A parametrized Borsuk–Ulam theorem for a product of spheres with free $\mathbb{Z}_p$–action and free $S^1$–action

Abstract

In this paper, we prove parametrized Borsuk–Ulam theorems for bundles whose fibre has the same cohomology (mod $p$) as a product of spheres with any free ${\mathbb{Z}}_p$–action and for bundles whose fibre has rational cohomology ring isomorphic to the rational cohomology ring of a product of spheres with any free $S^1$–action. These theorems extend the result proved by Koikara and Mukerjee in [A Borsuk–Ulam type theorem for a product of spheres, Topology Appl. 63 (1995) 39–52]. Further, in the particular case where $G={\mathbb{Z}}_p$, we estimate the “size” of the ${\mathbb{Z}}_p$–coincidence set of a fibre-preserving map.