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 \(X / \tau \).

在这项工作中，我们分析了曲面间多重贴图的 Borsuk-Ulam 定理类型的有效性。我们开发了一种涉及辫子群的代数技术来研究n值映射的这个问题。作为第一个应用，我们描述了 Borsuk-Ulam 定理在以下两种情况下适用于分裂和非分裂多重映射\(\phi :X \multimap Y\) ： ( i ) X是配备对映体的 2 球体对合，Y是闭曲面或欧几里德平面； ( ii ) X是一个封闭曲面，不同于配备自由对合\(\tau \)的 2-球面，Y是欧几里德平面。结果是详尽的，并且在情况 ( ii ) 中用涉及轨道空间\(X / \tau \)的第一积分同调群的代数条件来描述。