Crosscap slides and the level 2 mapping class group of a nonorientable surface

TL;DR

This paper investigates the subgroup generated by crosscap slides in the mapping class group of a nonorientable surface, showing it equals the level 2 subgroup acting trivially on homology and is generated by involutions.

Contribution

It proves that the subgroup generated by all crosscap slides equals the level 2 subgroup and is generated by involutions, clarifying their algebraic structure.

Findings

Crosscap slides generate the level 2 subgroup of the mapping class group.

The subgroup generated by crosscap slides acts trivially on H_1(N;Z_2).

This subgroup is generated by involutions.

Abstract

Crosscap slide is a homeomorphism of a nonorientable surface of genus at least 2, which was introduced under the name Y-homeomorphism by Lickorish as an example of an element of the mapping class group which cannot be expressed as a product of Dehn twists. We prove that the subgroup of the mapping class group of a closed nonorientable surface N generated by all crosscap slides is equal to the level 2 subgroup consisting of those mapping classes which act trivially on H_1(N;Z_2). We also prove that this subgroup is generated by involutions.