#### Volume 9, issue 2 (2009)

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The first cohomology of the mapping class group with coefficients in algebraic functions on the $\mathrm{SL}_2(\mathbf{C})$ moduli space

### Jørgen Ellegaard Andersen and Rasmus Villemoes

Algebraic & Geometric Topology 9 (2009) 1177–1199
##### Abstract

Consider a compact surface of genus at least two. We prove that the first cohomology group of the mapping class group with coefficients in the space of algebraic functions on the ${SL}_{2}$(C) moduli space vanishes. In the genus one case, this cohomology group is infinite dimensional.

##### Keywords
group cohomology, mapping class group, flat connection, moduli space
##### Mathematical Subject Classification 2000
Primary: 20J06
Secondary: 57M07, 57M60