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

## TL;DR

This paper investigates the integral cohomology of pure mapping class groups using FI-modules, providing explicit bounds and new stability results, especially for non-orientable surfaces.

## Contribution

It introduces linear bounds for the presentation degree of FI-modules associated with these groups and extends representation stability results to non-orientable surfaces.

## Key findings

- Explicit linear bounds for presentation degree of FI-modules.
- Inductive descriptions of FI-modules for pure mapping class groups.
- New representation stability results for non-orientable surfaces.

## Abstract

In this paper we study the integral cohomology of pure mapping class groups of surfaces, and other related groups and spaces, as 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 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.