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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
DOI: https://doi.org/10.1090/proc/12435
Published electronically: September 11, 2015
MathSciNet review: 3415604
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Abstract | References | Similar Articles | Additional Information

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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  • [ADÜ04] Alejandro Adem, James F. Davis, and Özgün Ünlü, Fixity and free group actions on products of spheres, Comment. Math. Helv. 79 (2004), no. 4, 758-778. MR 2099121 (2006a:57034), https://doi.org/10.1007/s00014-004-0810-4
  • [AS01] Alejandro Adem and Jeff H. Smith, Periodic complexes and group actions, Ann. of Math. (2) 154 (2001), no. 2, 407-435. MR 1865976 (2002i:57031), https://doi.org/10.2307/3062102
  • [DW96] James F. Davis and Shmuel Weinberger, Obstructions to propagation of group actions, Bol. Soc. Mat. Mexicana (3) 2 (1996), no. 1, 1-14. MR 1395907 (97i:57038)
  • [Ham06] Ian Hambleton, Some examples of free actions on products of spheres, Topology 45 (2006), no. 4, 735-749. MR 2236376 (2007b:57065), https://doi.org/10.1016/j.top.2006.02.002
  • [Hat02] Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354 (2002k:55001)
  • [Hir94] Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, vol. 33, Springer-Verlag, New York, 1994. Corrected reprint of the 1976 original. MR 1336822 (96c:57001)
  • [HM74] Morris W. Hirsch and Barry Mazur, Smoothings of piecewise linear manifolds, Annals of Mathematics Studies, No. 80, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0415630 (54 #3711)
  • [Hus94] Dale Husemoller, Fibre bundles, 3rd ed., Graduate Texts in Mathematics, vol. 20, Springer-Verlag, New York, 1994. MR 1249482 (94k:55001)
  • [KS77] Robion C. Kirby and Laurence C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, With notes by John Milnor and Michael Atiyah, Annals of Mathematics Studies, No. 88, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1977. MR 0645390 (58 #31082)
  • [Kla11] Michele Klaus, Constructing free actions of $ p$-groups on products of spheres, Algebr. Geom. Topol. 11 (2011), no. 5, 3065-3084. MR 2869452, https://doi.org/10.2140/agt.2011.11.3065
  • [MTW76] I. Madsen, C. B. Thomas, and C. T. C. Wall, The topological spherical space form problem. II. Existence of free actions, Topology 15 (1976), no. 4, 375-382. MR 0426003 (54 #13952)
  • [Mil65] John Milnor, Lectures on the $ h$-cobordism theorem, Notes by L. Siebenmann and J. Sondow, Princeton University Press, Princeton, N.J., 1965. MR 0190942 (32 #8352)
  • [RS82] Colin Patrick Rourke and Brian Joseph Sanderson, Introduction to piecewise-linear topology, Springer Study Edition, Springer-Verlag, Berlin-New York, 1982. Reprint. MR 665919 (83g:57009)
  • [ÜY10] Özgün Ünlü and Ergün Yalçın, Quasilinear actions on products of spheres, Bull. Lond. Math. Soc. 42 (2010), no. 6, 981-990. MR 2740018 (2012b:57056), https://doi.org/10.1112/blms/bdq072
  • [ÜY13a] Özgün Ünlü and Ergün Yalçin, Constructing homologically trivial actions on products of spheres, Indiana Univ. Math. J. 62 (2013), no. 3, 927-945. MR 3164851, https://doi.org/10.1512/iumj.2013.62.5007
  • [ÜY13b] Özgün Ünlü and Ergün Yalçin, Fusion systems and group actions with abelian isotropy subgroups, Proc. Edinb. Math. Soc. (2) 56 (2013), no. 3, 873-886. MR 3109762, https://doi.org/10.1017/S0013091513000345
  • [Wal66] C. T. C. Wall, Finiteness conditions for $ {\rm CW}$ complexes. II, Proc. Roy. Soc. Ser. A 295 (1966), 129-139. MR 0211402 (35 #2283)

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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

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