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Differential Cohomology

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Differential Topology

Part of the book series: C.I.M.E. Summer Schools ((CIME,volume 73))

Abstract

This is a slightly expanded version of two lectures given at the Institute for Advanced Study of Princeton in the fall 1972. Some of this material was supposed to be included in a joint paper with R. Bott on smooth cohomology.

These notes do not contain any concrete new result. We just try to explain the philosophy of differentiable cohomology.

Namely one would like to compute the cohomology of classifying spaces Br of some topological groupoids T which plays an important role in the problem of constructing foliations (cf. [45]et[24] ). Unfortunetely this seems to be extremely difficult.

Now the classifying space BΓ(or rather the groupoïd ʓ) carry also a softer topology with some kind of differentiable structure. In the complex of cochains giving the cohomology of BΓ, one can single out the subcomplex of smooth cochains whose cohomology, called the differentiable or smooth cohomology of BΓ, is computed in many important cases, because it is isomorphic Go the cohomology of some Lie algebra associated to Γ. This is proved as the theorem of Van Est (cf. [12] ) relating the differentiable cohomology of a Lie group to the cohomology of its Lie algebra. The titel of this course could also have been : –Variations on a theorem of Van Est–.

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References

  1. Atiyah-Wall. - Cohomology of groups. Algebraic number theory edited by Cassels and Frölich - Academic Press (1967), 94–115.

    Google Scholar 

  2. I.N. Bernstein and B.I..Rosenfeld- On Characteric classes of foliations - Functional Analysis and applications, 6 (1972), 68–69

    Google Scholar 

  3. A. Borel - Compact Clifford-Klein Forms of Symmetric Spaces, Topology 2 (1963), 111–122.

    Article  MathSciNet  MATH  Google Scholar 

  4. R. Bott - A topological obstruction to integrability - Proc. Symp. Pure Math. Vol XVI (1970) 127–131.

    MathSciNet  Google Scholar 

  5. R. Bott - Lectures on Characteristic Classes and Foliations -Springer Lectures Notes No 279 (1972).

    Google Scholar 

  6. R. Bott and A. Haefliger - On Characteristic Classes of Γ-foliations, Bull. AMS (1972), 1039–1044.

    Google Scholar 

  7. R. Bott - On the Chern-Weil homomorphism and the continuous cohomology of Lie groups. Advances in Math 11 (1973), p. 289–303.

    Article  MathSciNet  MATH  Google Scholar 

  8. J.P. Buffet-J.C. Lor - Une construction d'un universel pour une classe assez large de Γ-structures - C.R. Acad. Sc. Paris 270 A 640–2.

    Google Scholar 

  9. C. Chevalley-S. Eilenberg. Cohomology theory of Lie groups and Lie algebras - Trans. AMS, 63 (1948) 85–124.

    Article  MathSciNet  MATH  Google Scholar 

  10. Dupont - Simplicial de Rham cohomology and characteristic classes of flat bundles, Topology 15 (1976) 233–245.

    Article  MathSciNet  MATH  Google Scholar 

  11. C. Ehresmann. Les connexions infinitesimales dans un fibré différentiable Colloque de Topologie de Bruxelles 1950. CBRM, 29–55.

    Google Scholar 

  12. Van Est - On the algebraic cohomology concepts in Lie groups. Nedel Akad. Wetensch. Proc. Ser. A 58 (1955) I, 225–233 ; II, 286–294.

    MathSciNet  Google Scholar 

  13. Van Est - Une application d'une méthode de Cartan-Leray, Indag. Math. 18 (1955), p. 542–544.

    MathSciNet  Google Scholar 

  14. I.M. Gelfand-D.B. Fuchs. The cohomologies of the Lie algebra of the vector fields on the circle. Funct. Analiz. 3 No 4 (1968) 92–93.

    Google Scholar 

  15. I.M. Gelfand-D.B. Fuchs. The cohomology of the Lie algebra of formal vector fields, Izvestia An. CGCR - 34 (1970), 322–337.

    Google Scholar 

  16. I.M. Gelfand-D.B. Fuchs. The cohomology of the Lie algebra of tangent vector fields on a smooth manifold, I and II - Functional Analysis - Vol. 3 No 3 (1969), 32–52 and vol. 4 No 2 (1970) 23–32.

    Google Scholar 

  17. I.M. Gelfand-B.L. Feigin-D.B. Fuchs. Cohomologies of the Lie Algebra of Formal vector fields with coefficients in its adjoint space and variations of characteristic classes of foliations. funct. Analysis 8 (No 2) (1974) 13–29.

    Google Scholar 

  18. G. Godbillon-J.Vey. Un invariant des feuilletages de codimension un, C.R. Acad. Sc. Paris Juin 1971.

    Google Scholar 

  19. G, Godbillon. Cohomologie d'algebre de Lie de champs de vecteurs formels, Seminaire Bourbaki 1972–1973, expose 421.

    Google Scholar 

  20. R. Godement. Theorie des faisceaux, Hermann Paris (1964).

    Google Scholar 

  21. A. Grotlendieck. Sur quelques points d'algebre homologique. Tohoku Math. Journ. 9 (1957), 119–183.

    Google Scholar 

  22. V. Guillemin-S. Sternberg. Deformation Theory of Pseudogroup structures, Memoires AMS No 64 (1966).

    Google Scholar 

  23. V. Guillemin, Cohomology of vector fields on Manifolds, Advance in Math 10, 192–220 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Haefliger, Homotopy and Integrability - Lecture Notes in Math. No 197 Springer-Verlag (1971, 133–163.

    Google Scholar 

  25. A. Haefliger. Sur les classes caractéristiques des feuilletages Seminaire Bourbaki annee 1971–1972, No 412.

    Google Scholar 

  26. A. Haefliger. Sur la cohomologie de Gelfand-Fuchs.Differential topology and geometry - Dijon 1974- Springer-Verlag- Lecture Notes No 484.

    Google Scholar 

  27. J. Heitsch - Variation of Secondary classes (to appear).

    Google Scholar 

  28. G. Hochschild - The structure of Lie groups. Holden-Day Inc. 1965.

    MATH  Google Scholar 

  29. G. Hochshild-G.D. Mostow - Cohomology of Lie groups - Illinois Journ. 6 (1962) 367–401.

    Google Scholar 

  30. F. Kamber-P. Tondeur. Non trivial Characteristic Invariants of homogeneous foliated Bundles, Ann. Ec. Norm. Sup. 8. (1975) 433–486.

    MathSciNet  MATH  Google Scholar 

  31. F. Kamber-P. Tondeur. On the linear independence of certain cohomology classes of Br. To appear.

    Google Scholar 

  32. Kobayashi-Nomizu. Foundations of differential geometry I and II Interscience Tracts in Pure and Applied Math. (1963).

    Google Scholar 

  33. J.L. Koszul. Homologie et Cohomologie des algebres de Lie. Bull. Soc. Math. de France 78 1950, 65–127.

    MathSciNet  MATH  Google Scholar 

  34. J.L. Koszul. Homologie des formes différentielles d'ordre superieur. Ann. Sc. Ec. Norm. Sup. 7 (1974) 139–153.

    MathSciNet  MATH  Google Scholar 

  35. M.V. Losik, Cohomology of the Lie algebra of vector fields with coefficients in a trivial unitary representation -Functional Analysis 6 (1972), 24–36.

    MathSciNet  Google Scholar 

  36. S. MacLane, Homology. Grundlehren der mathematishen Wiss, Springer 1967.

    Google Scholar 

  37. J. Mather. On Haefliger's classifying Space I. Bull. AMS 77 (1971) 1111–1115.

    Article  MathSciNet  MATH  Google Scholar 

  38. J. Mather. Integrability in codimension I, Coram, Math, Helv, 48 (1973) 195–233.

    Article  MathSciNet  MATH  Google Scholar 

  39. J. Mather. Commutators of diffeomorphisms I and II, Comm. Math. Helv. 49, (1974), p. 512–528 and 50 (1975) 33–40

    Article  MathSciNet  MATH  Google Scholar 

  40. G.D.Mostow. Cohomology of Topological groups and Solvmanifolds. Ann. of Math, (1961), 20–48.

    Google Scholar 

  41. P. Molino. Classes caracteristiques et obstructions d'Atiyah pour les fibres principaux feuilletes. C.R. Acad. Sc. Paris 272 (1971).

    Google Scholar 

  42. C. Roger. Méthodes homotopiques et cohomologiques en théorie des feuilletages. Thèse (to appear).

    Google Scholar 

  43. W. Thurston. Non Cobordant Foliations of S3, Bull. AMS 78 (1972), 511–514.

    Article  MathSciNet  MATH  Google Scholar 

  44. W. Thurston. Foliations and groups of diffeomorphisms, Bull. AMS 81 (1975).

    Google Scholar 

  45. W. Thurston. The theory of foliations in codimension greater than one, Comm. Math. Helv. 49 (1974) 214–231.

    Article  MathSciNet  MATH  Google Scholar 

  46. W. Thurston. Existence of codimension one foliations, Ann. of Math. (to appear).

    Google Scholar 

  47. G. Segal. Classifying spaces and spectral sequences. IHES - Publications mathematiques No 34 (L l05–112.

    Google Scholar 

  48. N. Steenrod. The Topology of Fiber Bundles. Princeton University Press 1951.

    Google Scholar 

  49. J. Vey, Sur la cohomologie des champs de vecteurs symplectiques formels C.R. Acad. Sc. Paris 280 (1975) 805.

    Google Scholar 

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Authors and Affiliations

  1. Section de mathématiques, Université dè Geneve, 2-4, rue du Lièvre, 124, IL, Geneve

    André Haefliger

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  1. André Haefliger
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Vinicio Villani (Coordinatore)

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Haefliger, A. (2010). Differential Cohomology. In: Villani, V. (eds) Differential Topology. C.I.M.E. Summer Schools, vol 73. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11102-0_3

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