A computation of H^1(Γ, H_1(Σ))

Let Σ= Σ_{g,1} be a compact surface of genus g at least 3 with one boundary component, Γits mapping class group and M = H_1(Σ, Z) the first integral homology of Σ. Using that Γis 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(Γ, M) is either trivial or isomorphic to Z. Using Wajnryb's presentation for Γin terms of the Humphries generators we can show that it is not trivial.

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