2 JONATHAN ROSENBERG AND CLAUDE SCHOCHET

To this end, we henceforth assume that the compact Lie group G

is connected and has torsion-free fundamental group; we call this

the Hodgkin condition, to recognize the pioneering work of L.

Hodgkin [Ho]. Suppose that one knows KLfA) and K (B) . Does this

determine K^tA^B)? If G is the trivial group and A is suitably

restricted then we have shown [Sc2] that there is a Kunneth short

exact sequence

0 • K^A^K^B) —+ K#(A0B) • Tor*(K# (A) ,K* (B) ) • 0

which splits unnaturally. Here is the equivariant analogue. The

category BQ is defined at the beginning of Section 2. It

contains, in particular, all commutative G-algebras. We also

define a somewhat larger category CG, containing all Type I

G-algebras. For G = T or SU(2), BQ = CG-

THEOREM 5.1. (Kunneth Spectral Sequence). Let G be a compact Lie

group satisfying the Hodgkin condition. For A € BQ and B a

G-algebra, there is a spectral sequence of R(G)-modules strongly

converging to K#(A®B) with

Kj#„ =

TorJG)(K°(A)/K°(B)).

The spectral sequence has the canonical grading, so that

Tor*(G)(K?(A),Kw(B)) G has total degree p+s+t (mod 2). The spectral

p s

sequence is natural with respect to pairs (A,B) in the category.

2 r-f2

oo

If G has rank r then E =0 for pr+l and E = E .

COROLLARY. Let G be a compact Lie group satisfying the Hodgkin

~ G G

condition. Suppose that A € BQ and that K^fA) or K^B) is

R(G)-free (or more generally R(G)-flat). Then there is a natural

isomorphism

ct(A,B):

KG(A)SR(G)KG(B)

K^(A®B) .

The spectral sequence (5.1) was known to

Hodgkin-Snaith-McLeod [Ho,Mc,Sn] for A and B commutative (with

minor restrictions on the spaces involved which we have removed).

Localized versions of the spectral sequence also hold. For

instance, if G is a compact Lie group satisfying the Hodgkin