Mathematische Annalen

, Volume 341, Issue 4, pp 845–858 | Cite as

Smooth circle actions on highly symmetric manifolds

  • Krzysztof Pawałowski


We construct for the first time smooth circle actions on highly symmetric manifolds such as disks, spheres, and Euclidean spaces which contain two points with the same isotropy subgroup whose representations determined on the tangent spaces at the two points are not isomorphic to each other. This allows us to answer negatively a question of Hsiang and Hsiang [Some Problems in Differentiable Transformation Groups, Springer, Berlin, Problem 16, p. 228, 1968].

Mathematical Subject Classification (2000).

57S15 57S25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Assadi, A.: Finite group actions on simply-connected manifolds and CW complexes. Mem. Am. Math. Soc. No. 257 (1982)Google Scholar
  2. 2.
    Bredon, G.E.: Introduction to Compact Transformation Groups. Pure and App. Math, vol. 46. Academic Press, (1972)Google Scholar
  3. 3.
    Conner, P.E., Montgomery, D.: An example for SO(3). Proc. Natl. Acad. Sci. USA 48, 1918– (1962)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    tom Dieck, T.: Transformation Groups, de Gruyter Studies in Math. 8. Walter de Gruyter, 1987Google Scholar
  5. 5.
    Edmonds, A.L., Lee, R.: Fixed point sets of group actions on Euclidean space. Topology 14, 339– (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Edmonds, A.L., Lee, R.: Compact Lie groups which act on Euclidean space without fixed points. Proc. Am. Math. Soc. 55, 416– (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Hsiang, W.-C., Hsiang, W.-Y.: Differentiable actions of compact connected classical groups I. Am. J. Math. 89, 705– (1967)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Hsiang, W.-C., Hsiang, W.-Y.: Some Problems in Differentiable Transformation Groups, pp. 223–236. Springer, Heidelberg (1968)Google Scholar
  9. 9.
    Jackowski, S., Pawałowski, K. (ed.): Transformation Groups, Poznań 1985. Lecture Notes in Math., vol. 1217. Springer, Heidelberg (1986)Google Scholar
  10. 10.
    Kawakubo, K. (ed.): Transformation Groups, Osaka 1987. Lecture Notes in Math., vol. 1375. Springer, Heidelberg (1986)Google Scholar
  11. 11.
    Kawakubo, K.: The Theory of Transformation Groups. Oxford University Press, (1991)zbMATHGoogle Scholar
  12. 12.
    Milgram, R.J. (ed.): Algebraic and Geometric Topology. Proc. Sympos. Pure Math., vol. 32 (1978)Google Scholar
  13. 13.
    Milnor, J.: Lectures on the h-Cobordism Theorem. Princeton Mathematical Notes, vol. 1. Princeton University Press, Princeton (1965)Google Scholar
  14. 14.
    Morimoto, M., Pawałowski, K.: The equivariant bundle subtraction theorem and its applications. Fund. Math. 161, 279– (1999)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Morimoto, M., Pawałowski, K.: Smooth actions of finite Oliver groups on spheres. Topology 42, 395– (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Mostert, P.S. (ed.): Proceedings on the Conference on Transformation Groups (New Orleans, 1967). Springer, Heidelberg (1968)Google Scholar
  17. 17.
    Pawałowski, K.: Group actions with inequivalent representations at fixed points. Math. Z. 187, 29– (1984)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Pawałowski, K.: Equivariant thickening for compact Lie group actions. Mathematica Gottingensis, Heft 71 (1986)Google Scholar
  19. 19.
    Pawałowski, K.: Fixed point sets of smooth group actions on disks and Euclidean spaces. Topology 28, 273–289 (1989); Corrections: ibid. 35, 749–750 (1996)Google Scholar
  20. 20.
    Pawałowski, K.: Nonlinear smooth group actions on disks, spheres, and Euclidean spaces, Max-Planck-Institut für Mathematik, Bonn, MPI/89-30 (1989)Google Scholar
  21. 21.
    Pawałowski, K.: Manifolds as fixed point sets of smooth compact Lie group actions. Current Trends in Transformation Groups, K-Monographs in Mathematics 7, pp. 79–104. Kluwer, Dordrecht (2002)Google Scholar
  22. 22.
    Schultz, R. (ed.): Group Actions on Manifolds. Contemp. Math. 36 (1985)Google Scholar
  23. 23.
    Smith, P.A.: New results and old problems in finite transformation groups, Bull. Amer. Math. Soc.. Bull. Amer. Math. Soc. 66, 401– (1960)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Stallings, J.: The piecewise-linear structure of Euclidean space. Proc. Cambridge Phil. Soc. 58, 481– (1962)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Stein, E.: On the orbit type in a cirlce action. Proc. Am. Math. Soc. 66, 143– (1977)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceAdam Mickiewicz UniversityPoznanPoland

Personalised recommendations