Grupos de tranças de superfícies e espaços de recobrimento

Abstract

Given M a compact and connected surface without boundary, we define the braid groups of M, denoted by B_n(M), geometrically. We also explore its relation with the configuration space and the mapping class group of the same surface. In a more detailed manner, we present some relevant algebraic and geometric aspects of the braid groups of three specific surfaces, namely the closed disk, the sphere and the real projective plane. Later we consider p \colon \tilde{M} \rightarrow M a d-fold covering map and discuss the existence of an embedding from B_n(M)$ to $B_{dn}(\tilde{M}). In the possession of such result, we study the classification of the finite subgroups of B_n(\mathbb{R}P^2) and the mapping class group of the real projective plane. We conclude with the study of the algebraic realization of the finite dicyclic subgroups of B_n(\mathbb{R}P^2).