Ext*(Q)(K*(X),R(G)) Kjj(X),
which seems to be new even when X is a smooth manifold with a
differentiate G-action. This could be important in understanding
the index theory of G-equivariant elliptic operators on X, since
(at least "roughly") KQ(X) classifies G-vector bundles over X and
K^fX) classifies elliptic G-operators over X. For instance, the
fact that
K*(X) S HomR(Q)(K*(X),R(G))
when KQ(X) is R(G)-free generalizes [Pe, Part II, Proposition 3.9
and Theorem 5.2]. In fact, several geometric applications of this
spectral sequence were given in [IP], although they dealt only
with the simplest possible case: G = T and R(G) localized to make
it a principal ideal domain. Presumably the general Universal
Coefficient spectral sequence could be used for similar
applications with other compact groups, or for analyzing
phenomena even in the case of T that can be traced to ExtR,G*.
In order to make the discussion somewhat more concrete, we
pause to discuss some interesting cases where we can control
Kr(X), X a (locally) compact G-space, and hence get some
a c*
information about K^JX) and the KK -equivalence class of C(X).
Suppose that V is a finite-dimensional complex vector space
with a linear G-action, and let X = P(V), the projective space of
V. By the equivariant Bott periodicity theorem, Kp(V) 2 R(G)
(concentrated in degree 0) and KQ(X) is a free R(G)-module
(concentrated in degree 0), so all spectral sequences collapse.
Similarly, if S is the unit sphere in V and G acts by isometries,
so that S is a G-space, then the equivariant short exact
0 Co((0,oo)xS) CQ(V) C({0}) 0
implies that there is an exact sequence
0 KJ(S) H(G) —2—• R(G) K°(S) 0.
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