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Any finite group acts freely and homologically trivially on a product of spheres


Author: James F. Davis
Journal: Proc. Amer. Math. Soc. 144 (2016), 379-386
MSC (2010): Primary 57S25; Secondary 57Q40, 57R80
Published electronically: September 11, 2015
MathSciNet review: 3415604
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Abstract: The main theorem states that if $ K$ is a finite CW-complex with finite fundamental group $ G$ and universal cover homotopy equivalent to a product of spheres $ X$, then $ G$ acts smoothly and freely on $ X \times S^n$ for any $ n$ greater than or equal to the dimension of $ X$. If the $ G$-action on the universal cover of $ K$ is homologically trivial, then so is the action on $ X \times S^n$. Ünlü and Yalçın recently showed that any finite group acts freely, cellularly, and homologicially trivially on a finite CW-complex which has the homotopy type of a product of spheres. Thus every finite group acts smoothly, freely, and homologically trivially on a product of spheres.


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Additional Information

James F. Davis
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email: jfdavis@indiana.edu

DOI: https://doi.org/10.1090/proc/12435
Received by editor(s): September 21, 2012
Received by editor(s) in revised form: December 28, 2013
Published electronically: September 11, 2015
Additional Notes: This research was supported by the National Science Foundation grant DMS-1210991. The research was inspired by a visit to Boğaziçi University, where the visit was supported by the Boğaziçi University Foundation.
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2015 American Mathematical Society