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.
Dovermann
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Department of Mathematics, University of Chicago, Chicago, Illinois 60637.
Petrie
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