Fixed point free actions of spheres and equivariant maps

Abstract

The concept of index and co-index of a paracompact Hausdorff space X equipped with free involutions relative to the antipodal action on spheres were introduced by Conner and Floyd [2]. In this paper, we extend the notion of index and co-index for free G-spaces X, where X is a finitistic space and G=S1 (with complex multiplication) and G=S3 (with quaternionic multiplication). We prove that the index of X is less than or equal to the mod 2 cohomology index of X. We also compute the ring cohomology of the orbit space X/G, where G=S1 or G=S3 and X is a finitistic space whose cohomology ring is the same as the product of spheres Sn×Sm,1≤n≤m. Using these cohomological calculations, we obtain some Borsuk-Ulam type results.