A computation of H^1(Γ, H_1(Σ))

Details A computation of H^1(Γ, H_1(Σ)) A computation of H^1(Γ, H_1(Σ))

2011-02-23

arxiv.org/abs/1102.4809v1

Mathematics

Algebraic Topology

9 pages, 1 figure

Scientific paper

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.

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