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

Davis, James

doi:10.1090/proc/12435

Any finite group acts freely and homologically trivially on a product of spheres
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by James F. Davis

Proc. Amer. Math. Soc. 144 (2016), 379-386

DOI: https://doi.org/10.1090/proc/12435

Published electronically: September 11, 2015

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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.

References

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, DOI 10.1007/s00014-004-0810-4
Alejandro Adem and Jeff H. Smith, Periodic complexes and group actions, Ann. of Math. (2) 154 (2001), no. 2, 407–435. MR 1865976, DOI 10.2307/3062102
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
Ian Hambleton, Some examples of free actions on products of spheres, Topology 45 (2006), no. 4, 735–749. MR 2236376, DOI 10.1016/j.top.2006.02.002
Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, vol. 33, Springer-Verlag, New York, 1994. Corrected reprint of the 1976 original. MR 1336822
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
Dale Husemoller, Fibre bundles, 3rd ed., Graduate Texts in Mathematics, vol. 20, Springer-Verlag, New York, 1994. MR 1249482, DOI 10.1007/978-1-4757-2261-1
Robion C. Kirby and Laurence C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Annals of Mathematics Studies, No. 88, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1977. With notes by John Milnor and Michael Atiyah. MR 0645390
Michele Klaus, Constructing free actions of $p$-groups on products of spheres, Algebr. Geom. Topol. 11 (2011), no. 5, 3065–3084. MR 2869452, DOI 10.2140/agt.2011.11.3065
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 426003, DOI 10.1016/0040-9383(76)90031-8
John Milnor, Lectures on the $h$-cobordism theorem, Princeton University Press, Princeton, N.J., 1965. Notes by L. Siebenmann and J. Sondow. MR 0190942
Colin Patrick Rourke and Brian Joseph Sanderson, Introduction to piecewise-linear topology, Springer Study Edition, Springer-Verlag, Berlin-New York, 1982. Reprint. MR 665919
Özgün Ünlü and Ergün Yalçın, Quasilinear actions on products of spheres, Bull. Lond. Math. Soc. 42 (2010), no. 6, 981–990. MR 2740018, DOI 10.1112/blms/bdq072
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, DOI 10.1512/iumj.2013.62.5007
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, DOI 10.1017/S0013091513000345
C. T. C. Wall, Finiteness conditions for $\textrm {CW}$ complexes. II, Proc. Roy. Soc. London Ser. A 295 (1966), 129–139. MR 211402, DOI 10.1098/rspa.1966.0230