The level $d$ mapping class group of a compact non-orientable surface

Abstract

Let $N_{g,n}$ be a genus $g$ compact non-orientable surface with $n$ boundaries. We explain about relations on the level $d$ mapping class group ${\mathcal{M}}_d(N_{g,0})$ of $N_{g,0}$ and the level $d$ principal congruence subgroup ${\Gamma }_d(g-1)$ of $\mathrm{SL}(g-1;\mathbb{Z})$. As applications, we give a normal generating set of ${\mathcal{M}}_d(N_{g,n})$ for $g\ge 4$ and $n\ge 0$, and finite generating sets of ${\mathcal{M}}_d(N_{g,n})$ for some $d$, any $g\ge 4$ and $n\ge 0$.

21 pages, 10 figures