Geometric Topology Seminar

Guest Speaker: Becca Winarski

Title: Mapping Class Groups and Covering Spaces

Abstract: A fundamental question of the study of mapping class groups is finding maps between mapping class groups of different surfaces. Given a covering space of surfaces, one may wish to relate the mapping class groups of the two surfaces. We say a cover of S over X has the Birman-Hilden property if there is a finite index subgroup of the mapping class group of X that injects into the mapping class group of S, modulo the deck transformations. In the 1970s, Birman and Hilden proved that regular covers have the Birman-Hilden property. We extend their work to certain irregular branched covers. As new applications, we prove: (1) Simple covers do not have the Birman-Hilden property and (2) All covers with at most one branch point do have the Birman-Hilden property.