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Article Dans Une Revue Topological Methods in Nonlinear Analysis Année : 2022

The Borsuk-Ulam property for homotopy classes of maps between the torus and the Klein bottle -- part 2

Résumé

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 $f: M \to N$ 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 $T^2$ to the Klein bottle $K^2$ that possess the Borsuk-Ulam property with respect to any free involution of $T^2$ for which the orbit space is $K^2$. Our results are given in terms of a certain family of homomorphisms involving the fundamental groups of $T^2$ and $K^2$. This completes the analysis of the Borsuk-Ulam problem for the case $M=T^2$ and $N=K^2$, and for any free involution $\tau$ of $T^2$.
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Dates et versions

hal-03280987 , version 1 (07-07-2021)

Identifiants

Citer

Daciberg Lima Gonçalves, John Guaschi, Vinicius Casteluber Laass. The Borsuk-Ulam property for homotopy classes of maps between the torus and the Klein bottle -- part 2. Topological Methods in Nonlinear Analysis, 2022, 60 (2), pp.491-516. ⟨10.12775/TMNA.2022.005⟩. ⟨hal-03280987⟩
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