A finite generating set for the level 2 twist subgroup of the mapping class group of a closed non-orientable surface

TL;DR

This paper establishes a finite generating set for the level 2 twist subgroup of the mapping class group of a closed non-orientable surface, and computes its first homology group for certain genera.

Contribution

It provides the first finite generating set for the level 2 twist subgroup and shows how it is normally generated within the mapping class group.

Findings

Finite generating set for the level 2 twist subgroup.

Normal generation of the subgroup by a single crosscap pushing map for certain genus.

Calculation of the first homology group of the subgroup for specific genera.

Abstract

We obtain a finite generating set for the level 2 twist subgroup of the mapping class group of a closed non-orientable surface. The generating set consists of crosscap pushing maps along non-separating two-sided simple loops and squares of Dehn twists along non-separating two-sided simple closed curves. We also prove that the level 2 twist subgroup is normally generated in the mapping class group by a crosscap pushing map along a non-separating two-sided simple loop for genus $g\geq 5$ and $g=3$. As an application, we calculate the first homology group of the level 2 twist subgroup for genus $g\geq 5$ and $g=3$.