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

TL;DR

This paper computes the first homology group with twisted coefficients for mapping class groups of non-orientable surfaces with boundary, advancing understanding of their algebraic structure.

Contribution

It provides the first explicit calculations of the first homology with twisted coefficients for these mapping class groups, filling a gap in the algebraic topology of non-orientable surfaces.

Findings

Determined the structure of the first homology group with twisted coefficients for various non-orientable surface mapping class groups.

Extended previous results from orientable to non-orientable surfaces.

Enhanced understanding of the algebraic properties of non-orientable surface mapping class groups.

Abstract

We determine the first homology group with coefficients in $H_1(N;\mathbb{Z})$ for various mapping class groups of a non--orientable surface $N$ with punctures and/or boundary.