Farrell cohomology of the pure mapping class group of non-orientable surfaces

TL;DR

This paper computes the p-primary Farrell cohomology of pure mapping class groups of non-orientable surfaces, classifying subgroup conjugacy classes and extending topological equivalence notions for surface kernel epimorphisms.

Contribution

It introduces a classification of conjugacy classes of order p subgroups and extends topological equivalence concepts to surfaces with marked points.

Findings

Determined the p-primary Farrell cohomology for non-orientable surfaces of genus p.

Classified conjugacy classes of order p subgroups in the pure mapping class group.

Extended topological equivalence notions to include surfaces with marked points.

Abstract

For an odd prime $p$, we determine the $p$-primary component of the Farrell cohomology of the pure mapping class groups of a non orientable surface of genus $p$ with $k\geqslant 1$ marked points. To do this, we classify conjugacy classes of subgroups of order $p$ of the pure mapping class group of a non orientable surface of any genus with marked points. This is obtained by extending the notion of topological equivalence for surface kernel epimorphisms of non Euclidean crystallographic groups, adapting it to the setting of surfaces with marked points.