The center of some braid groups and the Farrell cohomology of certain pure mapping class groups

TL;DR

This paper investigates the centers of certain braid groups and computes the Farrell cohomology of specific pure mapping class groups, revealing their algebraic structures and cohomological properties.

Contribution

It demonstrates that many low genus surface braid groups have centers as direct factors and describes centralizers and normalizers in pure mapping class groups using automorphism groups.

Findings

Centers of many low genus surface braid groups are direct factors.

The p-primary part of Farrell cohomology groups are elementary abelian.

Explicit computations of Farrell cohomology for some pure mapping class groups.

Abstract

In this paper we first show that many braid groups of low genus surfaces have their centers as direct factors. We then give a description of centralizers and normalizers of prime order elements in pure mapping class groups of surfaces with spherical quotients using automorphism groups of fundamental groups of the quotient surfaces. As an application, we use these to show that the $p$-primary part of the Farrell cohomology groups of certain mapping class groups are elementary abelian groups. At the end we compute the $p$-primary part of the Farrell cohomology of a few pure mapping class groups.