Linear representation stable bounds for the integral cohomology of pure mapping class groups

Abstract

In this paper we study the integral cohomology of pure mapping class groups of surfaces, and other related groups and spaces, as $\mathsf{FI}$-modules. We use recent results from Church, Miller, Nagpal and Reinhold to obtain explicit linear bounds for their presentation degree and to give an inductive description of these $\mathsf{FI}$-modules. Furthermore, we establish new results on representation stability, in the sense of Church and Farb, for the rational cohomology of pure mapping class groups of non-orientable surfaces.