Mathematische Zeitschrift

, Volume 270, Issue 3–4, pp 939–959 | Cite as

Fusion systems and constructing free actions on products of spheres

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Abstract

We show that every rank two p-group acts freely and smoothly on a product of two spheres. This follows from a more general construction: given a smooth action of a finite group G on a manifold M, we construct a smooth free action on \({M \times \mathbb S^{n_1}\times \dots \times \mathbb S^{n_k}}\) when the set of isotropy subgroups of the G-action on M can be associated to a fusion system satisfying certain properties. Another consequence of this construction is that if G is an (almost) extra-special p-group of rank r, then it acts freely and smoothly on a product of r spheres.

Mathematics Subject Classification (2000)

Primary 57S25 Secondary 20D20 
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References

  1. 1.
    Adem A., Davis J.F., Ünlü O.: Fixity and free group actions on products of spheres. Comment. Math. Helv. 79, 758–778 (2004)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Adem A., Smith J.H.: Periodic complexes and group actions. Ann. Math. 154(2), 407–435 (2001)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Benson, D.J.: Representations and cohomology I: basic representation theory of finite groups and associative algebras. In: Cambridge Studies in Advanced Mathematics, vol. 30. Cambridge University Press, Cambridge (1998)Google Scholar
  4. 4.
    Benson, D.J.: Representations and cohomology II: cohomology of groups and modules. In: Cambridge Studies in Advanced Mathematics, vol. 31. Cambridge University Press, Cambridge (1998)Google Scholar
  5. 5.
    Bredon, G.: Equivariant Cohomology Theories. Lecture Notes in Mathematics, vol. 34. Springer, Berlin (1967)Google Scholar
  6. 6.
    Bredon G.: Introduction to Compact Transformation Groups. Academic Press, Burlington (1972)MATHGoogle Scholar
  7. 7.
    Broto C., Levi R., Oliver R.: The homotopy theory of fusion systems. J. Am. Math. Soc. 16(4), 779–856 (2003)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Bouc, S.: Biset Functors for Finite Groups. Lecture Notes in Mathematics, vol. 1990. Springer, Berlin (2010)Google Scholar
  9. 9.
    Tom Dieck, T.: Transformation Groups. Studies in Mathematics, vol. 8. de Gruyter, Berlin (1987)Google Scholar
  10. 10.
    Hambleton I.: Some examples of free actions on products of spheres. Topology 45, 735–749 (2006)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Hambleton I., Ünlü O.: Examples of free actions on products of spheres. Q. J. Math. 60, 461–474 (2009)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Hambleton I., Ünlü O.: Free actions of finite groups on \({\mathbb S^n\times \mathbb S^n}\). Trans. Am. Math. Soc. 362, 3289–3317 (2010)MATHCrossRefGoogle Scholar
  13. 13.
    Heller A.: A note on spaces with operators. Ill. J. Math. 3, 98–100 (1959)MathSciNetMATHGoogle Scholar
  14. 14.
    Hirsch M.W.: Differential Topology. Springer, New York (1976)MATHGoogle Scholar
  15. 15.
    Lee J.M.: Introduction to Smooth Manifolds. Graduate Texts in Mathematics, vol. 218. Springer, New York (2003)Google Scholar
  16. 16.
    Lück, W.: Transformation Groups and Algebraic K-Theory. Lecture Notes in Mathematics, vol. 1408 (1989)Google Scholar
  17. 17.
    Lück, W.: Survey on classifying spaces for families of subgroups, Infinite groups: geometric, combinatorial and dynamical aspects, vol. 248, pp. 269–322, Progr. Math. Birkhäuser, Basel (2005)Google Scholar
  18. 18.
    Lück W., Oliver R.: The completion theorem in K-theory for proper actions of a discrete group. Topology 40(3), 585–616 (2001)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Madsen I., Thomas C.B., Wall C.T.C.: The topological spherical space form problem II. Topology 15, 375–382 (1978)MathSciNetCrossRefGoogle Scholar
  20. 20.
    May, J.P. et al.: Equivariant homotopy and cohomology theory, CBMS regional conference series in mathematics, vol. 91. American Mathematical Society, Providence, RI (1996)Google Scholar
  21. 21.
    Milnor J.: Groups which act on \({\mathbb S^n}\) without fixed points. Am. J. Math. 79, 623–630 (1957)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Osborn H.: Vector Bundles, vol. 1. Foundations and Stiefel-Whitney Classes. Academic Press, New York (1982)MATHGoogle Scholar
  23. 23.
    Park S.: Realizing a fusion system by a single group. Arch. Math. 94, 405–410 (2010)MATHCrossRefGoogle Scholar
  24. 24.
    Ragnarsson, K., Stancu, R.: Saturated fusion systems as idempotents in the double Burnside ring (preprint)Google Scholar
  25. 25.
    Smith P.A.: Permutable periodic transformations. Proc. Natl. Acad. Sci. 30, 105–108 (1944)MATHCrossRefGoogle Scholar
  26. 26.
    Spanier E.H.: Algebraic Topology. Springer, New York (1966)MATHGoogle Scholar
  27. 27.
    Ünlü, Ö.: Doctoral dissertation, UW-Madison (2004)Google Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of MathematicsBilkent UniversityAnkaraTurkey

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