Topological aspects of symmetries on surfaces

The homeomorphism group of a topological surface \(Σ\), Homeo(\(Σ\)), admits a topology known as the compact-open topology, with which it becomes a topological group. In this work, we provide a self-contained proof of this fact. Moreover, we use elementary tools to prove that Homeo(\(Σ\)) is a Polish group (i.e., it is separable and completely metrizable). We translate the isotopy relation in Homeo(\(Σ\)) as path-connectedness in Homeo(\(Σ\)) and, denoting by Homeo0(\(Σ\)) the identity path component, we use classic results in Descriptive Set Theory to prove that the Extended Mapping Class Group of \(Σ\), Mod±(\(Σ\)) := Homeo(\(Σ\))/ Homeo0(\(Σ\)), is a Polish group with the quotient topology. At the end of this survey, we discuss an alternative proof of this result based on realizing the Extended Mapping Class Group as the automorphism group of the complex of curves; this connection arises as one of the most important and beautiful in the theory of Mapping Class Groups.