ON THE GROUPS JO(G)
The purpose of this memoir is to classify real representations
of a compact Lie group G under stable J-equivalence. Two representa-
tions V,W of G are said to be stably J-equivalent if there exist
equivariant maps S (V © U) - S (W © U), S (W © U) - S(V © U), both of
degree one, for some representation U, where S(V © U) denotes the unit
sphere in V © U. Denote by JO(G) the quotient group of the represen-
tation ring RO(G), modulo stable J-equivalence. It is shown (Theorem
3-20) that V,W are stably J-equivalent if and only if dim V = dim W
for every cyclic subgroup C of G such that C/C H G is a p-group, p a
prime where V denotes the subspace of V fixed under C, or equiva-
lently if and only if the difference character, Xv - Xrj, is constant
on each connected component of G which has prime power order G/G , and
L Xw(g) = I XTT(g) for every finite cyclic subgroup C of a prime
g£C V g£C W
power order. In addition, the above theorem has a localized version
at any collection if primes. The sufficiency of these conditions is
shown via a Thom-Pontryagin type construction and the necessity follows
from simple equivariant transversality arguments. The crucial fact
necessary to handle arbitrary compact Lie groups is that for any x in
G the centralizer and the conjugacy class of x meet transversally.
Some of the interesting consequences of these results are 1) J0(G) is
a free abelian group, 2) J0(G) injects into the product n J0(G )
where G is the inverse image of a Sylow p-subgroup of G/G , 3) if G
is connected, then J0(G) ~ R0(G). Moreover, these methods provide
another proof of the Atiyah-Tall theorem J0(G) ~ R0(G)r.
Conversations with G. Bredon, J. McLaughlin and D. Wigner were
very helpful. The authors would also like to thank Denise Cotter for
typing the first draft of this paper and Jean Whipple who prepared
the final manuscript.
Both authors were supported in part by the National Science
Foundation during the preparation of this memoir.
AMS(MOS) 1970 Classification 55E50
KEY WORDS. Compact Lie groups, linear representations, equivariant
maps, stable J-equivalence.