On equivariant Kasparov theory and Spanier-Whitehead duality

K-Theory 6: 363-385, I992. © 1992 Kluwer Academic Publishers. Printed in the Netherlands. 363

On Equivariant Kasparov Theory and Spanier-Whitehead Duality

C L A U D E S C H O C H E T Mathematics Department, Wayne State University, Detroit, MI 48202, U.S.A.

(Received: October 1991, revised: May 1992)

Abstract. Suppose that G is a second countable compact Lie group and that A and B are commutative G-C*-algebras. Then the Kasparov group KK~(A, B) is a bifunctor on G-spaces. It is computed here in terms of equivariant stable homotopy theory. This result is a consequence of a more general study of equivariant Spanier-Whitehead duality and uses in an essential way the extension of the Kasparov machinery to the setting of a-G-C*-algebras. As a consequence, we show that if (X, xo) is a based separable compact metric G-ENR (such as a smooth compact G-manifold) and (Y, yo) is a based countable G-CW-complex then there is a natural isomorphism

KK~,(C(X, xo),C( Y, Yo)) ~ K~('Z A FX)

where FX is the functional equivariant Spanier-Whitehead dual of X, This specializes when Y is trivial to yield a naturaI isomorphism

KK~,(C(X, Xo), ~) ~ ~K~(X)

where ~K,G(-) denotes equivariant Steenrod K-homology theory. This result is new even for X a finite G-CW-complex, in which case Steenrod K-homology coincides with the usual topological equivariant K-homology K~(X).

Key words and phrases. Kasparov groups, equivafiant KK-theory, Spanier-Whitehead duality, limits of C*-algebras, a-C*-algebras, pro-C*-algebras, equivafiant K-theory.

O. Introduction

The Kasparov groups KKa.(A, B) were defined by Kasparov [4, 5] for G a locally compact group and for A and B G-C*-algebras over the field ~ or C, denoted ~c generically. In the present paper, we consider in some detail the groups which result when A and B are taken to be commutative and based, so that A -- C(X, xo) and B = C(Y, Yo) for based spaces (X, xo) and (K Yo).* In that situation we establish the following Theorem.

*Here (X, Xo) denotes the space X with basepoint xo sometimes denoted as • ~ X. When it is clear that X is based then we frequently write X rather than (X,)Co). Let C(X, xo) = {~b s C(X): qS(xo)= 0}. Given based spaces (X, xo) and (Y, Yo), their smash product (X/'\ Y, *) is defined to be the quotient space (X x Y)/(X V Y). Note that

c(x A v,,) _~ c(x, xo) ~ c(~ yo).

When spectra are involved then all spaces and maps are understood to be based.

364 CLAUDE SCHOCHET

T H E O R E M 5.10. Let G be a second countable compact Lie group. Then there is a natural isomorphism

KK~,(C(X, xo), C(Y, Yo)) ~ K~(Y/ ' , FX)

which holds for Y a based countable G-CW-comptex and for X a based separable compact metric G-EN R.*

Here FX denotes the equivariant Spanier-Whitehead dual of X in the equivariant

stable category of [6]. Specializing to the case where Y is trivial yields:

T H E O R E M 5.11. Let G be a second countable compact Lie group. Then for all based separable compact metric G-ENR spaces X, there is a natural isomorphism

KK,~(C(X, Xo), ~) ~- "K~,(X).

This theorem generalizes the result of [3] in the nonequivariant setting. Here SKG,(X) denotes the equivariant Steenrod extension of topological K-homology to compact metric spaces. We believe that this theorem is new even when X is a finite

G-CW-complex, in which case

SK ,(X)

the usual equivariant K-homology for finite G-CW-complexes. Theorem 5.10, as stated, does not leave the realm of G-C*-algebras. However, the

main technique for proving this and related theorems involves a KK-pairing with duality elements which arise on function spaces (actually, function spectra). These function spaces are usually not locally compact, but they are of the homotopy type of countable G-CW-complexes. This leads one quickly to the necessity of extending the equivariant Kasparov groups to ¢~-G-C*-algebras so as to give a home to these

classes. This extension was carried out in [11]. The proofs of the principal theorems require the use of the equivariant stable

homotopy category. The real point is that to form an equivariant Spanier- Whitehead dual for a G-space X one must take into account all of the irreducible representations of the group G, and this is precisely what the functional dual FX is

set up to do. These results are all consequences of a generalized Spanier-Whitehead Duality

Theorem (5.12) of the form

KK~,(A Q C(X, xo),B) ~ KK°.(A, C(FX) Q B)

which holds quite generally, as described in Section 5.

*A G-space X is a G-ENR (Euclidean neighborhood retract) if it can be embedded as a retract of an open subset of some G-representation V. Any separable metric G-ENR is finite-dimensional and has the homotopy type of a G-CW-complex but not necessarily the homotopy type of a finite G-CW-complex, even stably. However, compact G-ENR's are finitely dominated and, hence, are wedge summands of finite G-CW-complexes stably ([6], p. 142).

EQUIVARIANT KASPAROV THEORY 365

The remainder of the paper is organized as follows. In Section 1, we recall the algebraic topological prerequisites for our work. Section 2 consists of a quick summary of results from [ t l j which are required, and in Section 3 these results are extended to G-spectra and the equivariant stable homotopy category. Section 4 consists of a statement of equivariant Spanier-Whitehead duality in a framework appropriate for our purposes; these results are described in more detail in [10]. With all of these tools on hand, it becomes possible in Section 5 to establish Theorem 5.12, which is the Spanier-Whitehead theorem in full generality in the setting of G-spectra.

1. G-Spaces and G-Spectra

In this section, we briefly review the algebraic topological prerequisites for our work. The basic reference is [6], from which we have borrowed freely. As we wish to approximate G-spectra by G-CW-spectra within the category, the basic equivariant zero-cell G/H (H a closed subgroup of G) must be a finite CW-complex. Accordingly, we immediately restrict and assume that G is a second countable compact Lie group for the remainder of the paper.

Let ~fq/denote the category of compactly generated weak Hausdorff left G-spaces. The weak Hausdorff condition stipulates that the diagonal is closed in the compactly generated product. This is the largest category of topological spaces suitable for homotopy theory. Furthermore, every commutative unital ¢-G-C*-algebra is of the form C(X) for some X e ~ g . ([7], Remark 2.10) In this paper, 'G-space' will signify a space in ~ . Let (¢ J denote the category of based G-spaces, where G acts trivially on the basepoint. An unbased G-space X may be regarded as a based space by the addition of a disjoint basepoint on which G acts trivially; the resulting based G-space is denoted X +. These categories are closed under the formation of (compactly generated) products and function spaces, where G acts diagonally on products and by conjugation on function spaces yx. Write F(X, Y) for the function space of based maps with the compactly generated topology.

In the equivariant setting the basic n-sphere is the based G-space

s h = ( 6 / H ) + A S ° - ( G / H x S " ) / ( G / H x . ) ,

where S" denotes the standard n-sphere with trivial G-action, H is a closed subgroup of G, and G/H has its natural left G-space structure. Let ~(X, Y)G denote the set of based homotopy classes of based maps in N~-. The equivariant homolopy groups of the G-space X are defined by

~ y ( x ) = ~ . (x ") = ~(s", x ) . = ~(s~, x)G,

where X H denotes the subspace of X fixed pointwise by H. A G-CW-complex is a G-space X which is the union of an expanding sequence of

sub-G-spaces X" such that X ° is a disjoint union of orbits G/H and X "+1 is obtained

366 CLAUDE SCHOCHET

from X" by attaching the cells (G /H)x e "+1 by means of attaching G-maps G/H x S" ~ X". When X is a based G-space, the basepoint is required to be a vertex and X ° is a wedge of 0-spheres S °. The successive quotients X"/X"-1 for n ~> 1 are wedges of n-spheres S~, one for each attaching map. A based G-CW-complex is a based G-space X ~ f f ~ which is the union of an expanding sequence of sub-G-spaces X" such that the basepoint is contained in X ° and X "+1 is the cofibre of a based G-map from a wedge of n-spheres S~ to X". The following theorem illustrates the central role played by G-CW-complexes.

THEOREM 1.1. (1) [12] There is a functor F from G-spaces to G-CW-complexes and a natural

transformation 7x: FX ~ X such that each )'x is a weak equivalence. (2) [14] I f X is a compact G-space and Y has the homotopy type of a G-

CW-complex then so does y x and (if X and Y are based) F(X, Y). (3) [13] A smooth compact G-manifold is triangulable as a finite G-CW-complex.

[]

In the nonequivariant world, spectra are indexed by integers. When one turns to the world of G-spaces, the most satisfactory indexing is by elements of the representation ring R(G) (which denotes the real or complex representation ring, depending upon Y). This appears, for instance, in [4] where the groups

w KvKG (A, B) are introduced; here V and W are finite-dimensional representations of G and the Kasparov group depends only upon the class IV] - [W] s R(G). If V is a finite-dimensional real inner product space, let S v denote its one-point compactification with basepoint at infinity. If G acts through isometrics on V then the action extends to a based G-action on S v. For any based G-space X,

define

ZV x = X A S v

and

~)V x = F(S v, X),

These 'suspensions' and 'loop spaces' are based G-spaces. If V is a subspace of W then let W - V denote the orthogonal complement.

Fix once and for all an ambient real inner product space H upon which G is represented with each irreducible representation acting by isometrics and appear- ing infinitely often and so that H is the direct sum of some finite-dimensional G-invariant sub-inner product spaces. Topologize 3(f as the colimit of these spaces. An indexing space is a finite-dimensional G-invariant sub-inner product space of W. An indexin9 set ~t/" is a set of indexing spaces which contains an expanding sequence of indexing spaces {V~} with Vo = {0} and such that W is the union of the Vii. Fix u/ once and for all so that each a ~ R(G) is represented by some specified formal difference IV] - [W] of indexing spaces. Henceforth, all indexing is understood to

be with respect to U.


DEFINITION t.2. A G-prespectrum (E, a) consists of based G-spaces E V for Ve ~/~ and based G-maps

o: ZW-V E V -~ E W

for V c W such that the following conditions hold:

(1) o-: Z ° E W ~ E W i s the identity map. (2) For V c W c Z the following diagram commutes:

z Z - W Z W - V E V :cz-% Z Z - W E W

l °1 EZ-V EV -. ~ ~ EZ

The G-map E V ~ ~ W - V E W which is adjoint to o- is denoted & A G-prespectrum (E, o-) is said to be a G-spectrum if each 6 is a homeomorphism. A map f : E ~ E' of G-prespectra is a system of based G-maps f V : E V ~ E ' V which respect the structural maps. Denote by ~5 ~ the category of G-spectra. [To be precise, one should indicate the dependence of ~ c j upon the indexing set ~ . However, a change in indexing set yields an equivalent homotopy category of G-spectra, so we suppress the issue.]

The category ~ Y of G-spectra is of primary interest. One needs prespectra because such basic constructions as colimits and smash products are obtained by carrying out the construction on the level of G-prespectra and then applying a suitable functor L from G-prespectra to G-spectra which is left adjoint to the forgetful functor. This matter is explained thoroughly in [6]. In addition, the version of Brown Representability used here (3.5) involves a special sort of G-prespectrum.

If E is a G-prespectrum and X is a based G-space, then the smash product E A X is defined to be the G-prespectrum

(E A x ) ( v ) = E v A x

with structural maps

a = a A I :Ew-v(EV A X) =- z W - V E V A X--* E W A X.

If EeNSP then E A X is defined to be L(E A X); that is, regard E as a G- prespectrum, form the G-prespectrum E A X and then take the associated G- spectrum. In particular, the cylinder E A I + is defined, and so it makes sense to define a homotopy to be a map h:E A I + ~ E ' or equivalently as a map ~': E --* F(I +, E'). Homotopy is an equivalence relation which respects composition, and so it is possible to form the homotopy category h~fY. Morphisms in h~¢5 p are homotopy classes of maps in N5 P. Denote by re(E, E')a the set h~fSP[E, E '] of homotopy classes of maps E ~ E'.

368 CLAUDE SCHOCHET

The formal relationship between G-spaces and G-spectra is given by a pair of adjoint flmctors. To any prespectrum E assign the space E(0). The restriction of this functor to ~5 ~ is denoted

Spaces and maps in the image of this functor are called infinite loop G-spaces and

maps. In the other direction, given a based G-space X there is a suspension G-prespectrum {YYX} with structural maps the natural identifications. Define the

suspension G-spectrum functor

Z~: f l y ~ ffSP

by

= L{ vx}.

Then Z ~ is left adjoint to ~ . When there is no risk of confusion write X for the

G-spectrum Z~X. The n-sphere G-spectrum S" is defined by

S" = Z~S " = Z"Z~S ° f o rn>10 ,

S" = A - " E ~ S ° for n < 0,

where A-" denotes translation desuspension. Abbreviate S = S °. Then S(V) = Q(S v) where Q(X) = colim f lyEr(x) . For a closed subgroup H c G, define the generalized

sphere G-spectrum S~ by

S~u = (G/H) + A S",

For H c G, n e Z, and E e f~,Y, define the equivariant homotopy groups of the

G-spectrum E by

# . ' ( E ) = E ) 0 =

Next, we briefly outline the theory of G-CW-spectra. We continue to work in ~5 P. Write • for the trivial G-spectrum (each E V a point) and write C f for the cofibre of a

G-map f : E ~ E'.

D E F I N I T I O N 1.3. A G-cell spectrum is a G-spectrum E together with a sequence of subspectra E, and maps j,: J , --* E. such that J , is a wedge of sphere spectra S~, Eo = *, E, + 1 = Cj. for n > 0, and E is the union of the E,. The map from the cone on a wedge summand of J , to E is a cell. The restriction of j , to a wedge summand is an attaching map. The spectrum E is finite if it contains only finitely many cells, countable if it contains at most a countable number of cells, and finite-dimensional if it contains cells in only finitely many dimensions. A G- CW-spectrum is a G-cell spectrum such that each attaching map S~ ~ E. factors through a cell subspectrum containing only cells of dimension at most q. Let ~qcd


denote the category of G-CW-spectra and cellular maps and h~C~ its homotopy category.

A map f: E-* E' of G-spectra is a weak equivalence if for each n ~ Z and each closed subgroup H c G the induced map

f ," ~ff(E) --* ~ff(E')

is an isomorphism. This is equivalent to insisting that the induced maps be spacewise weak equivalences. As [6] points out, the Whitehead theorem holds in this context.

THEOREM 1.4 (Whitehead) [6, page 30]. I f e: E ~ E' is a weak equivalence of

G-spectra, then

e,: re(D, E)G ~ ~(D, E')G

is a bijection for every G-cell spectrum D. I f E and E' are themselves G-cell spectra, then e is an equivalence. []

As one would expect from experience with spaces and spectra (cf. 1.1 (1)), each G-spectrum may be replaced by a weakly equivalent G-CW-spectrum. More precisely,

THEOREM 1.5 (cf. [6, 5.12]). There is a functor F: h(g5 p --+ hN~ and a natural

transformation

~E: F E -~ E

for each E E hN5 ~ such that each 7r is a weak equivalence.

DEFINITION 1.6 [6], pages 30-32. The equivariant stable category fff#5 P is the category which is obtained from the homotopy category hf#5 ° by formally inverting the weak equivalences.

Write [E,E']G for the set of morphisms h-N50[E,E '] in the category ]~ f j . Similarly, for based G-spaces X and X', write IX, X']G for the set obtained from ~z(X, X')G by formally inverting weak equivalences. The category ffNSP has arbitrary homotopy limits and homotopy colimits, fibration and cofibration sequences, and dual Milnor lim 1 exact sequences. Further, the category h-N~ has a smash product and with respect to that product it is a closed symmetric monoidal category (that is, the product is associative, commutative, with unit the sphere spectrum, all up to coherent natural isomorphism). If E is a G-CW-spectrum, then there is a natural isomorphism

~(E, E')z --~ [E, E']~

370 CLAUDE SCHOCHET

Similarly, if X has the homotopy type of a G-CW-complex, then there is a natural isomorphism

rc(X,X% ~-, [X,X']G.

The natural transformation F: hf#6 e ~ hf#~ induces a natural equivalence of categories F: fiN6 e --* hf#~

For each representation V E ~/~, the functors E v and f~v are inverse self-

equivalences in the category, so that one may desuspend by arbitrary representations in ~//~. Write y y - w = Evf~w Thus, Z" is defined for the fixed

choices of representations a = IV] - [W] and, hence, for each a ~ R(G). Write S a = x a s 0.

With a view to applications, we introduce the functional dual spectrum FX which is defined by FX(V) = F(X, Q(SV)) with the obvious structural maps. Note that if X is a based compact G-space then each F(X, Q(SV)) is of the homotopy type of a based

G-CW-complex, by (1.1 (2)), and

L({F(X, Sv)}) ~ FX.

D E F I N I T I O N 1.7. Given a G-spectrum E, its associated homology and cohomotogy theory are defined on some G-spectrum Y by

E~(Y) = IS a, Y A E]G

and

Ea(Y) = [Y, Z"E]~.

For a based G-space X, the (reduced) homology and cohomology groups are defined

by

E , ( x ) =

and

E*(X) = E*(Z X)

These theories satisfy the expected homotopy, suspension, and exactness axioms for G-CW-spectra and there is a Milnor lira 1 sequence for the theory E*

[6 p. 34].

2. Equivariant Kasparov Theory for a-G-C*-Algebras

As explained in the Introduction, it is necessary to extend the domain of the equivariant Kasparov groups to a much larger category than that of their initial definition by Kasparov. The first step in this procedure is the extension of the KK-groups to limits of G-C*-algebras. ~r This extension was accomplished in [11],


following the nonequivariant work of [15, 16]. The extension makes sense for G locally compact, but the best exactness and additivity results hold only when G is compact. Since the concern in the present work is with compact groups only, the results below are stated in that case. Recall that B z denotes either the real or complex field.

THEOREM 2.1 [I1, Proposition 2.7]. Let G be a second countable compact group. Then for each A and B pro-G-C*-aIgebras and a ~ R(G), there is an Abelian group KK~(A, B). The assignment

(A, B) KK ,(A, B)

is a bifunctor on the category of pro-G-C*-aIgebras to the category of R(G)-graded Abetian groups which is contravariant in A, covariant in B and which extends the Kasparov groups defined for G-C*-atgebras. The bifunctor is homotopy-invariant in each variable. []

Henceforth we assume that all operator algebras are countable at infinity ~'~" when necessary to form the requisite products.

THEOREM 2.2 [11, Theorems 3.4 and 3.6]. Let G be a second countable compact group. Then there is a naturally defined Kasparov product on pro-G-C*-algebras of the form

KK~(A1, B~ @ D) ® K K ~(D @ A2, B2) ®% KK~+b(A~ Q A2, B1 @ B2)

provided that A1 is separable and that B1 and A2 are countable at infinity. The Kasparov product has the following properties:

(1) The product is natural in each variable and it is associative. (2) I f B is countable at infinity then a G-map (p: A ~ B defines an element [(p]

KK°(A, B), and [cp~ = ~ , [ l a ] . (3) The product generalizes composition of maps. (4) Kasparov product with a K K °-invertibte element is an isomorphism. (5) i f D is countable at infinity then the natural map

Homo(A, B) --* Homo(A Q D, B @ D)

induces a homomorphism of groups

aD: KK°(A, B) ~ KK°(A @ D, B @ D)

and similarly for (D @ A, D ~ B).

~'For general information on pro-C*-atgebras and cr-G-C*-algebras, the reader should consult [7]. "k~XA pro-G-C*-algebra is countable at infinity if each of its quotient G-C*-algebras has a countable approximate unit.

372 CLAUDE SCHOCHET

Let A1 and A2 be separable. Then the Kasparov KK ° product has the following additional properties:

(6) The product is natural in D in the following sense: if ~p: D1 ~ D2 then

q?*(gl) @D2~2 = 0~1 @DI q):t:(~2)"

(7) The element 1A ~ KKG(A, A) is a unit jor the product. (8) The ring R(G)= KK~o(~,~) is commutative and for A separable each

KK~,(A, B) is a two-sided R(G)-module. (9) I f A~, A2, D2 are separable and D1 is ~-unital then

Jot e KK,%4 , B, ® m, s ® ®

(10) I f A1, A2, and D1 are separable then

Jor c KKG,(AI, B~ @ D), f le KKG,(D @ A2, B2). []

Let C,, denote the nth Clifford algebra.

THEOREM 2.3. Bott Periodicity (Kasparov). Suppose that the group G acts on ~" and the action commutes with the canonical spinor representation. Then the Bott element

fin e KK~(g:, Co(N n) @ Cn)

is KK~-invertible with KKG-inverse

an ~ KK~o(Co(N n) @ C., ~:).

Kasparov product with the Bott element defines an isomorphism

KK~+.(A ~ Co(JR"), B) ~ KK~.(A, B) ~- KK~a-.(A, B ~ Co(R")). []

3. Extension of KKa, to spectra

The functor KK~,(A ~ C(X), B Q C(Y)) is defined for X and Y countably compactly generated* G-spaces, since these spaces correspond to the most general commutative unital a-G-C*-algebras [7, Prop. 5.7.]. In this section we extend this functor and the associated Kasparov product to G-spectra,

*A space X is eountably compactly generated if there is a countable family {Kn} of compact subsets of X such that a subset C of X is closed iI and only if the set C c~ K. is closed for each n. We may take K1 = Kz c .... Then X is countably compactly generated if and only if it is a countable colimit of compact spaces.

E Q U I V A R I A N T KASPAR OV T H E O R Y 373

Henceforth, X and Y are understood to be based G-spaces and maps preserve basepoints, so that we are working in the category N~. Any statement involving both spectra and spaces requires working with C(X, xo) if X is a based compact G-space or with Co(X)= C(J~, *) where 32 is the one-point compactification of the locally compact unbased G-space X with basepoint the added point. Fix A and B and regard

KK~,(A ~ C(X), B ® C(Y))

as a bifunctor in X and in Y. It is homotopy invariant in each variable and satisfies the usual suspension and exactness axioms. Limits are respected as follows:

PROPOSITION 3.1. Limits in the X variable. Suppose that A is a separable pro-G-C*-algebra, B is a G-C*-algebra, X is a colimit of separable compact metric G-spaces Xj and either every Xj ~ Xj+ 1 is a cofibration or A is separable nuclear.

(1) Suppose that Y is a locally compact G-space. Then there is a natural isomorphism of R(G)-moduIes

KK~(A Q C(X), B Q Co(Y)) ~ cotim KKG,(A ~b C(Xj), B Q Co(Y)).

(2) Suppose that (IT, yo) is a based countably compactly generated G-space. Then there is a natural isomorphism of R(G)-modules

KKG,(A Q C(X), B Q C(Y, Yo)) ~- colim KKa,(A Q C(Xj), B ~ C(Y, Yo)).

Pro@ This follows from [11], Theorem 7.1. []

PROPOSITION 3.2. Limits in the Y variable. I f A is a separable pro-G-C*-algebra, (X, xo) is a based separable compact metric G-space, (Y, Yo) is a colimit of based compact G-spaces (Yj, *) and either

(1) each map Yj -~, ~+ I is a cofibration or (2) B is a separable nuclear and each space Yj is compact metric,

then there is a natural short exact sequence of R(G)-modules

0 ~ lim I KKa,+ I(A ~ C(X, xo), B Q C(Yj, *))

--* KKy(A Q C(X, Xo), B Q C( Y, Yo))

--* lira KKy(A ~ C(X, xo), B @ C(Yj, *)) ~ 0

Proof. (1) If each map ~-*Yj+I is a cofibration then so too is each map C(Yj+ b *) ~ C(Y~, *). This implies that each map

B @ C(Yj+ ~, .) - . B @ C(Yj, *)

is a cofibration,* so that [11], Theorem 7.1 implies the result.

*Here we use the fact [9] that if B -~ B ' is a cofibration then so is the map A @ B -* A @ B' for any A.

374 CLAUDE SCHOCHET

(2) If B is separable nuclear and each Yj is compact metric then each B ~ C(Yj, *) is separable nuclear, so that [11], Theorem 7.1 again implies the result. []

Fix separable pro-G-C*-algebras A and B. Given a based G-space Y, let

h*(Y) = KK~,(A, B ~ C(Y, yo)).

Then h* is an equivariant cohomology theory on based G-complexes. It satisfies a wedge axiom: if Y is the wedge of a countable collection of G-spaces Yj (wedged at the common base point), then B Q C(Y, Yo) is the product of the algebras B (~ C(Y i, *) and thus

h*(Y) ~- 1-1 h*(Yj) J

by [ t l ] , Theorem 6.9. To proceed it is necessary to apply an equivariant version of the J. F. Adams variant of E. H. Brown's Representability Theorem [1], which requires the use of yet another sort of spectrum. For this purpose, assume that attention has been restricted to an indexing sequence {V~} in ~.

D E F I N I T I O N 3.3 ([6], page 35). An f~G-prespectrum E is a G-prespectrum E indexed on ~U such that each

~: E(~) --, nW'E(V,+ 1)

is a weak equivalence, where Wi = Vi+l - Vi. A w-map f : E ~ E' of G-prespectra indexed on ~ consists of G-maps ff: E(V~) ~ E'(V~) such that the following diagrams are G-homotopy commutative:

XW, E(V~) xw'f~ , XW, E,(V,)

E(Vi+I) f~+l , E'(Vi+I).

Two w-maps are spacewise homotopic if fi ~-f} for all i, with no compatibility requirement on the homotopies.

Let w~¢~ denote the category of G-prespectra and spacewise homotopy classes of w-maps. The analog of Theorem 1.5 holds in this context, and it is the case that if E is an f~G-prespectrum then FE is an f~G-prespectrum with each structural map ~7 an equivalence. Let v~f#~ denote the category obtained from the category wf¢~ by formally inverting spacewise weak equivalences.

If E' (but not necessarily E) is an f~G-prespectrum, then

v;,fq~(E, E') = lim [E(V~), E'(V~)]G (3.4)

(by [6], page 36) where the limit is taken with respect to the maps

[E(V~ + t), E'(V~+ t)]~ • ' ' [Da'E(V;+ 1), ~Z~'E(V}+ 1)]G ___[g, (e3-t], [E(V~), E'(V~)] G.

The full subcategory # f ~ q ~ of ~ of f~G-prespectra is the representable equivalent of the category of cohomology theories on G-spaces. If H e # fE#~ then it


determines an equivariant cohomology theory H* by setting

HV'(y) = [II, H(V~)]o.

Such a theory satisfies the wedge axiom. The following theorem establishes the converse statement.

THEOREM 3.5. Brown Representability [1], [6], pages 34-35. Let G be a compact Lie group and let h* be an equivariant cohomoIogy theory on based G-complexes which satisfies the wedge axiom. Then there is an f~G-prespectrum H ~ # Y E ~ and a natural isomorphism

hv'(Y) -~ [Y, H(V,-)]o (3.6)

for each Vii in the indexing sequence which respects the structural maps. Proof. For each representation V, choose the minimal i such that V c V~. Then

hv( y) ~_ hV,(Zv~-v y).

Using Brown representability in the equivariant setting, represent each of the functors h v* by a G-CW-complex H(V3, so that

hv~(Y) ~ [Y, H(V~)]w

The suspension isomorphism implies that each

~: H(~) ~ n~H(~+ ~)

is a weak equivalence. Thus H = {H(Vi)} is an glG-prespectrum. []

In fact, by [6], page 35, Construction 6.3, we may assume that each 6 is an equivalence. Note that H is only unique up to noncanonical and nonunique equivalence.

Brown's theorem gives a method to extend equivariant cohomology theories from G-spaces to G-spectra. We carry out the construction in the KK-setting below, but the method works in general.

The equivariant KK-groups extend to G-spectra in the Y-variable as follows. Fix A and B. This determines an equivariant cohomology theory h* on based G- complexes by

hv(y) = KKCv(A, B Q C(Y))

which satisfies the wedge axiom. By Brown Representability (Theorem 3.5), there is an f~G-prespectrum H = HAB E W ~ which represents h*. Then for Y an f~G- prespectrum, define

KK~(A, B ~ C(Y)) = f f , 'aN~ [Y, XnH].

For a representation V~ ~, choose i minimal so that V c V~ and define

KK~(A, B ~ C(Y)) = KK~,(A, B @ C(EV'-Vv)).

376 CLAUDE SCHOCHET

Finally, if a = IV] - [W] in the fixed manner, define

KK°~(A, B @ C(Y)) = KK~.(A, B @ C(z-VY)).

This completes the definition of KK°,(A, B @ C(Y)) which includes as a special case the groups

KK~,(A ® C(X), B Q C(Y)) = ~ f # ~ [Y, E~HAxB]. (3.7)

PROPOSITION 3.8. Fix some G-spectrum Y. Then the functor KK~,(A, B @ C(Y)) is natural in A and in B.

Proof. This is not quite obvious, since (for instance), a map 0: B ~ B' will induce a map of f~G-prespectra

~9: HAB "* HAB, (3.9)

but the homotopy class of the map q~ is only unique modulo lirna choices. However, the induced map

q~,: KK°,(A, B @ C(Y)) ~ KK°,(A, B' @ C(Y))

is unique when Y is a finite G-spectrum. The functor KKG,(A, B ~ C(Y)) satisfies a lira I sequence in the Y-variable [6, page 34]. Suppose that Y is a G-CW-spectrum. Then it is a colimit of finite G-spectra yi. Suppose that Z and t# are two choices for qS. Then there is a commuting diagram of liml-sequences with ~ rnono and 7r epi of the form

lim 1 KK°, + I(A, B (~ C(Yi)) ~ , KK°,(A, B ~ C(Y)) - ~ lim KK~,(A, B @ C(Y')) , i ipi)l --~,)~ lim(z,--tp,)~ limltz, - , ~ (Z,

timI KK~,+ I(A,B, ~ C(yi)) ~ G B' KK,(A, Q C(Y)) --~ lirn KK~,(A, B' ~ C(yi)).

Since (Z, -- 0,) = 0 on finite G-spectra, the left and right arrows are zero. The snake lernma implies that the middle arrow (Z, - ~b,) = 0 as well. Thus, the map ~b, is unique on G-CW-spectra. Any G-spectrum is canonically weakly equivalent to a G-CW-spectrum by (1.5) and weak equivalences have been inverted so that ~b, is uniquely defined in general. A similar argument shows that the functor is also

natural in A. []

Next we turn our attention to the X variable. Recall that if E = {E(V)} is a G-spectrum and h , is a generalized homology theory on G-spaces, then h, may be extended to a generalized homology theory on G-spectra by defining

hv(E) = colim hv + w(E(W), eo) W

where the colirnit is taken with respect to some cofinal subset of representations W

of G.


DEFINITION 3.10. A G-prespectrum or G-spectrum E is separable if for each G-space E(V) the associated algebra C(E(V), *) is separable. [For E(V) compact, this is equivalent to assuming that each E(V) is second countable.]

The theory KK~,(A Q C(X, xo), B Q C(Y)) extends to separable G-spectra in the X variable as follows. Suppose given a separable G-spectrum X which we shall assume is indexed by an increasing sequence of representations V~. For each i select a map of spectra

~i: HA(~ +~ - ~,)x(v,)B -+ I=IAx(v~ + ,)B

which induces the map given at the KK-level by the structural map of X. (The maps ~i are unique up to lira 1 as usual.) Define

KK~.(A Q C(X), B Q C(Y))

= colim KK~v + w(A ~ C(X(W), Xo), B Q C(Y)). (3.1 I) w

Note that the wedge axiom in the Y-variable no longer holds in general since countable products do not commute with colimits. Next the Kasparov pairing (2.2) must be extended to this context. This shall be accomplished in three steps:

(1) The pairing ®D will be extended to allow Y to be a separable G-spectrum, rather than a G-space;

(2) The pairing ®D will be extenoed to allow X to be a separable G-spectrum, rather than a G-space; and then

(3) The pairing ®, will be extended to allow D = C(Z) where Z is a separable G-spectrum.

Step I. Previous work gives us the pairing

KK~(A1, Bj Q D) ® KK°b(D Q Az, B2) ®D KK~+b(A1 Q A2, B1 Q B2)

provided that At is separable and that B1 and A2 are countable at infinity. In particular, there is a pairing

KK°(AI, B~ ~ C(Ya, *) Q D) ® KK~(D Q A2, B2 ~ C(Y2, *))

i®o O KK~+b(AI Q A2, B1 Q B2 Q C(Y1 A I+2, *))

which gives rise to a map on representing G-spectra in ~ 2 ~ of the form

H A I , B 1 6 D A H D 6 A z , B 2 @D ) HAz6Az,BI6B2

which is unique rood lira a as usual. This map induces a pairing

KK°~(AI, B~ Q C(Ya) ~ D) ® KK~(D ~ Az, B2 ~ C(Y2))

t®o G I¢Ko+ (Aa ® Az, ® Bz ® C(Ya A Vz))

3 7 8 C L A U D E S C H O C H E T

as desired, and it is clear that the various representing maps induce pairings which satisfy the coherence conditions of (2.2).

Step 2. Extension to the case where X is taken to be a separable G-spectrum is immediate, since both the domain and range of the pairing

KK~(A1 @ C(Xl), B1 @ C(Y~) (~ D) ® KK~(D ~ A2 @ C(X2), B2 @ C(Y2)) l®o

G KK,+b(A~ ~ A2 @ C(X~ A X2), B1 @ 82 @ C(Y~ A Y2))

are colimits of instances of the pairing that have been defined previously.

Step 3. Finally, we must consider what happens when D is replaced by C(Z) for some G-spectrum Z. For temporary notational convenience, hold all variables except Z fixed and write

N*(Z) = KK~,(A~ ® C(X~), B~ ® C(Y~) ® C(Z)),

:7',(Z) = KK~(C(Z) ® A 2 ® C(X2) , S ® C(Y2) )

and

M = KK,~(& ® A2 ® C(Xl) ® C(X2), B1 ® B2 @ C(Y1) ® C(Y2)).

The Kasparov pairing is defined as a mapping

~*(Z) ® ~,(Z) ®z , M

for Z a G-CW-complex. Suppose that Z is a G-spectrum, so that ~*(Z) and ~,(Z) are defined. There are natural KK-pairings

~2*(Z(V~)) ® ~9°,(Z(V~)) ®z~v~ M

which are compatible with the structural maps of Z. Fix i. If ~e~*(Z(V~+I)) and

fle ~,(Z(V/)) then

(~)*c~ ® z ~ / ~ = c~ ®~÷~ (0~),/~

by 2.2 (6)). Thus for j > i there is a natural pairing

N*(Z(Vj)) ® 9°,(Z(V~)) ®z~v,~, M

which respects structural maps. This implies that there is a natural pairing

lim [N*(Z(V;))] ® ~,(Z(V~)) ~ lim [N*(Z(Vj)) ® 5z,(Z(V~))] ---, M j J

which is adjoint to a pairing

9°(Z(V¢)) --* Horn (lim N*(Z(Vj)), M). (*) J

Compose (*) with the natural map

~*(Z) ~ lim ~2*(Z(V~)) J


to obtain the map

~,(Z(V~)) ~ Horn (N*(Z), M). (**)

The map (**) induces the map

2P,(Z) = colim ~,(Z(V~)) - , Horn (~*(Z), M) i

whose adjoint is the desired pairing

~*(Z) ® °c£,(Z) ®z, M.

We summarize the results of this section in the following theorem.

THEOREM 3.12. Thefunctor

KK°,(A ~ C(X), B Q C(Y))

is defined for A and B pro-G-C*-algebras and for X and Y separable G-spectra extending the definition for X and Y based finite G-complexes. The functor is covariant in X and B, contravariant in Y and A. It satisfies the obvious homotopy and periodicity axioms in each variable. I f AI, As and Z are separable then there is a Kasparov pairing

KK~,(AI @ C(Xl), B1 @ C(¥i A Z)) @ KK~,(A2 @ C(Z A X2) , B 2 @ C(V2)) ®c,q

KK.~(A1 ® A2 ~ C(XI A X,). B1 ~ ~. ~ C(Y1 A Y,))

with the usual ([11], Theorem 3.6) properties.

4. Spectra and Duality

For certain based G-spaces X, the functional dual spectrum FX acts as an equivariant Spanier-Whitehead dual of X in the stable category. This section is devoted to making that statement predse. In order to do so, we sketch a 'categorical duality theory' in the formulation found in [6, Ch. III.1] and described in [10, §4]. Categorical duality theory is due to others, particularly Dold and Puppe [2]; the reader should consult [61 for details of attribution.

Assume that cg is a closed category. Such a category comes equipped with a unit object S, a product A : (g x (g --~ c6 and an internal hom functor F: qf°P x cg ~ of. The product A must be unital (with unit S), associative, and commutative up to coherent natual isomorphism (with 7: X A Y ~ Y A X the natural isomorphism), and there is a natural adjunction

~(xA y , z ) ~_ ~(X,F(Y,Z)).

The adjunction and the product yield a natural map

v: F(X, Y) A Z -~ F(X, r A Z).

380 CLAUDE SCHOCHET

Write

~ I : X ~ F ( Y , X A Y ) and e : F ( X , Y ) A X ~ Y

for the unit and counit of the adjunction. The map s is viewed as an evaluation map. Define the dual F X of an object X to be F X = F(X, S).

DEFINITION 4.1. An object X s (g is said to be strongly dualizable if there exists a coevaluation map (p: S ~ X A F X such that the diagram

S 0 ,, X A F X

F(X, X) ~ - FX A X

commutes. This implies that the map v is an isomorphism, so that the coevaluation map is

characterized as the composite 7v-at/. The terminology is due to Dold and Puppe, who introduced a different but equivalent definition. The authors of [6] prefer to call such objects 'finite' but that word is too overworked in the present context to use without danger of being misunderstood. Note that any retract of a strongly dualizable object is strongly duatizable.

The equivariant stable category h(¢5 P is a closed category and finite G-CW-spectra and their wedge summands are strongly dualizable ([6], page 131, Theorem 2.7). This includes (the suspension spectra of) smooth compact G-manifolds and, more generally, compact G-ENR's. Not every G-spectrum is strongly dualizable [cf. [10, Remark 5.2]]. For example, if X is strongly dualizable in h(¢5 p then it is also strongly dualizable in the (non-equivariant) stable category, which implies that •jHj(X) is a finitely generated Abelian group. It is conjectured in [6] that the most general strongly dualizable G-spectrum is a wedge summand of a finite G-CW-spectrum.

The most important properties of a strongly dualizable spectrum for the purposes of this paper are given by the following theorems.

THEOREM 4.2 [6]. Suppose that (g is a closed category.

(1) I f X is strongly dualizable then so is F X and the natural map

~(W, Z A FX) ~ ~(W A X, Z)

is an isomorphism. (2) I f X is strongly dualizabIe then the natural map X ~ F F X is an isomorphism. (3) I f X or Z is strongly dualizabte then the natural adjunction map

v: F(X, Y) A Z -~ F(X, Y A Z)

is an isomorphism. (4) If X is strongly dualizable then the composites

X_ ~S A X ,,,,OA1, X A F X A X 1 A s > X A S _~ X

EQUIVARIANT KASPAROV THEORY 38t

and

F X = F X A S 1 A ¢ , F X A X A F X ~A1 . . . . . . , S A FX _~ FX

are identity maps. C]

THEOREM 4.3 [6]. In the equivariant stable category ~ f ~ if X is a strongly dualizable spectrum then for any spectrum E the natural map

FX A E ~ F(X, E)

is an isomorphism. Thus there are natural isomorphisms

E,(FX) ~ E*(X)

and

E*(FX) ~ E,(X). []

This theorem shows that on strongly dualizable G-spectra that FX does indeed play the role of the equivariant Spanier-Whitehead dual of X.

5. Equivariant Duality for KKC. In this section the main theorems of the paper are established. The section begins with the construction of the natural structural map

~c(w): KK°,(A @ C(X), B @ C(Y)) -, KK°,(A Q C(X A W), B @ C(Y A W)).

Then the Spanier-Whitehead duality class #x e KK~(~-, C(FX A X)) is defined. If X is strongly dualizable then/~x is shown to be KKO-invertible, and this implies the main Spanier-Whitehead Duality Theorem (5.12) and its corollaries. The section con- cludes with a demonstration of the natural isomorphism

KKo,(C(X, xo), IF) -~ SKG,(X)

announced in the introduction. If D is a pro-G-C*-algebra then there is a natural structural map

a = ~D: KK~,(A, B) ~ KK~,(A ~) D, B Q D). (5.t)

Specializing, this yields a structural map

ac(w): KK°,(A @ C(X, *), B @ C(Y, *))

KK~,(A @ C(X A W, *), B @ C(Y A W, *)) (5.2)

which if W is strongly dualizable may be written via the isomorphism (3.8) as

O'c(w/~f2~P[ I~ Z* Hx] ~ ~*?f2~f#[ Y A W, Z* Hx/,w]

#f2N~[Y, FW A Y,* HxAw]. (5.3)

3 8 2 C L A U D E S C H O C H E T

This map may be extended immediately to the case where Y is a G-spectrum. Indeed, the map (5.3) is induced by a map of G-spectra

a(X, W): Hx ~ FW A HxAw

which is unique rood lira 1 as usual. Suppose next that X and W are G-spectra, with W strongly dualizable. Then

KK~,(A ~ C(X), B Q C(Y)) --- eolim ~ f2~EY, Hx(v,)] i

and

KKG.(A Q C(X A W), B (~ C(Y A W))

colim ~ f ~ [Y A W, HtxA w)(v~)] i

colim ~ a J ~ [ Y , FW A H(xAw)(v~)] i

so it suffices to construct a coherent family of maps

Hx(vo ~ FW(Vj) A Hx(v0Aw(o.

These maps are just the maps a(X(V 3, W(Vj)) constructed above.

PROPOSITION 5.4. Let W be a separable strongly dualizable G-spectrum. Then there is a map

ac(w): KK~(A ~ C(X), B ~) C(Y)) ~ KKG.(A Q C(X A W), B (~ C(Y A W))

which satisfies properties 2.2(5) and 2.2(9). []

With these tools at our disposal it is possible to prove the pivotal duality result. We introduce some notation.

Let X be a G-spectrum. Then there is a natural evaluation map

e = ex: FX AX ~ S.

Let

d e KKo (IF, iF) = KKo ( , C(S))

denote the generator. Define the Spanier-Whitehead duality class #x by

i~ x = e*d ~ KK~o(~ :, C(FX A X)).

If X is strongly dualizable with coevaluation map

~ b : S ~ F X A X

then define the dual Spanier-Whitehead duality class 2x by

2x = Ox.d ~ KK~(C(FX A X), iF).


T H E O R E M 5.5. Suppose that X is a separable strongly dualizable G-spectrum. Then gx and 2x are KKG-inverse to each other. That is,

/ix ®c¢x)2x = t s KK~o(C(FX), C(FX))

and

2x ®C(FX)/~x = 1 6 KK~(C(X), C(X)).

Proof. This is a direct calculation:

/ix ®c~x) 2x = ac(rx)(e*(d)) ®c~vx)~ c(x)~ c(rx~ ~ctvx)(~bx,(d)) by 2.2(9),

= ~rc(vx~(d)®cCrx)ex,¢,(d) by (2.2(6)),

= ¢c(vx)(d)®c(vx)~c~vx)(d) since ex,¢x, = 1 by (2.2(3)),

= ¢c~rx)(d ®~: d) by (2.2(9)),

= ~c¢vx)(d) since d is the unit by 2.2(7),

= lc~vx) since a is a ring map by 2.2(10).

The second identity follows by symmetry. []

The main result of this section follows immediately from this theorem.

T H E O R E M 5.6. Let G be a second countable compact Lie group. Suppose that A is a separable G-C*-algebra, B is a separable a-G-C*-atgebra, and Z and Y are separable G-spectra. Let X be a separable strongly dualizable G-spectrum. Then the natural map #x @ccx)(-) induces the Spanier-Whitehead duality isomorphism

KK~,(A @ C(Z A X), B Q C(Y)) ~ KK6,(A ~ C(Z), B ~) C(FX A Y)) (5.7)

with inverse given by KKC-product with the class 2x. Taking A = B = F and Z = S, there are isomorphisms

KKG,(C(X), C(Y)) ~ KK,°(F, C(FX A Y)) ~ K; (Y A FX). [] (5.8)

C O R O L L A R Y 5.9. Under the hypotheses of Theorem 5.6, there is an isomorphism of ~G-prespectra

ffZ: Z A HAB x ~ HAB(X/\Z)

which is unique up to lim 1 perturbation and which induces the Spanier-Whitehead duality isomorphism 2x ®c(Fx) ( - ). []

Finally it is possible to state the space-level version of the theorem.

T H E O R E M 5.10. Let G be a second countable compact Lie group. Then there is a natural isomorphism

KK~,(C(X, xo), C(Y, Yo)) ~ K; (YA FX)

which holds for Y a based countable G-CW-complex and for X a based separable compact metric G-ENR.

384 CLAUDE SCHOCHET

Proof. Any compact G-ENR is a wedge summand of a finite G-CW-complex stably, and hence the suspension spectrum of X is strongly dualizable. Then apply Theorem 5.6. []

The equivariant Steenrod homology groups associated to a G-spectrum E are defined for based G-spaces (X, Xo) by

SEa(X ) = E-a(FX).

We believe that the following result is new even for the case when X is a finite G-CW-complex, in which case SKG,(X) is isomorphic to the usual equivariant topological K-homology theory.

THEOREM 5.11. Let G be a second countable compact Lie group. Then for all based separable compact metric G-ENR spaces X, there is a natural isomorphism

KK~,(C(X, Xo), F) ~ SKG,(X).

Proof. Theorem 5.11 is immediate from (5.10). []

Acknowledgement

It is a pleasure to acknowledge our dependence upon the work of G. G. Kasparov, N. C. Phillips, J. Weidner, and the trio of J. G. Lewis Jr., J. P. May, and M. Steinberger, with a special thanks to Peter May for continuing assistance when called upon over a period of 25 years.

References

1. Adams, J. F.: A variant of E. H. Brown's representability theorem, Topology 10 (1971), 185-198. 2. Dold, A. and Puppe, D.: Duality, trace, and transfer, in K. Borsuk and A. Kirkor (eds.), Proc. Intl.

Conf. Geometric Topology 1980. 3. Kahn, D. S., Kaminker, J. and Schochet, C.: Generalized homology theories on compact metric

spaces, Michigan Math. J. 24 (1977), 203-224. 4. Kasparov, G. G.: The Operator K-functor and extensions of C*-algebras, Math. USSR Izv. 16 (1981),

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in Mathematics, vol. 1213, Springer-Verlag, New York, 1986. 7. Phillips, N. C.: Inverse limits of C*-algebras, J. Operator Theory 19 (1988), 159-195. 8. Phillips, N. C.: Representable K-theory for cr-C*-algebras, K-Theory 3 (1989), 441-478. 9. Schochet, C.: Topological methods for C*-algebras III: axiomatic homology, Pacific J. Math. 114

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13. Verona, A.: Triangulation of stratified fibre bundles, Manuscripta Math. 30 (1980), 425-445. 14. Warier, S.: Equivariant homotopy theory and Milnor's Theorem, Trans. Amer. Mark Soc. 258 (1980),

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On equivariant Kasparov theory and Spanier-Whitehead duality

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