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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Free actions of finite groups on $ S^n \times S^n$


Authors: Ian Hambleton and Özgün Ünlü
Journal: Trans. Amer. Math. Soc. 362 (2010), 3289-3317
MSC (2010): Primary 57S17, 57R67
Published electronically: December 15, 2009
MathSciNet review: 2592957
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Abstract: Let $ p$ be an odd prime. We construct a non-abelian extension $ \Gamma$ of $ S^1$ by $ \mathbf{Z}/p \times \mathbf{Z}/p$, and prove that any finite subgroup of $ \Gamma$ acts freely and smoothly on $ S^{2p-1} \times S^{2p-1}$. In particular, for each odd prime $ p$ we obtain free smooth actions of infinitely many non-metacyclic rank two $ p$-groups on $ S^{2p-1} \times S^{2p-1}$. These results arise from a general approach to the existence problem for finite group actions on products of equidimensional spheres.


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Additional Information

Ian Hambleton
Affiliation: Department of Mathematics & Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1
Email: ian@math.mcmaster.ca

Özgün Ünlü
Affiliation: Department of Mathematics & Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1
Address at time of publication: Department of Mathematics, Bilkent University, Ankara, Turkey 06800
Email: unluo@fen.bilkent.edu.tr

DOI: http://dx.doi.org/10.1090/S0002-9947-09-05039-9
PII: S 0002-9947(09)05039-9
Received by editor(s): April 10, 2008
Received by editor(s) in revised form: March 4, 2009
Published electronically: December 15, 2009
Additional Notes: This research was partially supported by NSERC Discovery Grant A4000.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.