Free cyclic actions on surfaces and the Borsuk-Ulam theorem

TL;DR

This paper generalizes the Borsuk-Ulam theorem to free group actions on surfaces, providing algebraic criteria involving braid groups to determine when a homotopy class of maps has the Borsuk-Ulam property.

Contribution

It introduces a Borsuk-Ulam-type property for maps from spaces with free group actions, and offers algebraic criteria using braid groups to identify such maps on surfaces.

Findings

Provides an algebraic criterion involving braid groups for the Borsuk-Ulam property.

Determines conditions based on surface orientability, group order modulo 4, and homology for free actions of cyclic groups.

Offers examples with symmetric group actions and partial results for maps to .

Abstract

Let $M$ and $N$ be topological spaces, let $G$ be a group, and let $\tau \colon\thinspace G \times M \to M$ be a proper free action of $G$. In this paper, we define a Borsuk-Ulam-type property for homotopy classes of maps from $M$ to $N$ with respect to the pair $(G,\tau)$ that generalises the classical antipodal Borsuk-Ulam theorem of maps from the $n$-sphere $\mathbb{S}^n$ to $\mathbb{R}^n$. In the cases where $M$ is a finite pathwise-connected CW-complex, $G$ is a finite, non-trivial Abelian group, $\tau$ is a proper free cellular action, and $N$ is either $\mathbb{R}^2$ or a compact surface without boundary different of $\mathbb{S}^2$ and $\mathbb{RP}^2$, we give an algebraic criterion involving braid groups to decide whether a free homotopy class $\beta \in [M,N]$ has the Borsuk-Ulam property. As an application of this criterion, we consider the case where $M$ is a compact surface without boundary equipped with a free action $\tau$ of the finite cyclic group $\mathbb{Z}_n$. In terms of the orientability of the orbit space $M_\tau$ of $M$ by the action $\tau$, the value of $n$ modulo $4$ and a certain algebraic condition involving the first homology group of $M_\tau$, we are able to determine if the single homotopy class of maps from $M$ to $\mathbb{R}^2$ possesses the Borsuk-Ulam property with respect to $(\mathbb{Z}_n,\tau)$. Finally, we give some examples of surfaces on which the symmetric group acts, and for these cases, we obtain some partial results regarding the Borsuk-Ulam property for maps whose target is $\mathbb{R}^2$.