2022 The Borsuk-Ulam property for homotopy classes of maps from the torus to the Klein bottle - part 2
Daciberg Lima Gonçalves, John Guaschi, Vinicius Casteluber Laass
Topol. Methods Nonlinear Anal. 60(2): 491-516 (2022). DOI: 10.12775/TMNA.2022.005

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 class 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 any free involution of $\mathbb{T}^2$ for which the orbit space is $\mathbb{K}^2$. Our results are given in terms of a certain family of homomorphisms involving the fundamental groups of $\mathbb{T}^2$ and $\mathbb{K}^2$. This completes the analysis of the Borsuk-Ulam problem for the case $M=\mathbb{T}^2$ and $N=\mathbb{K}^2$, and for any free involution $\tau$ of $\mathbb{T}^2$.

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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 - part 2." Topol. Methods Nonlinear Anal. 60 (2) 491 - 516, 2022. https://doi.org/10.12775/TMNA.2022.005

Information

Published: 2022
First available in Project Euclid: 8 September 2022

MathSciNet: MR4563245
Digital Object Identifier: 10.12775/TMNA.2022.005

Keywords: Borsuk-Ulam Theorem , braid groups , homotopy class , surfaces

Rights: Copyright © 2022 Juliusz P. Schauder Centre for Nonlinear Studies

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Vol.60 • No. 2 • 2022
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