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

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 {\it 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 classes of maps from the \(2\)-torus \(T^2\) to the Klein bottle \(K^2\) that possess the Borsuk-Ulam property with respect to a free involution \(\tau_1\) of \(T^2\) for which the orbit space is \(T^2\). Our results are given in terms of a certain family of homomorphisms involving the fundamental groups of \(T^2\) and \(K^2\).