EQUIVARIANT K-THEORY AND KK-THEORY
21
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
Previous Page Next Page