Degree one cohomology with twisted coefficients of the mapping class group

TL;DR

This paper investigates the first cohomology group of the mapping class group with coefficients in the dual of algebraic functions on the $SL(2,C)$ moduli space, revealing non-trivial cohomological structures.

Contribution

It proves the non-triviality of the first cohomology group of the mapping class group with twisted coefficients related to the $SL(2,C)$ moduli space.

Findings

$H^1( ext{mapping class group}, O(M_{SL(2,C)})^*)$ is non-trivial

Establishes new connections between mapping class groups and algebraic geometry

Advances understanding of cohomological properties of surface groups

Abstract

Let $\Gamma$ be the mapping class group of an oriented surface $\Sigma$ of genus g with r boundary components. We prove that the first cohomology group $H^1(\Gamma, O(M_{SL(2, C)})^*)$ is non-trivial, where the coefficient module is the dual of the space of algebraic functions on the $SL(2, C)$ moduli space over $\Sigma$.