Skip to main content
SpringerLink
Log in

Characteristic invariants of foliated bundles

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

This paper gives a construction of characteristic invariants of foliated principal bundles in the category of smooth and complex manifolds or non-singular algebraic varieties. It contains a generalization of the Chern-Weil theory requiring no use of global connections. This construction leads for foliated bundles automatically to secondary characteristic invariants. The generalized Weil-homomorphism induces a homomorphism of spectral sequences. On the E1-level this gives rise to further characteristic invariants (derived characteristic classes). The new invariants are geometrically interpreted and examples are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Atiyah, M.F.: Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc.85, 181–207 (1957)

    Google Scholar 

  2. Baum, P., Bott, R.: On the zeroes of meromorphic vectorfields, Essays On Topology and Related Topics (dedicated to G. De Rham), 1st Ed. Berlin-Heidelberg-New York: Springer 1970

    Google Scholar 

  3. Borel, A.: Compact Clifford-Klein forms of symmetric spaces, Topology2, 111–122 (1963)

    Google Scholar 

  4. Bott, R.: A residue formula for holomorphic vector-fields, Differential Geometry1, 311–330 (1967)

    Google Scholar 

  5. Bott, R.: On a topological obstruction to integrability, Proc. Symp. Pure Math., Vol.XVI, 127–131 (1970)

    Google Scholar 

  6. Bott, R.: Lectures on characteristic classes and foliations, Springer Lecture Notes279 (1972)

  7. Bott, R.: On the Lefschetz formula and exotic characteristic classes, Proc. of the Diff. Geom. Conf. Rome (1971)

  8. Bott, R., Haefliger, A.: On characteristic classes of Γ-foliations, Bull. Amer. Math. Soc., to appear

  9. Cartan, H.: Cohomologie réelle d'un espace fibré principal différentiable, Séminaire Cartan, exposés 19 et 20 (1949/50)

  10. Chern, S.S.: Geometry of characteristic classes, Proc. Canad. Math. Congress Halifax (1971), to appear

  11. Chern, S.S., Simons, J.: Some cohomology classes in principal fibre bundles and their applications to Riemannian geometry, Proc. Nat. Acad. Sc. USA68, 791–794 (1971)

    Google Scholar 

  12. Chern, S.S., Simons, J.: Characteristic forms and transgression I, to appear

  13. Conlon, L.: Transversally parallelizable foliations of codimension two, to appear

  14. Gelfand, I.M., Fuks, D.B.: The cohomology of the Lie algebra of tangent vector fields of a smooth manifold, I and II, Funct. Anal.3, 32–52 (1969), and4, 23–32 (1970)

    Google Scholar 

  15. Gelfand, I.M., Fuks, D.B.: The cohomology of the Lie algebra of formal vectorfields, Izv. Akad. Nauk SSR34, 322–337 (1970)

    Google Scholar 

  16. Godbillon, C.: Cohomologies d'algèbres de Lie de champs de vecteurs formels, Séminaire Bourbaki (novembre 1972), exposé 421

  17. Godbillon, C., Vey, J.: Un invariant des feuilletages de codimension un, C. R. Ac. Sc. Paris, t.273, 92–95 (1971)

    Google Scholar 

  18. Godement, R.: Théorie des faisceaux, 1st Ed.: Hermann Paris 1958

    Google Scholar 

  19. Greub, W., Halperin, S., Vanstone, R.: Connections, curvature and cohomology, Vo. III, Academic Press, to appear

  20. Grothendieck, A.: Sur quelques points d'algèbre homologique, Tohôku Math. J.9, 119–221 (1957)

    Google Scholar 

  21. Grothendieck, A., Dieudonné, J.: Eléments de géométrie algébrique, Chap. III, Part 1, Publ. Math. IHES 11 (1961), Part 2, ibid. 17(1963)

  22. Grothendieck, A.: On the De Rham cohomology of algebraic varieties, Publ. Math. IHES29, 95–103 (1966)

    Google Scholar 

  23. Grothendieck, A.: Classes de Chern et représentations linéaires des groupes discrets, Six exposés sur la cohomologie des schémas, North Holland, Amsterdam, exp. VIII, 215–305 (1968)

    Google Scholar 

  24. Haefliger, A.: Feuilletages sur les variétés ouvertes, Topology9, 183–194 (1970)

    Google Scholar 

  25. Haefliger, A.: Sur les classes caractéristiques des feuilletages, Séminaire Bourbaki (juin 1972), exposé 412

  26. Heitsch, J.L.: Deformations of secondary characteristic classes, Topology, to appear

  27. Illusie, L.: Complexe cotangent et déformations, I et II, Lecture Notes in Math. 239 (1971) and 283 (1972) Springer

  28. Kamber, F., Tondeur, Ph.: Flat manifolds, Springer Lecture Notes67, 1–53 (1968)

    Google Scholar 

  29. Kamber, F., Tondeur, Ph.: Invariant differential operators and cohomology of Lie algebra sheaves, Differentialgeometrie im Grossen, Juli 1969, Berichte aus dem Math. Forschungsinstitut Oberwolfach, Heft 4, Mannheim, 177–230(1971)

  30. Kamber, F., Tondeur, Ph.: Invariant differential Operators and the cohomology of Lie algebra sheaves, Memoirs Amer. Math. Soc.113, 1–125 (1971)

    Google Scholar 

  31. Kamber, F., Tondeur, Ph.: Characteristic classes of modules over a sheaf of Lie algebras, Notices Amer. Math. Soc.19, A-401 (February 1972)

    Google Scholar 

  32. Kamber, F., Tondeur, Ph.: Characteristic invariants of foliated bundles, preprint University of Illinois (August 1972)

  33. Kamber, F., Tondeur, Ph.: Derived characteristic classes of foliated bundles, preprint University of Illinois (August 1972)

  34. Kamber, F., Tondeur Ph.: Cohomologie des algèbres de Weil relatives tronquées, C. R. Ac. Sc. Paris, t.276, 459–462 (1973)

    Google Scholar 

  35. Kamber, F., Tondeur, Ph.: Algèbres de Weil semi-simpliciales, C. R. Ac. Sc. Paris, t.276, 1177–1179 (1973); Homomorphisme caractéristique d'un fibré principal feuilleté, ibid. t.276, 1407–1410 (1973); Classes caractéristiques dérivées d'un fibré principal feuilleté, ibid. t.276, 1449–1452 (1973)

    Google Scholar 

  36. Katz, N.M., Oda, T.: On the differentiation of De Rham cohomology classes with respect to parameters, J. Math. Kyoto Univ.8-2, 199–213 (1968)

    Google Scholar 

  37. Koszul, J.L.: Déformations et connexions localement plates, Ann. Inst. Fourier, Grenoble18, 103–114 (1968)

    Google Scholar 

  38. Lehmann, D.: J-homotopie dans les espaces de connexions et classes exotiques de Chern-Simons, C. R. Ac. Sc. Paris, t.275, 835–838 (1972)

    Google Scholar 

  39. Lehmann, D.: Classes caractéristiques exotiques et J-connexité des espaces de connexions, to appear

  40. Martinet, J.: Classes caractéristiques des systèmes de Pfaff, to appear

  41. Matsushima, Y.: On Betti numbers of compact locally symmetric Riemannian manifolds, Osaka Math. J.14, 1–20 (1962)

    Google Scholar 

  42. Molino, P.: Connexions et G-structures sur les variétés feuilletées, Bull. Soc. Math. France92, 59–63 (1968)

    Google Scholar 

  43. Molino, P.: Classes d'Atiyah d'un feuilletage et connexion transverses projetables, C. R. Ac. Sc. Paris, t.272, 779–781 (1971).

    Google Scholar 

  44. Molino, P.: Classes caractéristiques et obstructions d'Atiyah pour les fibrés principaux feuilletés, C. R. Ac. Sc. Paris, t.272, 1376–1378 (1971)

    Google Scholar 

  45. Molino, P.: Propriétés cohomologiques et propriétés topologiques des feuilletages à connexion transverse projetable, to appear

  46. Pasternack, J. S.: Foliations and compact Lie group actions, Comment. Math. Helv.46, 467–477 (1971)

    Google Scholar 

  47. Reinhart, B. L.: Foliated manifolds with bundle-like metrics, Ann. of Math.69, 119–132 (1959)

    Google Scholar 

  48. Reinhart, B. L.: Harmonic integrals on foliated manifolds, Amer. J. of Math.81, 529–536 (1959)

    Google Scholar 

  49. Vaisman, I.: Sur la cohomologie des variétés Riemanniennes feuilletées, C. R. Ac. Paris, t.268, 720–723 (1969)

    Google Scholar 

  50. Vaisman, I.: Sur une classe de complexes de cochaines, Math. Ann.194, 35–42 (1971)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Illinois, 61801, Urbana, Illinois

    Franz W. Kamber & Philippe Tondeur

  2. Forschungsinstitut für Mathematik, Eidg. Technische Hochschule, 8006, Zürich

    Franz W. Kamber & Philippe Tondeur

Authors
  1. Franz W. Kamber
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Philippe Tondeur
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Text of lectures given during the meeting on “Exotic Characteristic Classes” in Lille, February 1973.

This work was partially supported by a grant from the National Science Foundation and by the Forschungsinstitut für Mathematik of the ETH in Zürich.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kamber, F.W., Tondeur, P. Characteristic invariants of foliated bundles. Manuscripta Math 11, 51–89 (1974). https://doi.org/10.1007/BF01189091

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01189091

Keywords

Search

Navigation