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2020 The Borsuk-Ulam property for homotopy classes of maps from the torus to the Klein bottle
Daciberg Lima Gonçalves, John Guaschi, Vinicius Casteluber Laass
Topol. Methods Nonlinear Anal. 56(2): 529-558 (2020). DOI: 10.12775/TMNA.2020.003

Abstract

Let $M$ be a topological space that admits a free involution $\tau$, and let $N$ be a topological space. A homotopy class $\beta \in [ M,N ]$ is said to have the Borsuk-Ulam property with respect to $\tau$ if for every representative map $f\colon M \to N$ of $\beta$, there exists a point $x \in M$ such that $f(\tau(x))= f(x)$. In this paper, we determine the homotopy classes of maps from the $2$-torus $\mathbb{T}^2$ to the Klein bottle $\mathbb{K}^2$ that possess the Borsuk-Ulam property with respect to a free involution $\tau_1$ of $\mathbb{T}^2$ for which the orbit space is $\mathbb{T}^2$. Our results are given in terms of a certain family of homomorphisms involving the fundamental groups of $\mathbb{T}^2$ and $\mathbb{K}^2$.

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Daciberg Lima Gonçalves. John Guaschi. Vinicius Casteluber Laass. "The Borsuk-Ulam property for homotopy classes of maps from the torus to the Klein bottle." Topol. Methods Nonlinear Anal. 56 (2) 529 - 558, 2020. https://doi.org/10.12775/TMNA.2020.003

Information

Published: 2020
First available in Project Euclid: 24 December 2020

Digital Object Identifier: 10.12775/TMNA.2020.003

Rights: Copyright © 2020 Juliusz P. Schauder Centre for Nonlinear Studies

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Vol.56 • No. 2 • 2020
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