Fixed sets of involutions
Frank L. Capobianco · Pacific Journal of Mathematics · 1978
In their work Differentiable Periodic Maps, Conner and Floyd posed the following question: Given a closed smooth n -manifold M", for what values of k does there exist a closed (n +/c)-manifold V n+k with smooth involution T whose fixed point set is diffeomorphic to M"?In this paper we show that for many values of k there is a closed manifold with involution (T, V n+k ) whose fixed point set is cobordant to M n .We begin by defining I k n to be the set of classes in the n -dimensional unoriented cobordism group 3l n which are represented by an n -manifold which is the fixed point set of a closed (n + k)-manifold with smooth involution.Some properties of I k are easy to see-for instance, that I k is a subgroup of 31 m that I° n = 3l m and that 1$ = S; =o I* is an ideal in 9?*.It follows from [4] that if the manifold with involution (T, V n+1 ) has fixed point set F n , then F n bords; hence P n = (0).It is well-known that if the manifold with involution (T, V n+k ) has fixed point set F n , then the mod 2 Euler characteristics w n (F n ) and w n+k (V n+k ) are equal; hence for k odd I k is contained in Xn, the subgroup of classes in 3l n with zero Euler characteristic.The main result of this paper is the following:THEOREM.For 2^-k^n and k even, I k =3l n ; for 2 n.In fact, the techniques employed in Section 2 of this paper originally appeared in a dissertation written under the suprevision of R. E. Stong at the University of Virginia which verified this conjecture for n ^ 5.In this regard, the author wishes to express his gratitude and indebtedness to Professor Stong for the