A presentation for the mapping class group of a non-orientable surface from the action on the complex of curves

Abstract

We study the action of the mapping class group $\mathcal{M}(F)$ on the complex of curves of a non-orientable surface $F$. Following the outline of [1] we obtain, using the result of [4], a presentation for $\mathcal{M}(F)$ defined in terms of the mapping class groups of the complementary surfaces of collections of curves, provided that $F$ is not sporadic, i.e. the complex of curves of $F$ is simply connected. We also compute a finite presentation for the mapping class group of each sporadic surface.