Curve Pushing Maps and Homological Representations of Mapping Class Groups

Abstract

In this paper we discuss one type of finite dimensional representation of the mapping class group using push maps. Mapping class groups, denoted Mod(X), are well studied and appear in many areas of mathematics, including the study of braid groups and 3−manifolds. One area that still has much work to be done is the finite dimensional representation theory. One type of finite dimensional representation of Mod(X) is a homological representation. Homological representations are a class of finite dimensional representations of Mod(X). For every finite cover Y → X there is an associated homological representation of Mod(X). Push maps are homeomorphisms of X that push a point or a curve along the surface returning them to their original position. Since they are homeomorphisms, they belong to a class in Mod(X). This work describes the image of push maps under homological representations.