The first homology group with twisted coefficients for the mapping class group of a non-orientable surface of genus three with two boundary components

TL;DR

This paper computes the first homology group with twisted coefficients for the mapping class group of a specific non-orientable surface, advancing understanding of its algebraic structure.

Contribution

It provides the first explicit calculation of the first homology group with twisted coefficients for this class of non-orientable surfaces.

Findings

Determined the structure of the first homology group with twisted coefficients.

Extended knowledge of mapping class groups for non-orientable surfaces.

Contributed to algebraic topology and geometric group theory.

Abstract

We determine the first homology group with coefficients in $H_1(N;\mathbb{Z})$ for the mapping class group of a non-orientable surface $N$ of genus three with two boundary components.