Differentiable and algebroid cohomology, Van Est isomorphisms, and characteristic classes

Commentarii Mathematici Helvetici

Volumn 78, Issue 4, 2003, Pages 681-721

  Crainic, Marius ^a  

^a UTRECHT UNIVERSITY   (Netherlands)

Author keywords

Characteristic classes; Cohomology; Groupoids; Lie algebroids; Poisson geometry; Van Est isomorphism

Indexed keywords

EID: 0242319576     PISSN: 00102571     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00014-001-0766-9     Document Type: Article

References (51)

1. J. Bernstein and V. Lunts, Equivariant Sheaves and Functors, LNM 1578, 1991.
2. J. M. Bismut and J. Lott, Flat vector bundles, direct images and higher real analytic torsion, J. Amer. Math. Soc. 8 (1995), 291-363.
3. A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Mathematics Studies 94.
4. R. Bott, Lectures on characteristic classes and foliations, Springer LNM 279, 1-94.
5. R. Bott and L. W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics 82, Springer, 1982/1995.
6. H. Cartan, Notions d'algebre différentielle; applications aux groupes de Lie et aux . . ., in: Colloque de Topologie, 15-27, C.B.R.M. Bruxelles, 1950.
7. A. S. Cattaneo and G. Felder, Poisson sigma models and symplectic groupoids, Preprint available as math.SG/0003023
8. A. Connes, A survey of foliations and operator algebras, Proc. Sympos. Pure Math., AMS Providence, 32 (1982), 521-628.
9. M. Crainic, Cyclic cohomology and characteristic classes for foliations, PhD Thesis, Utrecht University, April 2000.
10. M. Crainic, Chern characters via connections up to homotopy, Preprint available as math.DG/0008064
11. M. Crainic, Connections up to homotopy and characteristic classes, Preprint available as math.DG/0010085
12. M. Crainic and I. Moerdijk, A homology theory for étale groupoids, to appear in Crelle's J. Reine Angew. Math. 521 (2000).
13. M. Crainic and I. Moerdijk, Foliation groupoids and their cyclic homology, Advances of Mathematics, to appear.
14. P. Dazord and G. Hector, Integration symplectique des varietes de Poisson totalement aspheriques, in: Symplectic Geometry, Groupoids and Integrable Systems, 37-72, MSRI Publ. 20, 1991.
15. A. Douady and M. Lazard, Espaces fibrés en algèbres de Lie et en groupes, Inv. Math. 1 (1966), 133-151.
16. J. Duistermaat and J. A. C. Kolk, Lie group, Universitext. Springer-Verlag, Berlin, 2000.
17. W. T. van Est, Group cohomology and Lie algebra cohomology in Lie groups I, II, Proc. Kon. Ned. Akad. 56 (1953), 484-504.
18. W. T. van Est, On the algebraic cohomology concepts in Lie groups I, II, Proc. Kon. Ned. Akad. 58 (1955), 225-233, 286-294.
19. W.T. van Est, Une application d'une méthode de Cartan-Leray, Proc. Kon. Ned. Akad. 58 (1955), 542-544.
20. S. Evens, J. H. Lu and A. Weinstein, Transverse measures, the modular class, and a cohomology pairing for Lie algebroids, Quart. J. Math. Oxford 50 (1999), 417-436.
21. R. L. Fernandes, Connections in Poisson Geometry I: Holonomy and Invariants, to appear in J. of Differential Geometry (available as math.DG/0001129)
22. R. L. Fernandes, Lie Algebroids, Holonomy and Characteristic Classes, preprint DG/0007132
23. V. L. Ginzburg and J. H. Lu, Poisson cohomology of Morita-equivalent Poisson manifolds, Internat. Math. Res. Notices (1992), no. 10, 199-205.
24. V. L. Ginzburg and A. Golubev, Holonomy on Poisson Manifolds and the Modular Class, preprint DG/9812153
25. V. L. Ginzburg, Grothendieck Groups of Poisson Vector Bundles, Preprint available as math.DG /0009124
26. A. Haefliger, Differentiable cohomology, "Differential Topologi (CIME, Varenna 1976) Ligouri, Naples", 1979, 19-70.
27. A. Haefliger, Groupoides d'holonomie et espaces classifiants, Astérisque 116 (1984), 70-97.
28. J. Heitsch, A cohomology for foliated manifolds, Bull. Amer. Math. Soc. 79 (1973), 1283-1285.
29. F. Kamber and P. Tondeur, Foliations and metrics, Progress in Mathematics 32, Differential Geometry (1983), 103-152.
30. F. Kamber and P. Tondeur, Foliated Bundles and Characteristic Classes, Springer LNM 493.
31. N. P. Landsman, Mathematical Topics Between Classical and Quantum Mechanics, Springer, New York, 1998.
32. K. Mackenzie, Lie groupoids and Lie algebroids in differential geometry, Lecture Notes Series 124, London Mathematical Society, 1987.
33. K. Mackenzie, Double Lie algebroids and the double of a Lie bialgebroid, Preprint DG/9808081
34. K. Mackenzie and P. Higgins, Algebraic constructions in the category of Lie Algebroids, J. of Algebra 129 (1990), 194-230.
35. K. Mackenzie and P. Xu Integration of Lie bialgebroids, Topology 39 (2000), 445-467.
36. I. Moerdijk, Proof of a conjecture of A. Haefliger, Topology 37 (1998), 735-741.
37. I. Moerdijk and J. Mrcun, On integrability of infinitesimal actions, preprint math.DG/0006042
38. P. Molino, Riemannian foliations, Progress in Mathematics 73, Birkhäuser, Boston, 1988.
39. Moore and Schochet, Global Analysis on Foliated Spaces, Springer-Verlag, 1988.
40. P. Muhly, J. Renault, and D. Williams, Equivalence and isomorphism for groupoid C*-algebras, J. Operator Theory 17 (1987), 3-22.
41. J. Mrcun, Functoriality of the bimodule associated to a Hilsum-Skandalis map, K-Theory 18 (1999), 235-253.
42. V. Nistor, Cyclic cohomology of crossed products by algebraic groups, Inventiones Mathematicae 112 (1993), 615-638.
43. V. Nistor, Groupoids and the integration of Lie algebroids, to appear.
44. V. Nistor, A. Weinstein and P. Xu, Pseudodifferential operators on differential groupoids, Pacific J. Math. 189 (1999), 117-152.
45. J. Renault, A Groupoid Approach to C*-Algebras, Springer LNM 793, 1980.
46. G. Segal, Classifying spaces and spectral sequences, Publ. Math. IHES 34 (1968), 105-112.
47. J. L. Tu, La conjecture de Novikov pour les feuilletages hyperboliques, K-Theory 16 (1999), 129-184.
48. I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Birkhäuser, 1994.
49. A. Weinstein, The local structure of Poisson manifolds, J. Differential Geometry 18 (1983), 523-557.
50. A. Weinstein and P. Xu, Extensions of symplectic groupoids and quantization, J. Reine Angew. Math. 417 (1991), 159-189.
51. P. Xu, Morita Equivalence of Poisson Manifolds, Commun. Math. Phys. 142 (1991), 493-509.