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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

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 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 S1–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 = p, we estimate the “size” of the p–coincidence set of a fibre-preserving map.

parametrized Borsuk–Ulam theorem, characteristic polynomials, free action, equivariant map, product of spheres
Mathematical Subject Classification 2000
Primary: 55M20
Secondary: 55R91, 55R25
Received: 5 February 2007
Revised: 2 October 2007
Accepted: 4 October 2007
Published: 18 December 2007
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
Edivaldo L dos Santos
Departamento de Matemática
Universidade Federal de São Carlos
Caixa Postal 676
São Carlos, SP 13565-905