2021/07/08 by Daciberg Lima Gonçalves, Gonçalves, Daciberg Lima, John Guaschi +3
Computer Science · Mathematics · #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2107.03682
openalex publication_date 2021/07/08 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
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 f: M → 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 T2 to the Klein bottle K2 that possess the Borsuk-Ulam property with respect to any free involution of T2 for which the orbit space is K2. Our results are given in terms of a certain family of homomorphisms involving the fundamental groups of T2 and K2. This completes the analysis of the Borsuk-Ulam problem for the case M=T2 and N=K2, and for any free involution τ of T2.