The Borsuk-Ulam theorem for n-valued maps between surfaces

TL;DR

This paper investigates a Borsuk-Ulam type theorem for n-valued maps between surfaces, using algebraic braid group techniques to determine when the theorem holds for specific surface involutions.

Contribution

It introduces an algebraic approach with braid groups to analyze Borsuk-Ulam theorems for multimaps between surfaces under various involutions.

Findings

Characterized when the Borsuk-Ulam theorem holds for splits and non-splits multimaps

Provided algebraic conditions involving homology groups for the case of free involutions

Developed a comprehensive algebraic framework for multimaps between surfaces

Abstract

In this work we analysed the validity of a type of Borsuk-Ulam theorem for multimaps between surfaces. We developed an algebraic technique involving braid groups to study this problem for $n$-valued maps. As a first application we described when the Borsuk-Ulam theorem holds for splits and non-splits multimaps $\phi \colon X \multimap Y$ in the following two cases: $(i)$ $X$ is the $2$-sphere eqquiped with the antipodal involution and $Y$ is either a closed surface or the Euclidean plane; $(ii)$ $X$ is a closed surface different of the $2$-sphere eqquiped with a free involution $\tau$ and $Y$ is the Euclidean plane. The results are exhaustive and in the case $(ii)$ are described in terms of an algebraic condition involving the first integral homology group of the orbit space $X / \tau$.