Skip to main content Accessibility help

Fusion systems and group actions with abelian isotropy subgroups

  • Özgün Ünlü (a1) and Ergün Yalçin (a1)


We prove that if a finite group G acts smoothly on a manifold M such that all the isotropy subgroups are abelian groups with rank ≤ k, then G acts freely and smoothly on M × × … × for some positive integers n1, …, nk. We construct these actions using a recursive method, introduced in an earlier paper, that involves abstract fusion systems on finite groups. As another application of this method, we prove that every finite solvable group acts freely and smoothly on some product of spheres, with trivial action on homology.



Hide All
1.Adem, A., Davis, J. F. and Ünlü, Ö., Fixity and free group actions on products of spheres, Comment. Math. Helv. 79 (2004), 758778.
2.Bouc, S., Biset functors for finite groups, Lecture Notes in Mathematics, Volume 1990 (Springer, 2010).
3.Heller, A., A note on spaces with operators, Illinois J. Math. 3 (1959), 98100.
4.Holm, P., Microbundles and Thom classes, Bull. Am. Math. Soc. 72 (1966), 549554.
5.Klaus, M., Constructing free actions of p-groups on products of spheres, Alg. Geom. Top. 11 (2011), 30653084.
6.Lück, W. and Oliver, R., The completion theorem in K-theory for proper actions of a discrete group, Topology 40 (2001), 585616.
7.Madsen, I., Thomas, C. B. and Wall, C. T. C., The topological spherical space form problem, II, Topology 15 (1978), 375382.
8.Milnor, J., Groups which act on without fixed points, Am. J. Math. 79 (1957), 623630.
9.Oliver, R., Free compact group actions on products of spheres, in Algebraic topology, Lecture Notes in Mathematics, Volume 763, pp. 539548 (Springer, 1979).
10.Park, S., Realizing a fusion system by a single group, Arch. Math. 94 (2010), 405410.
11.Ray, U., Free linear actions of finite groups on products of spheres, J. Alg. 147 (1992), 456490.
12.Smith, P. A., Permutable periodic transformations, Proc. Natl Acad. Sci. USA 30 (1944), 105108.
13.Spanier, E. H., Algebraic topology (Springer, 1966).
14.Ünlü, Ö. and Yalçin, E., Quasilinear actions on products of spheres, Bull. Lond. Math. Soc. 42 (2010), 981990.
15.Ünlü, Ö. and Yalçin, E., Fusion systems and constructing free actions on products of spheres, Math. Z. 270 (2012), 939959.


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed