Skip to main content

On topological actions of finite, non-standard groups on spheres

Abstract

The standard actions of finite groups on spheres \(S^d\) are linear actions, i.e. by finite subgroups of the orthogonal groups \(\mathrm{O}(d+1)\). We prove that, in each dimension \(d>5\), there is a finite group G which admits a faithful, topological action on a sphere \(S^d\) but is not isomorphic to a subgroup of \(\mathrm{O}(d+1)\). The situation remains open for smooth actions.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Boileau, M., Leeb, B., Porti, J.: Geometrization of 3-dimensional orbifolds. Ann. Math. 162, 195–250 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Cannon, J.W.: The recognition problem: what is a topological manifold. Bull. Am. Math. Soc. 84, 832–866 (1978)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Chen, W., Kwasik, S., Schultz, R.: Finite symmetries of \(S^4\). Forum Math. 28, 295–310 (2016)

  4. 4.

    Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Oxford University Press, Oxford (1985)

    MATH  Google Scholar 

  5. 5.

    Davis, M.W.: A survey of results in higher dimensions. In: Morgan, J.W., Bass, H. (eds.) The Smith Conjecture, pp. 227–240. Academic Press, New York (1984)

    Google Scholar 

  6. 6.

    Dinkelbach, J., Leeb, B.: Equivariant Ricci flow with symmetry and applications to to finite group actions on 3-manifolds. Geom. Topol. 13, 1129–1173 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Dotzel, R.M., Hamrick, G.C.: \(p\)-Group actions on homology spheres. Invent. math. 62, 437–442 (1981)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Fulton, W., Harris, J.: Representation Theory: A First Course. Graduate Texts in Mathematics, vol. 129. Springer, New York (1991)

    MATH  Google Scholar 

  9. 9.

    Guazzi, A., Zimmermann, B.: On finite simple groups acting on homology spheres. Monatsh. Math. 169, 371–381 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Mecchia, M., Zimmermann, B.: On finite simple and nonsolvable groups acting on homology 4-spheres. Topol. Appl. 153, 2933–2942 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Mecchia, M., Zimmermann, B.: On finite groups acting on homology 4-spheres and finite subgroups of \({\rm SO}(5)\). Topol. Appl. 158, 741–747 (2011)

  12. 12.

    Milgram, R.J.: Evaluating the swan finiteness obstruction for finite groups. In: Algebraic and Geometric Topology. Lecture Notes in Mathematics, vol. 1126, pp. 127-158, Springer (1985)

  13. 13.

    Milnor, J.: Groups which act on \(S^n\) without fixed points. Am. J. Math. 79, 623–630 (1957)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Munkres, J.R.: Elements of Algebraic Topology. Addison-Wesley Publishing Company, Menlo Park (1984)

    MATH  Google Scholar 

  15. 15.

    Serre, J.-P.: Linear Representations of Finite Groups. Graduate Texts in Mathematics, vol. 42. Springer, New York (1977)

    Book  Google Scholar 

  16. 16.

    Zimmermann, B.: Some results and conjectures on finite groups acting on homology spheres. Sib. Electron. Math. Rep. 2, 233–238 (2005). http://semr.math.nsc.ru

  17. 17.

    Zimmermann, B.: On the classification of finite groups acting on homology 3-spheres. Pac. J. Math. 217, 387–395 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Zimmermann, B.: On finite simple groups acting on homology spheres with small fixed point sets. Bol. Soc. Mat. Mex. 20, 611–621 (2014)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Bruno P. Zimmermann.

Additional information

Communicated by A. Constantin.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zimmermann, B.P. On topological actions of finite, non-standard groups on spheres. Monatsh Math 183, 219–223 (2017). https://doi.org/10.1007/s00605-016-0959-0

Download citation

Keywords

  • Finite groups acting on spheres
  • Non-linear actions
  • Non-standard groups

Mathematics Subject Classification

  • 57S25