The Borsuk-Ulam property for homotopy classes of maps from the torus to the Klein bottle - part 2 - Gonçalves, Daciberg Lima; Guaschi, John; Laass, Vinicius Casteluber (Topological Methods in Nonlinear Analysis, 2022)
Let $M$ be a topological space that admits a free involution $τ$, and let $N$be a topological space. A homotopy class $β \in [ M,N ]$ is said to have{\it the Borsuk-Ulam property with respect to $τ$} if for every representative map $f\colon M→ N$ of $β$, there exists a point $x \in M$ such that $f(τ(x))= f(x)$. 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 $τ$ of $\mathbb{T}^2$.