EQUIVARIANT K-THEORY AND KK-THEORY

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C((G/H)X(R ,T), C((G/H)X(R ,T) ^((G/HJXIR^Z)

res.

res

C((G/H)X{0},T)o • C((G/H)X{0},T) H*((G/H)X{0},Z) 0

and the five-lemma applied to the corresponding long exact

cohomology sequence imply that res induces an isomorphism on

2

HM(G,-) if resQ does. Consider the commuting diagram

H°({G/H)XIRn,Z) C((G/H)XIRn,IR) C((G/H)XIR ,T)

res. res.

0 — IT((G/H)X{0},Z) —C((G/H)X{0},[K) — C((G/H)X{0},T)

Q

— 0.

By "averaging" of cocycles, Hr,(G,V) = 0 for j0 and V any Frechet

space with linear G-action (such as C((G/H)xM,IR) for M a trivial

G-space) so that res0 is an isomorphism on H^(G,-) for j0. By

2

the five-lemma, res. is an isomorphism on HM(G#-). Thus we deduce

that the restriction map res induces an isomorphism

Hj(G,C((G/H)XIRn,T)) - H^(G/C((G/H)X{0}/T))

Since W with action w is constructed as the quotient of A with

action a, the image of the obstruction is zero. This completes

the proof of Proposition 2.9 and thus of Theorem 2.8. •

For arbitrary compact groups G (even satisfying the Hodgkin

condition), identifying explicitly all the G-algebras in a

G-fundamental family requires being able to list all the closed

o

subgroups H of G and being able to compute HM(H,T) for each. For

G simply connected (e.g., SU(N), Spin(N), Sp(N)), the algebras

C(GX„End(V)) (V a finite-dimensional projective representation of

H, H a closed subgroup of G) form a G-fundamental family. This is

conceptually satisfactory and perhaps it is the best possible

result at this level of generality. However, further progress is