6 JONATHAN ROSENBERG AND CLAUDE SCHOCHET

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

2

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

sequence

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.