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

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.

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Keywords

braid group, mapping class group, normalizer, centralizer, cohomology of groups

Mathematical Subject Classification 2000

Primary: 20F36, 20J06

References

Publication

Received: 14 March 2007

Revised: 16 September 2007

Accepted: 5 October 2007

Published: 26 December 2007

Authors

Yu Qing Chen
Department of Mathematics and Statistics Wright State University Dayton OH 45435-0001

Henry H Glover
Department of Mathematics Ohio State University Columbus OH 43210-1174

Craig A Jensen
Department of Mathematics University of New Orleans New Orleans LA 70148

Volume 7, issue 4 (2007)

Yu Qing Chen, Henry H Glover and Craig A Jensen

Abstract