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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Any finite group acts freely and homologically trivially on a product of spheres
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by James F. Davis PDF
Proc. Amer. Math. Soc. 144 (2016), 379-386 Request permission

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
  • MR Author ID: 194576
  • Email: jfdavis@indiana.edu
  • 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
  • © Copyright 2015 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 144 (2016), 379-386
  • MSC (2010): Primary 57S25; Secondary 57Q40, 57R80
  • DOI: https://doi.org/10.1090/proc/12435
  • MathSciNet review: 3415604