Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

$G$-foliations and their characteristic classes


Authors: Franz W. Kamber and Philippe Tondeur
Journal: Bull. Amer. Math. Soc. 84 (1978), 1086-1124
MSC (1970): Primary 57D20; Secondary 57D30
DOI: https://doi.org/10.1090/S0002-9904-1978-14546-7
MathSciNet review: 508449
Full-text PDF

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • [A] V. I. Arnold, Characteristic class entering in quantization conditions, Functional Anal. Appl. 1 (1967), 1-13. MR 211415
  • [BA] P. Baum, On the cohomology of homogeneous spaces, Topology 7 (1968), 15-38. MR 219085
  • [BK] D. Baker, On a class of foliations and the evaluation of their characteristic classes, Bull. Amer. Math. Soc. 83 (1977), 394-396. MR 438351
  • [BO] A. Borel, Compact Clifford-Klein forms of symmetric spaces, Topology 2 (1963), 11-122. MR 146301
  • [BR1] I. N. Bernstein and B. I. Rosenfeld, Characteristic classes of foliations, Functional Anal. Appl. 6 (1972), 68-69. MR 296967
  • [BR2] I. N. Bernstein and B. I. Rosenfeld, Homogeneous spaces of infinite-dimensional Lie algebras and characteristic classes of foliations, Russian Math. Surveys 28 (1973), 107-142. MR 415633
  • [B1] R. Bott, Lectures on characteristic classes and foliations, Lectures on Algebraic and Differential Topology, Lecture Notes in Math., vol. 279, Springer-Verlag, Berlin and New York, 1972, pp. 1-94. MR 362335
  • [B2] R. Bott, The Lefschetz formula and exotic characteristic classes, Sympos. Math. Vol. X, Rome, 1972, pp. 95-105. MR 362336
  • [B3] R. Bott, On the Chern-Weil homomorphism and the continuous cohomology of Lie groups, Advances in Math. 11 (1973), 289-303. MR 345115
  • [B4] R. Bott, Gelfand-Fuchs cohomology and foliations, Proc. of the 11th Ann. Holiday Sympos. at New Mexico State University (Dec. 1973), 1975, pp. 1-220.
  • [B5] R. Bott, On characteristic classes in the framework of Gelfand-Fuchs cohomology, Société Mathématique de France, Astérisque 32-33 (1976), 113-139. MR 402768
  • Raoul Bott, On some formulas for the characteristic classes of group-actions, Differential topology, foliations and Gelfand-Fuks cohomology (Proc. Sympos., Pontifícia Univ. Católica, Rio de Janeiro, 1976) Lecture Notes in Math., vol. 652, Springer, Berlin, 1978, pp. 25–61. MR 505649
  • [BH] R. Bott and A. Haefliger, On characteristic classes of Γ-foliations, Bull. Amer. Math. Soc. 78 (1972), 1039-1044. MR 307250
  • [CA] H. Cartan, Cohomologie réelle d'un espace fibré principal différentiable, Séminaire Cartan, Exposés 19 et 20 (1949/50).
  • [CE] H. Cartan and S. Eilenberg, Homological Algebra, Princeton Univ. Press, Princeton, N. J., 1956. MR 77480
  • [CH] S. S. Chern, The geometry of G-structures, Bull. Amer. Math. Soc. 72 (1966), 167-219. MR 192436
  • [CS] S. S. Chern and J. Simons, Characteristic forms and geometric invariants, Ann. of Math. (2) 99 (1974), 48-69. MR 353327
  • [CL] L. Conlon, Transversally parallelizable foliations of codimension two, Trans. Amer. Math. Soc. 194 (1974), 79-102. MR 370617
  • [D] T. Duchamp, Characteristic invariants of G-foliations, Ph.D. thesis, University of Illinois, Urbana, Illinois, 1976.
  • [DU] J. L. Dupont, Simplicial De Rham cohomology and characteristic classes of flat bundles, Topology 15 (1976), 233-245. MR 413122
  • [F1] D. B. Fuks, Finite-dimensional Lie algebras of formal vectorfields and characteristic classes of homogeneous foliations, Izv. Akad. Nauk SSSR 40 (1976), 55-62. MR 413125
  • [F2] D. B. Fuks, Non-trivialité des classes caractéristiques de g-structures. Applications aux classes caractéristiques de feuilletages, C.R. Acad. Sci. Paris 284 (1977), 1017-1019. MR 433469
  • [F3] D. B. Fuks, Non-trivialité des classes caractéristiques de g-structures. Applications aux variations des classes caractéristiques de feuilletages, C.R. Acad. Sci. Paris 284 (1977), 1105-1107. MR 433470
  • [GF1] I. M. Gelfand and D. B. Fuks, The cohomology of the Lie algebra of tangent vectorfields on a smooth manifold, Functional Anal. Appl. 3 (1969), 32-52; ibid. 4 (1970), 23-32. MR 287589
  • [GF2] I. M. Gelfand and D. B. Fuks, The cohomology of the Lie algebra of formal vectorfields, Izv. Akad. Nauk SSSR 34 (1970), 322-337. MR 266195
  • [GB] C. Godbillon, Cohomologies d'algèbres de Lie de champs de vecteurs, Séminaire Bourbaki, 25e année (1971/72), no. 421.
  • [GV] C. Godbillon and J. Vey, Un invariant des feuilletages de codimension un, CR. Acad. Sci. Paris 273 (1971), 92-95. MR 283816
  • [H1] A. Haefliger, Feuilletages sur les variétés ouvertes, Topology 9 (1970), 183-194. MR 263104
  • [H2] A. Haefliger, Sur les classes caractéristiques des feuilletages, Séminaire Bourbaki, 24e année (1971/72), n° 412.
  • [H3] A. Haefliger, Sur la cohomologie de Gelfand-Fuks, Dijon (1974), Springer Lecture Notes 484 (1975), 121-152. MR 394698
  • [H4] A. Haefliger, Sur la cohomologie de l'algèbre de Lie des champs de vecteurs, Ann. Sci. École Norm. Sup. 9 (1976), 503-532. MR 448367
  • [H5] A. Haefliger, Differentiable cohomology, Cours donné au C.I.M.E., 1976.
  • [HT1] J. Heitsch, Deformations of secondary characteristic classes, Topology 12 (1973), 381-388. MR 321106
  • James L. Heitsch, Derivatives of secondary characteristic classes, J. Differential Geom. 13 (1978), no. 3, 311–339. MR 551563
  • [HT3] J. Heitsch, Independent variation of secondary characteristic classes (to appear).
  • [K1] J. L. Koszul, Homologie et cohomologie des algèbres de Lie, Bull. Soc. Math. France 78 (1950), 65-127. MR 36511
  • [K2] J. L. Koszul, Sur un type d'algèbres différentielles en rapport avec la transgression, Coll. de Topologie, Bruxelles (1950), pp. 73-81. MR 42428
  • [K3] J. L. Koszul, Déformations et connexions localement plates, Ann. Inst. Fourier (Grenoble) 18 (1968), 103-114. MR 239529
  • [KT1] F. Kamber and Ph. Tondeur, Flat manifolds, Lecture Notes in Math., vol. 67, Springer-Verlag, Berlin and New York, 1968. MR 238342
  • [KT2] F. Kamber and Ph. Tondeur, Invariant differential operators and cohomology of Lie algebra sheaves, Differentialgeometrie im Grossen, Juli 1969, Berichte aus dem Math. Forschungsinstitut Oberwolfach, Heft 4, Mannheim (1971), 177-230. MR 370616
  • [KT3] F. Kamber and Ph. Tondeur, Invariant differential operators and cohomology of Lie algebra sheaves, Mem. Amer. Math. Soc. No. 113 (1971). MR 413129
  • [KT4] F. Kamber and Ph. Tondeur, Characteristic classes of modules over a sheaf of Lie algebras, Notices Amer. Math. Soc. 19 (1972), A-401.
  • [KT5] F. Kamber and Ph. Tondeur, Cohomologie des algèbres de Weil relatives tronquées, C.R. Acad. Sci. Paris 276 (1973), 459-462. MR 321127
  • [KT6] F. Kamber and Ph. Tondeur, Algèbres de Weil semi-simpliciales, C.R. Acad. Sci. Paris 276 (1973), 1177-1179; Homomorphisme caractéristique d'un fibré principal feuilleté, ibid. 1407-1410; Classes caractéristiques dérivées d'un fibré principal feuilleté, ibid. 1449-1452. MR 321128
  • [KT7] F. Kamber and Ph. Tondeur, Characteristic invariants of foliated bundles, Manuscripta Math. 11 (1974), 51-89. MR 334237
  • [KT8] F. Kamber and Ph. Tondeur, Semi-simplicial Weil algebras and characteristic classes for foliated bundles in Čech cohomology, Proc. Sympos. Pure Math. Vol. 27, Part 1, Amer. Math. Soc., Providence, R. I., 1975, pp. 283-294. MR 385886
  • [KT9] F. Kamber and Ph. Tondeur, Foliated bundles and characteristic classes, Lecture Notes in Math., vol. 493, Springer-Verlag, Berlin and New York, 1975, pp. 1-208. MR 402773
  • [KT10] F. Kamber and Ph. Tondeur, Classes caractéristiques generalisées des fibrés feuilletés localement homogènes, C.R. Acad. Sci. Paris 279 (1974), 847-850. Quelques classes caractéristiques generalisées nontriviales de fibrés feuilletés, ibid. 921-924. MR 368030
  • [KT11] F. Kamber and Ph. Tondeur, Non-trivial characteristic invariants of homogeneous foliated bundles, Ann. Sci. École Norm. Sup. 8 (1975), 433-486. MR 394700
  • [KT12] F. Kamber and Ph. Tondeur, On the linear independence of certain cohomology classes of BГ, Studies in algebraic topology, Advances in Mathematics Supplementary Studies, Vol. 5, 1979, 213—263.
  • Franz W. Kamber and Philippe Tondeur, Semi-simplicial Weil algebras and characteristic classes, Tôhoku Math. J. (2) 30 (1978), no. 3, 373–422. MR 509023, https://doi.org/10.2748/tmj/1178229977
  • [KT14] F. Kamber and Ph. Tondeur, Classes caractéristiques et suites spectrales d'Eilenberg-Moore, C.R. Acad. Sci. Paris 283 (1976), 883-886. MR 431201
  • Franz W. Kamber and Philippe Tondeur, Characteristic classes and Koszul complexes, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 159–166. MR 520532
  • [KT16] F. Kamber and Ph. Tondeur, Cohomology of g-DG-algebras (to appear).
  • [KT17] F. Kamber and Ph. Tondeur, Characteristic classes of complex analytic foliated bundles (to appear).
  • [L1] H. B. Lawson, Foliations, Bull. Amer. Soc. 80 (1974), 369-418. MR 343289
  • [L2] H. B. Lawson, Lectures on the quantitative theory of foliations, CBMS Regional Conf. Series in Math., vol. 27, 1977. MR 448368
  • [LN] D. Lehmann, Classes caractéristiques exotiques et J-connexité des espaces de connexions, Ann. Inst Fourier (Grenoble) 24 (1974), 267-306. MR 362342
  • [LP1] C. Lazarov and J. Pasternack, Secondary characteristic classes for Riemannian foliations, J. Differential Geometry 11 (1976), 365-385. MR 445513
  • [LP2] C. Lazarov and J. Pasternack, Residues and characteristic classes for Riemannian foliations, J. Differential Geometry 11 (1976), 599-612. MR 445514
  • [MO1] P. Molino, Classes d'Atiyah d'un feuilletage et connexions transverses projetables, C.R. Acad. Sci. Paris 272 (1971), 779-781. MR 281224
  • [MO2] P. Molino, Classes caractéristiques et obstructions d'Atiyah pour les fibres principaux feuilletés, C.R. Acad. Sci. Paris 272 (1971), 1376-1378. MR 287571
  • [MO3] P. Molino, Feuilletages et classes caractéristiques, Sympos. Math. Vol. X, Rome (1972), 199-209. MR 358806
  • [MO4] P. Molino, Propriétés cohomologiques et propriétés topologiques des feuilletages a connexion transverse projetable, Topology 12 (1973), 317-325. MR 353332
  • [MR] S. Morita, A remark on the continuous variation of secondary characteristic classes of foliations, J. Math. Soc. Japan 29 (1977), 253-260. MR 442952
  • [MS] V. P. Maslov, Théorie des perturbations et méthodes asymptotiques, Gauthier-Villars, Paris, 1972.
  • [MT] Y. Matsushima, On Betti numbers of compact locally symmetric Riemannian manifolds, Osaka J. Math. 14 (1962), 1-20. MR 141138
  • [NS] S. Nishikawa and H. Sato, On characteristic classes of Riemannian, conformal and projective foliations, J. Math. Soc. Japan 28 (1976), 223-241. MR 400255
  • [P] J. Pasternack, Foliations and compact Lie group actions, Comment Math. Helv. 46 (1971), 467-477. MR 300307
  • [RE] R. Reinhart, Foliated manifolds with bundle-like metrics, Ann. of Math. (2) 69 (1959), 119-132. MR 107279
  • [ST] H. Shulman and D. Tischler, Leaf invariants of foliations and the Van Est isomorphism, J. Differential Geometry 11 (1976), 535-546. MR 451262
  • [TH] W. Thurston, Noncobordant foliations of S, Bull. Amer. Math. Soc. 78 (1972), 511-514. MR 298692
  • [Y] K. Yamato, Examples of foliations with non-trivial exotic characteristic classes, Osaka J. Math. 12 (1975), 401-417. MR 397749

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1970): 57D20, 57D30

Retrieve articles in all journals with MSC (1970): 57D20, 57D30


Additional Information

DOI: https://doi.org/10.1090/S0002-9904-1978-14546-7

American Mathematical Society