical. Through the combination of paracompactness and properness we can find a partition of unity {φ i } for X consisting of G-invariant functions with cocompact support; it therefore suffices to consider only the case when X is cocompact. Since X is paracompact there exist Hermitian inner products on E; let h : E E C be one of them. As X is proper and cocompact, Lemma tells us that there is a positive f C c (X) such that γ G γ f = 1 > 0. Setting ĥ(u, v) = γ Gf(γ x) h(γ u, γ v) for u, v E x then yields a G-invariant Hermitian inner product for E. Unfortunately there are some results which have no direct equivariant generalization, the most notable being the properties of trivial bundles. Recall that a vector bundle on X is trivial if and only if it is isomorphic to a pullback over the map X pt; unfortunately, while this definition has an obvious generalization to G-vector bundles, it is largely useless when G is not compact. For one thing, there need not actually be any interesting G-vector bundles over a point. Indeed, a G-bundle over a point is simply a finite-dimensional representations of G, and there are several well-known examples of discrete groups which have no non-trivial finite-dimensional representations 1. Ultimately this is merely a symptom of a more severe problem for countable groups there is no finite-dimensional analogue of the regular representation. After all, the value of trivial bundles lies not in their mere existence, but in their containment properties in the fact that every vector bundle can be realized as a summand of a trivial bundle. Without some form of finite-dimensional regular representation there is no hope of extending this result to the equivariant case. Thankfully there is a way around these difficulties. Morally speaking, when G is infinite working with pullbacks from a point is a rather questionable affair after all, the action of G on a point is far from being proper. Thus a different notion of trivial bundle is required when dealing with infinite discrete groups. Definition Let X and Z be proper cocompact G-spaces, and a X : X Z a G-map. A complex (resp. real, resp. G-Spin c ) G-vector bundle on X is Z-trivial if it is isomorphic to the pullback over a X of a complex (resp. real, resp. G-Spin c ) G-vector bundle on Z. When G is finite, we have a universal choice for Z in the form of a single point, but when G is infinite there is usually no such option. One might hope to use the universal proper G-space EG, but in general it need not admit a cocompact model. Nonetheless, the following results of Lück and Oliver show that this notion of triviality has the necessary containment property. 1 If a finitely-generated group has a non-trivial finite-dimensional representation then a result of Mal cev [Mal65] implies that it contains a finite index normal subgroup. But there are numerous examples in the literature of infinite finitely-generated simple groups. 12

19 Proposition ([LO01]). Let Z be a proper cocompact G-space. Then there exists a complex G-vector bundle E Z such that for each z Z the fiber E z is a multiple of the regular representation of G z. Corollary ([LO01]). Let a : X Z be a G-map between proper cocompact G-spaces X and Z. Then for any complex G-vector bundle E over X there exists a complex G-vector bundle F such that E F is Z-trivial. Remark In fact Lück and Oliver only consider the case when X and Z are finite G-CW complexes; however it follows immediately from Lemma that the proposition also holds whenever Z is proper and cocompact. The results from earlier in this section then show that their proof of the corollary carries over without change. 2.3 Equivariant K-Theory Let X be a proper cocompact G-space. The set Vect C G (X) possesses a natural addition operation in the form of the Whitney sum, and we shall let K G (X) denote the corresponding Grothendieck group. The results of the previous section show that K G (X) is a G-homotopy invariant functor; moreover for any subgroup H G and H-space Y there is a natural isomorphism ind G H : K H(Y ) = K G (G H Y ). However, far more is true Lück and Oliver have shown that K G (X) extends to a Z 2 -graded cohomology theory on finite proper G-CW pairs. In fact, the theory even extends to proper cocompact G-pairs; unfortunately the arguments in [LO01] rely heavily on CW machinery, and thus must be redone for this more general case. On a positive note, given the results from the previous sections their argument for the following lemma carries over unchanged, and ultimately this is all we shall require. Lemma ([LO01]). Let X be a proper cocompact G-space with closed G-invariant subspaces A and B such that X = A B. Then the sequence is exact in the middle. K G (X) K G (A) K G (B) K G (A B) Remark In other words, given elements of K G (A) and K G (B) which agree on the intersection A B we can glue them to obtain a (not necessarily unique!) element of K G (X). In Chapter 4 we shall often be working with spaces of the form M Y, where M is a proper cocompact G-space and Y is a compact but generally not proper G-space. Note that the fact that M is proper implies the same of the product M Y, while the fact that Y is compact means that M Y is also cocompact. It therefore makes sense to consider the group K G (M Y ); the preceding lemma implies that the sequence K G (M Y ) K G (A Y ) K G (B Y ) K G ((A B) Y ) will be exact in the middle for any pair of closed G-subsets A and B which cover M. 13

20 2.4 Equivariant Spin c Structures Definition Let X be a proper G-space and G a topological group. A principal (G, G)- bundle over X consists of a principal G-bundle π : P X together with a commuting left action of G on P such that π is a G-map. Perhaps the simplest non-trivial example of a principle (G, G)-bundle is the orthonormal frame bundle of a Euclidean G-vector bundle (E, q). Specifically, let F(E, q) X be the principal O(n)- bundle whose fibre at a point p X consists of all orthonormal frames in E p. The G-invariance of the inner product q implies that the G-action on E lifts to a corresponding action on F(E, q); together this provides F(E, q) with the structure of a principal (G, O(n))-bundle. If E happens to possess a G-invariant orientation we can restrict ourselves to only the oriented orthonormal frames, and thus obtain a principal (G, SO(n))-bundle F + (E, q) F(E, q). In fact, the existence of a G-invariant orientation is equivalent to the existence of such a sub-bundle; this suggests that one way to impose additional structure on a G-vector bundle is to require the existence of a suitable principal (G, G)-bundle which maps to F(E, q). However, before delving further in this there is one technical point that should be addressed. The frame bundle F(E, q) quite clearly depends on the choice of the G-invariant inner product, and one might reasonably worry that a statement about F(E, q) might not hold for F(E, q ). The following lemma should put any such concerns to rest. Lemma Let E be a real G-vector bundle with G-invariant inner products q and q. Then there is a canonical isometric isomorphism φ : (E, q ) = (E, q). As a result there is a canonical (G, O(n))-bundle isomorphism F(φ) : F(E, q ) = F(E, q). Proof. Note that the G-invariant inner product q induces a G-equivariant isomorphism ˆq : E E such that q(u, v) = ˆq(u), v. Let ψ = q 1 q ; then q(ψ(u), u) = ( q ψ)(u), u = q (u), u = q (u, u) > 0. Thus ψ is a fibrewise positive operator and φ = ψ yields the desired map. For n 3 let Spin(n) denote the unique non-trivial double-cover of SO(n). The Spin groups play a central part in the development of real index theory; however our interests will lie with the corresponding complex theory, and for this we shall need their complex analogues. Definition For n 3 define the n th Spin c group to be Spin c (n) = Spin(n) Z2 S 1, where Z 2 Spin(n) is the kernel of the surjection onto SO(n). 14

21 Remark It will often be convenient to view Spin c (n) as an S 1 -bundle over SO(n). It is a rather non-trivial exercise in the cohomology of Lie groups to show that H 2 (SO(n); Z) = Z 2 when n 3; hence up to isomorphism Spin c (n) is the unique non-trivial line bundle over SO(n). Definition Let (E, q) be a Euclidean G-vector bundle of (locally constant) rank n over a proper G-space X. A G-Spin c structure for (E, q) consists of a principal (G, Spin c (n))-bundle P on X together with a G-equivariant bundle map σ : P F(E, q). Two G-Spin c structures (P, σ) and (P, σ ) are isomorphic if there exists a (G, Spin c (n))-bundle map τ : P P such that σ = τ σ. Note that thanks to Lemma 2.4.2, a G-Spin c structure for (E, q) induces a G-Spin c structure on (E, q ) we simply let P be the pullback P P F(E, q ) φ F(E, q). Definition Let E be a real G-vector bundle on a proper G-space X. A G-Spin c structure for E consists of a choice of G-Spin c structure for some (and hence every) G-invariant inner product on E. A G-Spin c structure (P, σ) endows the G-vector bundle E with a number of ancillary structures. For instance, every G-Spin c vector bundle is naturally oriented the quotient P/S 1 is a principal (G, SO(n))-bundle, and is identified under σ with a sub-bundle F + (E, q) F(E, q). Similarly, since Spin(n) Spin c (n) we can form the quotient L = P/Spin(n); as Spin c (n)/spin(n) is isomorphic to S 1 this quotient is naturally a principal (G, S 1 )-bundle over X. Definition Let X be a proper G-space. The equivariant Picard group Pic G (X) is the group of principal (G, S 1 )-bundles on X. Remark It is a well-known result of Hattori and Yoshida [HY76] that for a compact Lie group G and finite G-CW-complex X one has Pic G (X) = H 2 (EG G X; Z). A careful examination of the argument in [LMS83] shows that the same result also holds for a discrete group G acting on a finite proper G-CW-complex. The aforementioned (G, S 1 )-bundle L thus associates to any principle (G, Spin c (n))-bundle a Chern class c G 1 (P, σ) = L Pic G (X); when the choice of G-Spin c structure on E is clear we shall simply write c G 1 (E). Proposition Every Hermitian G-vector bundle (E, h) possesses a canonical G-Spin c structure whose Chern class is the determinant line bundle of E. 15

22 Proof. Let Ũ(n) denote the non-trivial double-cover of U(n). The inclusion U(n) SO(2n) and determinant U(n) S 1 lift to give maps Ũ(n) Spin(2n) and Ũ(n) S 1 = S 1 respectively. Together these define a map Ũ(n) Spin(2n) S1 which descends to give a homomorphism U(n) Spin c (2n); by construction the compositions U(n) Spin c (2n) SO(2n) U(n) Spin c (2n) S 1 are the inclusion U(n) SO(2n) and determinant U(n) S 1. Let F u (E, h) be the unitary frame bundle of E the principal (G, U(n))-bundle whose fibre at a point p B consists of all unitary frames of E p. Setting q = Re(h) defines a G-invariant inner product on the underlying real G-vector bundle E R, and one can identify F + (E R, q) with F u (E, h) U(n) SO(2n). Setting P = F u (E, h) U(n) Spin c (2n) then yields a principal (G, Spin c (2n))-bundle on X which maps to F(E R, q) via P = F u (E, h) U(n) Spin c (2n) F u (E, h) U(n) SO(2n) = F + (E R, q) F(E R, q). The statement about the Chern class now follows from the fact that the composition U(n) Spin c (2n) S 1 is the determinant. Proposition (Two-out-of-three). Let E 1, E 2 and E = E 1 E 2 be G-vector bundles on a proper G-space X. Then any choice of G-Spin c structures for two of these bundles induces a canonical G-Spin c structure on the third such that c G 1 (E) = cg 1 (E 1) c G 1 (E 2). Proof. Choose G-invariant inner products q 1 and q 2 for E 1 and E 2 ; together these provide a G-invariant inner product q on E. With these choices there is a natural map F(E 1, q 1 ) X F(E 2, q 2 ) F(E, q) and the frame bundle F(E, q) can be obtained as [F(E 1, q 1 ) X F(E 2, q 2 )] SO(n1) SO(n 2) SO(n). ( ) Now suppose that E 1 and E 2 have G-Spin c structures (P 1, σ 1 ) and (P 2, σ 2 ) respectively, and set P = [P 1 X P 2 ] Spin c (n 1) Spin c (n 2) Spin c (n). The various projections then yield an equivariant map from P to ( ), and hence to F(E, q); this defines the desired G-Spin c structure on E. On the other hand, suppose E 1 and E have G-Spin c structures (P 1, σ 1 ) and (P, σ). Let Q be the pullback Q P P 1 X F(E 2, q 2 ) F(E, q). 16

23 It is easily checked that Q is a principal (G, Spin c (n 1 ) Spin c (n 2 ))-bundle over X. Now set P 2 = Q/Spin c (n 1 ). Then P 2 is a principal (G, Spin c (n 2 ))-bundle over X, and the map Q P 1 X F(E 2, q 2 ) descends to the quotient to give σ 2 : P 2 F(E 2, q 2 ). Checking the Chern class formula is now simply a matter of chasing through the two constructions. This proposition allows one to make the following definition. Definition Let E be a G-Spin c vector bundle on a proper G-space X. The G-Spin c dual of E is the G-Spin c vector bundle E such that E E = E C as G-Spin c vector bundles, where E C is given the G-Spin c structure associated to its complex structure. Remarks (a) It follows from the two-out-of-three result that E exists and has c 1 (E ) = c 1 (E). (b) Let E be a complex G-vector bundle and let E denote its complex dual. Then since E R C = E E as complex G-vector bundles it follows that E is a model for the G-Spin c dual of E. As noted earlier, a G-Spin c structure for E naturally provides it with an orientation. However, it takes only a slight change to the G-Spin c structure to obtain the opposite orientation. To understand this, we first need the following lemma, which is an immediate consequence of the construction of the Spin-groups. Lemma The adjoint action Ad : O(n) Aut(SO(n)) lifts to a homomorphism Âd : O(n) Aut(Spin(n)). Note that Âd extends in an obvious way to give an action of O(n) on Spinc (n) which preserves the subgroups Spin(n) and S 1. Definitions (a) Let T n = diag(1,...,1, 1) O(n) and let α n represent the automorphism Âd(T n) Aut(Spin c (n)). Then from any principal (G, Spin c (n))-bundle P one can form a new principal (G, Spin c (n)) bundle P op which is isomorphic to P as a G-space but has a right Spin c (n)-action given by p op m = (p α n (m)) op for any p P, m Spin c (n). (b) Let (P, σ) be a G-Spin c structure for E and let α E : F(E, q) F(E, q) be the map which sends the frame (v 1, v 2,..., v n ) to (v 1, v 2,..., v n ). Then the opposite G-Spin c structure is that given by the pair (P op, α E σ). Note The opposite G-Spin c structure has the same Chern class as the original G-Spin c structure, but induces the opposite orientation on E. 17

24 2.5 Equivariant Bott Elements Let X be a proper cocompact G-space, and let (E, q) be a Euclidean G-vector bundle over X; though X is cocompact the same is obviously not true of E. In classical K-theory this is remedied by working with the one-point compactification E +, but since the induced G-action on E + has a fixed point this is usually not a proper G-space. We will therefore find it more useful to work with the unit sphere bundle X E = S(E 1) = {(v, t) E 1 : q(v, v) + t 2 = 1}, which has the twin virtues of being both proper and cocompact. Note We can also obtain X E by gluing two copies of the unit disc of E; namely, if we set D + = {(v, t) S(E 1) : t 0} = D(E) and D = {(v, t) S(E 1) : t 0} = D(E) then X E is simply the identification D + S(E) D. Now suppose that the bundle E is even-dimensional and possesses a G-Spin c structure (P, σ). Recall that the theory of Clifford algebras provides Spin c (n) with a complex Spin representation, which when n is even splits into two distinct complex 1 2 -Spin representations ±. Combined with the G-Spin c structure P these yield G-equivariant spinor bundles S ± = P Spin c (2n) ± on X. It follows from the particulars of the 1 2-Spin representations (see [LM89]) that the endomorphism bundle End(S) is naturally (graded) isomorphic to the Clifford bundle Cl(E), and therefore there is a G-equivariant Clifford multiplication ρ : E End(S, S). In particular this multiplication restricts to give a map ρ : E Hom(S +, S ) which associates an isomorphism to each non-zero v E. We ll now use this data to form an element of the group K G (X E ). To begin with, let p = p E : X E X be the obvious projection. Then by pulling the spinor bundles S ± back over p we obtain bundles H ± = p S ± on X E. At the same time, the pullback p E admits an obvious G-invariant section, namely s(v, t) = v; when combined with the Clifford multiplication this gives a G-invariant section ρ s of Hom(H +, H ) which consists of isomorphisms away from the poles ( 0, ±1) X E. In particular ρ s is an isomorphism when restricted to the equator S(E) S(E 1), and thus we can use it to glue the restrictions H + D+ and H D along their common boundary; we shall denote the resulting G-vector bundle by H. 18

25 Definition The Bott element associated to the G-Spin c vector bundle E is the class β E = [H ] [H ] K G (X E ). Remarks (a) This definition differs from that used by Baum and Douglas in [BD82]; they chose to work with vector bundles rather than K-theory classes, and thus only used the bundle H. On the other hand, the element defined here is far better behaved. Observe that there are natural sections i ± : X X E given by sending a point x X to the pole (0, ±1) E x 1 that lies above it. It follows immediately from the definition that i β E = 0 K G (X), and this vanishing will be essential in later arguments. (b) The suspicious reader might question the sudden appearance of duals in the definition of β E. This is a classic (and annoyingly clever) trick that makes a key index calculation (Corollary 3.5.6) turn out nicely. (c) Finally, although it is not clear from our notation, the Bott element depends on the choice of G-Spin c -structure on the vector bundle E. Note that although we have pursued this discussion under the assumption that G is a countable discrete group acting on a cocompact G-space, it applies just as well to the case of a compact group G acting on a compact G-space. One example of this is the G-action on a single point. In this context a Euclidean G-vector bundle is simply a representation G SO(n), and an equivariant Spin c -structure is just a lifting ρ : G Spin c (n). Meanwhile the associated sphere bundle is nothing more than the ordinary n-sphere S n R n+1. Paralleling our work above, suppose we focus on the even-dimensional case. Then associated to the lift G Spin c (2n) we have an element β ρ K G (S 2n ), where K G denotes the equivariant K-theory for compact groups defined by Segal in [Seg68]. The standard representation Spin c (2n) SO(2n) gives a universal example of this construction. The identity homomorphism Spin c (2n) Spin c (2n) is a lifting, and thus there is a natural class β 2n K Spin c (2n)(S 2n ); given a homomorphism ρ : G Spin c (2n) we then recover β ρ as the restriction res ρ β 2n. With this slight digression out of the way we are now in a position to explain a principal bundle construction which will underlie many of the arguments in the next two chapters. As usual, let X be a proper cocompact G-space. Continuing our notation from the last section we shall let G be a topological group and P a principal (G, G)-bundle P over a X. Given any compact G-space Z we can form the space P G Z; note that conditions on X and Z conspire to make P G Z a proper cocompact G-space. We have already seen one example of this construction earlier in the section. Indeed, suppose Z = S n and let G = Spin c (n) act via Spin c (n) SO(n) SO(n + 1). If E X is a real G-vector bundle with a given G-Spin c -structure (P, σ) then we can form the space P Spin c (n) S n ; a moment s thought about the structures involved reveals that this is simply the space M E = S(E 1). 19