The first cohomology of the mapping class group with coefficients in algebraic functions on the SL2(C) moduli space

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 SL2(C) moduli space vanishes. In the genus one case, this cohomology group is infinite dimensional. © 2009 Mathematical Sciences Publishers.