Generating the Mapping Class Group: A Geometric, Algebraic, and Historical Survey

Abstract

The aim of this thesis is to examine a problem that lies in the intersection of geometry, algebra, and the history of mathematics—the generation of the mapping class group. The group is a useful algebraic invariant of geometric surfaces. Given any surface, its mapping class group is the collections of distortions of the surface (smooth or topological) that preserve orientation, boundary points, and irregularities such as punctures or marked points. As Birman’s remarks imply, the study of the mapping class group draws together concepts from geometric and algebraic topology. To understand the algebraic structure of this group, we need to understand how distortions act on a surface up to isotopy. The ‘point pushing map’ and Birman exact sequence, which Birman proved in her doctoral dissertation in 1968, provides a case-in-point. Her work and subsequent research in this field relies on a combination of visual intuition and algebraic machinery—the relationship between which has long preoccupied the history of mathematics. This thesis will examine a specific aspect of this research—the generators of the mapping class group, i.e. the smallest set of distortions from which we can obtain the entire group. Our aim is to demonstrate that the mapping class group of a surface is generated by Dehn twists, which are a particular distortion obtained by cutting a surface about the neighborhood a curve, twisting the cut-out component in a full circle, and gluing the components back together again.