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

 

$SL_2$ action on the cohomology of a rank two abelian group with arbitrary coefficient domain


Authors: Eric Jespers and Alexander Zimmermann
Journal: Proc. Amer. Math. Soc. 130 (2002), 315-325
MSC (2000): Primary 20J06, 20C05, 20F29
Published electronically: June 19, 2001
MathSciNet review: 1862108
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Abstract:

A rank two abelian group $C_n\times C_n$ is in a natural way an $SL_2({\mathbb Z})$-module. This induces an action of $SL_2({\mathbb Z})$ on its group cohomology $H^m(C_n\times C_n,R)$ for any trivial coefficient domain $R$. In the present note we determine this module, including the question of when the universal coefficient theorem sequence splits.


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Additional Information

Eric Jespers
Affiliation: Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium
Email: efjesper@vub.ac.be

Alexander Zimmermann
Affiliation: LAMFA, Faculté de Mathématiques, Université de Picardie Jules Verne, 33 rue St Leu, 80039 Amiens Cedex, France
Email: Alexander.Zimmermann@u-picardie.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06087-7
PII: S 0002-9939(01)06087-7
Received by editor(s): May 19, 2000
Received by editor(s) in revised form: June 12, 2000
Published electronically: June 19, 2001
Additional Notes: This research was done while the authors collaborated at the “Mathematisches Forschungsinstitut Oberwolfach” financed by the “Research in Pairs” program of the “Volkswagen Stiftung”. The first-named author is also supported in part by Fonds voor Wetenschappelijk Ondezoek (Belgium) and Onderzoeksraad Vrije Universiteit Brussel.
Communicated by: Stephen D. Smith
Article copyright: © Copyright 2001 American Mathematical Society