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
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