Skip to main content

The pontrjagin numbers of an orbit map and generalized G-signature theorem

  • 471 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 1375)

Abstract

In this paper we study the relationships between the Pontrjagin numbers of an orbit map and the existence of a non-empty fixed point set. For smooth torus group actions G on M we obtain an explicit formula for the Pontrjagin numbers pI(M,z) and proving a generalized G-signature theorem. We also define the characteristic numbers of an orbit map in the topology category. We then show some results about the existence of non-empty fixed points for torus group and compact connected abelian group actions.

Keywords

  • Topology Category
  • Torus Group
  • Rational Cohomology
  • Pontrjagin Class
  • Finite Orbit

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   46.00
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A Borel, Seminar on Transformation Groups, Ann. of Math. Studies, No. 46, Princeton Univ. Press, 1960.

    Google Scholar 

  2. W. Browder, S1-actions on open manifolds, Contemporary Math. 37 (1985), 25–30.

    CrossRef  MathSciNet  Google Scholar 

  3. W. Browder and W. C. Hsiang, G-actions and the fundamental group, Invent. Math. 65 (1982), 411–424.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. W. Browder and F. Quinn, A surgery theory for G-manifolds and stratified sets, Manifolds Tokyo, Univ. of Tokyo Press, 1973, pp. 27–36.

    MATH  Google Scholar 

  5. P. E. Conner and E. E. Floyd, Differentiable Periodic Maps, Springer-Verlag, Berlin and New York, 1964.

    CrossRef  MATH  Google Scholar 

  6. W. Y. Hsiang, Cohomology Theory of Topological Transformation Groups, Springer-Verlag, Berlin and New York, 1975.

    CrossRef  MATH  Google Scholar 

  7. K. Kawakubo, Equivariant Riemannian-Rock type theorems and related topics, London Math. Soc. Lecture Note Ser. 26 (1977), 284–294.

    MathSciNet  Google Scholar 

  8. K. Kawakubo and F. Raymond, The index of manifolds with toral actions and geometric interpretations of the σ(∞,S1,Mn) invariant of Atiyah-Singer, Lecture Notes in Math., vol. 298, Springer-Verlag, Berlin and New York, 1972, pp. 228–233.

    Google Scholar 

  9. K. Kawakubo and F. Uchida, On the index of a semi-free S1-action, Proc. Japan Academy 46 (1970), 620–622.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. S. Kobayashi, Transformation Groups in Differential Geometry, Springer-Verlag, Berlin and New York, 1972.

    CrossRef  MATH  Google Scholar 

  11. H. T. Ku and M. C. Ku, Group actions on Ak-manifolds, Trans. Amer. Math. Soc. 245 (1978), 469–492.

    MathSciNet  MATH  Google Scholar 

  12. H. T. Ku and M. C. Ku, Group actions on aspherical Ak(N)-manifolds, Trans. Amer. Math. Soc. 278 (1983), 841–859.

    MathSciNet  MATH  Google Scholar 

  13. M. C. Ku, On the action of compact groups, Proc. of Conf. on Trans. Groups, New Orleans 1967, Springer-Verlag, Berlin and New York, pp. 381–400.

    Google Scholar 

  14. F. Raymond, The orbit space of totally disconnected groups of transformations on manifolds, Proc. Amer. Math. Soc. 12 (1961), 1–7.

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. J. C. Su, Integral weight system of S1 action on cohomology projective spaces, Chinese J. Math. 2 (1974), 77–112.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1989 Springer-Verlag

About this paper

Cite this paper

Ku, HT., Ku, MC. (1989). The pontrjagin numbers of an orbit map and generalized G-signature theorem. In: Kawakubo, K. (eds) Transformation Groups. Lecture Notes in Mathematics, vol 1375. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0085610

Download citation

  • DOI: https://doi.org/10.1007/BFb0085610

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51218-9

  • Online ISBN: 978-3-540-46178-4

  • eBook Packages: Springer Book Archive