On squares of Dehn twists about non-separating curves of a non-orientable closed surface

Abstract

The level 2 mapping class group of an orientable closed surface can be generated by squares of Dehn twists about non-separating curves (see [5]). On the other hand, the level 2 mapping class group M 2 (Ng ) of a nonorientable closed surface Ng can not be generated by only Dehn twists, and so it can not be generated by squares of Dehn twists about non-separating curves. In this paper, we prove that the Dehn twist subgroup of M 2 (Ng ) can not be generated by squares of Dehn twists about non-separating curves either. As an application, we give a finite generating set for the subgroup of M 2 (Ng ) generated by Dehn twist about separating curves and squares of Dehn twists about non-separating curves. Moreover, we examine about actions on non-separating simple closed curves of Ng by M 2 (Ng ).