A Theorem on Periodic Transformations of Homology Spheres

Abstract

In his theory of periodic transformations in a topological space' Smith proved the following theorem: THEOREM A2. Let X be a compact Hausdorff space of finite dimension which is a homology n-sphere over the cyclic group Jp of prime order p and let T be a periodic transformation on X of period p. Then the fixed point set is a homology r-sphere over Jp with -1 < r ? n. If X is an n-sphere and T acts analytically in X, it is well known that T preserves or reverses orientation according as the integer n r is even or odd. The question naturally arises [FP, p. 373]2 whether this remains true without this restriction that T is analytic. A partial solution of this problem has been given by Smith. He showed that,3 in the above theorem if p is an odd prime, and X is a compact metric space which has the same integral (Cech) cohomology groups as an n-sphere S' and the open subsets of which possess certain homology properties of those of S', then n r is even. The main aim of this paper is to establish the following theorem: THEOREM B. In Theorem A, let all the integral cohomology groups of X be finitely generated groups. Then T preserves or reverses orientation according as n r is even or odd. The meaning of orientation-preserving or reversing is defined here in terms of