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
Publication
Received: 5 February 2007
Revised: 2 October 2007
Accepted: 4 October 2007
Published: 18 December 2007
Authors
 Denise de Mattos Departamento de Matemática Instituto de Ciências Matemáticas e de Computação Universidade de São Paulo Caixa Postal 668 São Carlos, SP 13560-970 Brazil http://www.icmc.usp.br/~topologia/ Edivaldo L dos Santos Departamento de Matemática Universidade Federal de São Carlos Caixa Postal 676 São Carlos, SP 13565-905 Brazil http://www2.dm.ufscar.br/~edivaldo/