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
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
A Borel, Seminar on Transformation Groups, Ann. of Math. Studies, No. 46, Princeton Univ. Press, 1960.
W. Browder, S1-actions on open manifolds, Contemporary Math. 37 (1985), 25–30.
W. Browder and W. C. Hsiang, G-actions and the fundamental group, Invent. Math. 65 (1982), 411–424.
W. Browder and F. Quinn, A surgery theory for G-manifolds and stratified sets, Manifolds Tokyo, Univ. of Tokyo Press, 1973, pp. 27–36.
P. E. Conner and E. E. Floyd, Differentiable Periodic Maps, Springer-Verlag, Berlin and New York, 1964.
W. Y. Hsiang, Cohomology Theory of Topological Transformation Groups, Springer-Verlag, Berlin and New York, 1975.
K. Kawakubo, Equivariant Riemannian-Rock type theorems and related topics, London Math. Soc. Lecture Note Ser. 26 (1977), 284–294.
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.
K. Kawakubo and F. Uchida, On the index of a semi-free S1-action, Proc. Japan Academy 46 (1970), 620–622.
S. Kobayashi, Transformation Groups in Differential Geometry, Springer-Verlag, Berlin and New York, 1972.
H. T. Ku and M. C. Ku, Group actions on Ak-manifolds, Trans. Amer. Math. Soc. 245 (1978), 469–492.
H. T. Ku and M. C. Ku, Group actions on aspherical Ak(N)-manifolds, Trans. Amer. Math. Soc. 278 (1983), 841–859.
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.
F. Raymond, The orbit space of totally disconnected groups of transformations on manifolds, Proc. Amer. Math. Soc. 12 (1961), 1–7.
J. C. Su, Integral weight system of S1 action on cohomology projective spaces, Chinese J. Math. 2 (1974), 77–112.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights 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
