Algebraic & Geometric Topology

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 dos Santos

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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 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.

Article information

Source
Algebr. Geom. Topol., Volume 7, Number 4 (2007), 1791-1804.

Dates
Received: 5 February 2007
Revised: 2 October 2007
Accepted: 4 October 2007
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796773

Digital Object Identifier
doi:10.2140/agt.2007.7.1791

Mathematical Reviews number (MathSciNet)
MR2366178

Zentralblatt MATH identifier
05221960

Subjects
Primary: 55M20: Fixed points and coincidences [See also 54H25]
Secondary: 55R91: Equivariant fiber spaces and bundles [See also 19L47] 55R25: Sphere bundles and vector bundles

Keywords
parametrized Borsuk–Ulam theorem characteristic polynomials free action equivariant map product of spheres

Citation

de Mattos, Denise; dos Santos, Edivaldo. A parametrized Borsuk–Ulam theorem for a product of spheres with free $\mathbb{Z}_p$–action and free $S^1$–action. Algebr. Geom. Topol. 7 (2007), no. 4, 1791--1804. doi:10.2140/agt.2007.7.1791. https://projecteuclid.org/euclid.agt/1513796773


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