The Borsuk-Ulam property for homotopy classes of maps from the torus to the Klein bottle

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 $\mathbb{K}^2$.