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.