manuscripta mathematica

, Volume 30, Issue 4, pp 425–445

Triangulation of stratified fibre bundles

  • Andrei Verona
Article
  • 57 Downloads

Abstract

We prove that for any stratified fibre bundle p:A·M (A being the underlying space of an abstract prestratification and M a smooth manifold) and any triangulation of M there exists a triangulation of A such that p becomes linear with respect to these triangulations. In particular, any abstract prestratification is triangulable. As a corollary we obtain that the orbit space of a smooth action of a compact Lie group is triangulable.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    CAIRNS, S.S.: Triangulated manifolds and differentiable manifolds. Lectures in Topology, University of Michigan Press, Ann Arbor, Mich., 1941, pp. 143–157Google Scholar
  2. 2.
    GIBSON, C.G., WIRTHMÜLLER, K., DU PLESSIS, A.A., LOOIJENGA, E.J.N.: Topological stability of smooth mappings. Lecture Notes in Math. 552, Springer-Verlag, Berlin-Heidelberg-New York, 1976Google Scholar
  3. 3.
    GORESKY, M.: Triangulation of stratified objects. Proc. Amer. Math. Soc. 72, 193–200 (1978)Google Scholar
  4. 4.
    HARDT, R.M.: Triangulation of subanalytic sets and proper light subanalytic maps. Inventiones math. 38, 207–217 (1977)Google Scholar
  5. 5.
    HENDRICKS, E.: Triangulation of stratified sets. Thesis, M.I.T. 1973Google Scholar
  6. 6.
    HUDSON, J.F.RP.: Piecewise linear topology. Benjamin, New York, 1969Google Scholar
  7. 7.
    ILLMAN, S.: Smooth equivalent triangulations of G-manifolds for G a finite group. Math. Ann. 233, 199–220 (1978)Google Scholar
  8. 8.
    JOHNSON, F.E.A.: Triangulation of stratified sets and other questions in geometric topology. Thesis, University of Liverpool, 1972Google Scholar
  9. 9.
    KATO, M.: Elementary topology of analytic sets, Sugaku 25, 38–51 (1973)Google Scholar
  10. 10.
    LELLMAN, N.W.: Orbitraüme von G-Mannigfaltigkeiten und stratifizierte Mengen. Diplomarbeit, Bonn 1975Google Scholar
  11. 11.
    MATHER, J.: Notes on topological stability. Mimeographed Notes, Harvard, 1970Google Scholar
  12. 12.
    MUNKRESS, J.R.: Elementary differential topology. Annals of Math. Study 54, Princeton University Press, Princeton, 1966Google Scholar
  13. 13.
    PUTZ, H.: Triangulation of fibre bundles. Canadian Journal of Math. 19, 499–513 (1967)Google Scholar
  14. 14.
    SIEBENMANN, L.C.: Topological manifolds. In: Proc.I.C.M. Nice (1970), Vol.2, 133–163, Gauthiers-Villars, Paris, 1971Google Scholar
  15. 15.
    VERONA, A.: Homological properties of abstract prestratifications. Rev. Roum. Math. Pures et Appl. 17, 1109–1121 (1972)Google Scholar
  16. 16.
    YANG, C.T.: The triangulability of the orbit space of a differentiable transformatio group. Bull. Amer. Math. Soc. 69, 405–408 (1963)Google Scholar
  17. 17.
    MATUMOTO, T.: Equivariant stratification of a compact differentiable transformation group. Preprint 1977Google Scholar
  18. 18.
    THOM, R.: Ensembles et morphismes stratifies. Bull. Amer. Math. Soc. 75, 240–281 (1969)Google Scholar
  19. 19.
    THOM, R.: Stratified sets and morphisms: Local models. Lecture Notes in Math. 192, 153–164, Springer-Verlag, Berlin-Heidelberg-New York, 1971Google Scholar
  20. 20.
    ULLMAN, W.: Triangulability of abstract prestratified sets and the stratification of the orbit space of a G-manifold. Thesis, Universität Bonn, Mathematisches Institut, 1973Google Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Andrei Verona
    • 1
  1. 1.Department of MathematicsNational Institute for Scientific and Technical CreationBucharestRomania

Personalised recommendations