The first cohomology of the mapping class group with coefficients in algebraic functions on the SL(2, C) moduli space

TL;DR

This paper proves that the first cohomology group of the mapping class group with coefficients in algebraic functions on the SL(2,C) moduli space is zero for compact surfaces of genus at least two, revealing a cohomological rigidity.

Contribution

It establishes the vanishing of the first cohomology group with these specific coefficients, a novel result in the study of mapping class groups and moduli spaces.

Findings

First cohomology group vanishes for genus ≥ 2 surfaces

Cohomological rigidity of the mapping class group with these coefficients

Advances understanding of algebraic functions on moduli spaces

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.