ON THE GROUPS J0(G)

by

Chung-Nim Lee and Arthur G. Wasserman

INTRODUCTION

J. F. Adams raised the following question at the 1963 Seattle Con-

ference on Differential Topology [1]: given orthogonal representations

V, W of a compact Lie group G, when does there exist an equivariant map

f: S(V) + S(W) of degree k? (S(V) is the unit sphere in V.)

The problem was first considered by Atiyah and Tall in [3]. They

define two representations V, W of a finite group G to be J equivalent if

there are equivariant maps f: S (V) - S(W), h: S (W) - S(V) with both degree

f and degree h prime to the order of G; V, W are stably J equivalent if

there is a representation U of G such that V D U is J equivalent to W ® U.

Then the groups J0(G) may be defined as the quotient of the real repre-

sentations R0(G) by the subgroup T(G) = {V - W|V is stably J equivalent

to W}. Atiyah and Tall were able to compute the groups J0(G) for G a

finite p-group (p ± 2) in terms of the action of the Adams operation ty on

R0(G). Let R0(G)r= R0(G)/W0(G) where W0(G) is the subgroup generated by

V - \p V where k is prime to the order of G; their result is then J0(G) z

R0(G) for G a finite p-group (p f 2). Subsequently, Snaith extended the

result to include the case p = 2 for complex representations [9].

Received by the editors April 24, 1974

Research supported in part by the National Science Foundation

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