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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: Graham Hope and Ulrike Tillmann
Journal: Proc. Amer. Math. Soc. 137 (2009), 393-400
MSC (2000): Primary 57M60; Secondary 20J05, 57S05
Published electronically: September 3, 2008
MathSciNet review: 2439465
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Abstract: We study the unstable cohomology of the mapping class groups $ \mathcal{N}_g$ of non-orientable surfaces of genus $ g$. In particular, we determine for all genus $ g$ and all primes $ p$ when the group $ \mathcal{N}_g$ is $ p$-periodic.

To this purpose we show that $ \mathcal{N}_g$ is a subgroup of the mapping class group $ \Gamma_{g-1}$ of an orientable surface of genus $ g-1$ and deduce that $ \mathcal{N}_g$ has finite virtual cohomological dimension. Furthermore, we describe precisely which finite groups of odd order are subgroups of $ \mathcal{N}_g$.


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Additional Information

Graham Hope
Affiliation: Mathematical Institute, Oxford University, Oxford OX1 3LB, United Kingdom
Email: hope@maths.ox.ac.uk

Ulrike Tillmann
Affiliation: Mathematical Institute, Oxford University, Oxford OX1 3LB, United Kingdom
Email: tillmann@maths.ox.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09507-5
PII: S 0002-9939(08)09507-5
Received by editor(s): September 25, 2007
Received by editor(s) in revised form: January 18, 2008
Published electronically: September 3, 2008
Communicated by: Paul Goerss
Article copyright: © Copyright 2008 American Mathematical Society