The twist subgroup of the mapping class group of a nonorientable surface

TL;DR

This paper studies the twist subgroup of the mapping class group of nonorientable surfaces, providing a simple generating set and computing its first homology group, advancing understanding of its algebraic structure.

Contribution

It introduces a simple generating set for the twist subgroup T(N) and calculates its abelianization, offering new insights into its algebraic properties.

Findings

A simple generating set for T(N) is established.

The first homology group of T(N) is computed.

Results enhance understanding of the algebraic structure of T(N).

Abstract

Let T(N) be the subgroup of the mapping class group of a nonorientable surface N (possibly with punctures and/or boundary components) generated by twists about two-sided circles. We obtain a simple generating set for T(N). As an application we compute the first homology group (abelianization) of T(N).