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 ϕ : X ⊸ 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 τ 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 / τ .