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 [ M,N ] is said to have the Borsuk-Ulam property with respect to τ if for every representative map fM→ N of β, there exists a point x M such that f(τ(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 τ of T^2.