Artin relation for smooth representations.
Let G be a finite group. If G acts smoothly on a closed homotopy sphere S, we call S a smooth representation of G. The main result is: There is a function h(G) such that for every smooth representation S of G, dimension S(G) = h(G){dimension S(H)H proper subgroup of G} if and only if G has prime power order and G is not cyclic. In other words, only for a noncyclic p-group G is dimension S(G) a universal function of the dimensions of the fixed sets S(H) as H ranges over proper subgroups of G. This result is compared with an old theorem of Artin's dealing with dimensions of fixed sets of orthogonal representations of G.
Department of Mathematics, University of Chicago, Chicago, Illinois 60637.
K H
KH
Dovermann
T
T
Petrie