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

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 split and non-split multimaps \(\phi :X \multimap Y\) in the following two cases: (i) X is the 2-sphere equipped with the antipodal involution and Y is either a closed surface or the Euclidean plane; (ii) X is a closed surface different from the 2-sphere equipped 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 \(X / \tau \).