The Borsuk–Ulam property for homotopy classes of self-maps of surfaces of Euler characteristic zero

Abstract

Let M and N be topological spaces such that M admits a free involution $$\tau $$ . A homotopy class $$\beta \in [ M , N ] $$ is said to have the Borsuk–Ulam property with respect to $$\tau $$ if for every representative map $$f:\,M \rightarrow N$$ of $$\beta $$ , there exists a point $$x \in M$$ such that $$f ( \tau ( x) ) = f(x)$$ . In the case where M is a compact, connected manifold without boundary and N is a compact, connected surface without boundary different from the 2-sphere and the real projective plane, we formulate this property in terms of the pure and full 2-string braid groups of N, and of the fundamental groups of M and the orbit space of M with respect to the action of $$\tau $$ . If $$M=N$$ is either the 2-torus $$\mathbb {T}^2$$ or the Klein bottle $$\mathbb {K}^2$$ , we then solve the problem of deciding which homotopy classes of [M, M] have the Borsuk–Ulam property. First, if $$\tau :\,\mathbb {T}^2\rightarrow \mathbb {T}^2$$ is a free involution that preserves orientation, we show that no homotopy class of $$[ \mathbb {T}^2, \mathbb {T}^2]$$ has the Borsuk–Ulam property with respect to $$\tau $$ . Second, we prove that up to a certain equivalence relation, there is only one class of free involutions $$\tau :\,\mathbb {T}^2\rightarrow \mathbb {T}^2$$ that reverse orientation, and for such involutions, we classify the homotopy classes in $$[\mathbb {T}^2, \mathbb {T}^2]$$ that have the Borsuk–Ulam property with respect to $$\tau $$ in terms of the induced homomorphism on the fundamental group. Finally, we show that if $$\tau :\,\mathbb {K}^2\rightarrow \mathbb {K}^2$$ is a free involution, then a homotopy class of $$[\mathbb {K}^2, \mathbb {K}^2]$$ has the Borsuk–Ulam property with respect to $$\tau $$ if and only if the given homotopy class lifts to the torus.