The first homology group of the mapping class group of a nonorientable surface with twisted coefficients

TL;DR

This paper computes the first homology group of the mapping class group of a nonorientable surface with twisted coefficients, advancing understanding of its algebraic structure.

Contribution

It provides the first explicit calculation of the first homology group of M(N) with twisted coefficients, a previously unresolved problem.

Findings

Explicit description of H_1(M(N); H_1(N;Z))

New insights into the algebraic structure of mapping class groups

Foundation for further topological and algebraic studies

Abstract

We determine the first homology group of the mapping class group M(N) of a nonorientable surface N with coefficients in H_1(N;Z).