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

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.