Volume 7, issue 4 (2007)

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

Denise de Mattos and Edivaldo L dos Santos

Algebraic & Geometric Topology 7 (2007) 1791–1804
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 ${ℤ}_{p}$–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 $G={ℤ}_{p}$, we estimate the “size” of the ${ℤ}_{p}$–coincidence set of a fibre-preserving map.

Keywords
parametrized Borsuk–Ulam theorem, characteristic polynomials, free action, equivariant map, product of spheres
Mathematical Subject Classification 2000
Primary: 55M20
Secondary: 55R91, 55R25