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In this paper we study the integral cohomology of pure mapping class groupsof surfaces, and other related groups and spaces, as FI-modules. We use recentresults from Church, Miller, Nagpal and Reinhold to obtain explicit linearbounds for their presentation degree and to give an inductive description ofthese FI-modules. Furthermore, we establish new results on representationstability, in the sense of Church and Farb, for the rational cohomology of puremapping class groups of non-orientable surfaces.

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