EQUIVARIANT K-THEORY AND KK-THEORY 7
Most of the time a is injective and then K*(S) • 0 and K°(S) has
homological dimension 1. Otherwise a = 0 and KQ(S) is R(G)-free.
Here is another type of example. Let T be a torus, and let
us suppose that T is embedded as a maximal torus in a simply
connected compact Lie group G. Let H be any closed subgroup of G.
Then G/H is a T~space (one can get circle actions on
3-dimensional lens spaces, for instance), and
K*(G/H) 2 R(T)®R(G)K*(G/H) 2 R(T)*R(Q)R(H),
concentrated in degree 0 (though possibly with big homological
We briefly mention one other consequence of our results: for
groups G satisfying Hodgkin's condition one obtains a fairly
definitive answer to a question raised in [Pa]. By a theorem of
Green [Gr2] and Julg [Jul], one knows that for any compact group
G and any G-algebra A, there is a natural isomorphism
K*(A) 2 K#(GKA)
where GKA denotes the C -crossed product. (The case A = C(X) and
G finite had actually been treated much earlier by Atiyah.)
Paulsen points out (and in fact this is done much more generally
in [Ka3]) that the analogous result for the dual theory holds if
G is finite, but not in general, and he raises the question of
determining the precise relationship between KQ(A) and K (GKA).
We see in fact that for good connected groups, the relationship
is given by a spectral sequence
ExtjJ(Q)(K,,(GKA),R(G)) —» K*(A),
while we have a short exact sequence
0 —*Ext2(K#(GrA),2) •
,Z) —+ 0.
which splits, unnaturally. It's no wonder one sees no obvious
connection between K (GKA) and K_(A).