A computation of H^1(\Gamma , H_1(\Sigma))

TL;DR

This paper computes the first cohomology group of the mapping class group of a genus g surface with one boundary component, showing it is either trivial or isomorphic to Z, and confirms it is not trivial.

Contribution

It provides a detailed calculation of H^1(\Gamma, H_1(\Sigma)) using Humphries' generators and Wajnryb's presentation, establishing its non-triviality.

Findings

H^1(\Gamma, H_1(\Sigma)) is either trivial or Z.

Using Humphries' generators, the non-triviality of the cohomology is demonstrated.

The cohomology group is explicitly computed for genus g ≥ 3 surfaces.

Abstract

Let \Sigma = \Sigma _{g,1} be a compact surface of genus g at least 3 with one boundary component, \Gamma its mapping class group and M = H_1(\Sigma , Z) the first integral homology of \Sigma . Using that \Gamma is generated by the Dehn twists in a collection of 2g+1 simple closed curves (Humphries' generators) and simple relations between these twists, we prove that H^1(\Gamma , M) is either trivial or isomorphic to Z. Using Wajnryb's presentation for \Gamma in terms of the Humphries generators we can show that it is not trivial.