$G$-foliations and their characteristic classes
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- by Franz W. Kamber and Philippe Tondeur PDF
- Bull. Amer. Math. Soc. 84 (1978), 1086-1124
References
- V. I. Arnol′d, On a characteristic class entering into conditions of quantization, Funkcional. Anal. i Priložen. 1 (1967), 1–14 (Russian). MR 0211415, DOI 10.1007/BF01075861
- Paul F. Baum, On the cohomology of homogeneous spaces, Topology 7 (1968), 15–38. MR 219085, DOI 10.1016/0040-9383(86)90012-1
- Daniel Baker, On a class of foliations and the evaluation of their characteristic classes, Bull. Amer. Math. Soc. 83 (1977), no. 3, 394–396. MR 438351, DOI 10.1090/S0002-9904-1977-14288-2
- Armand Borel, Compact Clifford-Klein forms of symmetric spaces, Topology 2 (1963), 111–122. MR 146301, DOI 10.1016/0040-9383(63)90026-0
- I. N. Bernšteĭn and B. I. Rozenfel′d, Characteristic classes of foliations, Funkcional. Anal. i Priložen. 6 (1972), no. 1, 68–69 (Russian). MR 0296967
- I. N. Bernšteĭn and B. I. Rosenfel′d, Homogeneous spaces of infinite-dimensional Lie algebras and the characteristic classes of foliations, Uspehi Mat. Nauk 28 (1973), no. 4(172), 103–138 (Russian). MR 0415633
- Raoul Bott, Lectures on characteristic classes and foliations, Lectures on algebraic and differential topology (Second Latin American School in Math., Mexico City, 1971) Lecture Notes in Math., Vol. 279, Springer, Berlin, 1972, pp. 1–94. Notes by Lawrence Conlon, with two appendices by J. Stasheff. MR 0362335
- Raoul Bott, On the Lefschetz formula and exotic characteristic classes, Symposia Mathematica, Vol. X (Convegno di Geometria Differenziale, INDAM, Rome, 1971) Academic Press, London, 1972, pp. 95–105. MR 0362336
- R. Bott, On the Chern-Weil homomorphism and the continuous cohomology of Lie-groups, Advances in Math. 11 (1973), 289–303. MR 345115, DOI 10.1016/0001-8708(73)90012-1 [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.
- Raoul Bott, On characteristic classes in the framework of Gelfand-Fuks cohomology, “Analyse et Topologie” en l’Honneur de Henri Cartan (Orsay, 1974), Astérisque, No. 32-33, Soc. Math. France, Paris, 1976, pp. 113–139. MR 0402768
- 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 Mathematics, Vol. 652, Springer, Berlin, 1978, pp. 25–61. MR 505649
- R. Bott and A. Haefliger, On characteristic classes of $\Gamma$-foliations, Bull. Amer. Math. Soc. 78 (1972), 1039–1044. MR 307250, DOI 10.1090/S0002-9904-1972-13101-X [CA] H. Cartan, Cohomologie réelle d’un espace fibré principal différentiable, Séminaire Cartan, Exposés 19 et 20 (1949/50).
- Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
- S. S. Chern, The geometry of $G$-structures, Bull. Amer. Math. Soc. 72 (1966), 167–219. MR 192436, DOI 10.1090/S0002-9904-1966-11473-8
- Shiing Shen Chern and James Simons, Characteristic forms and geometric invariants, Ann. of Math. (2) 99 (1974), 48–69. MR 353327, DOI 10.2307/1971013
- Lawrence Conlon, Transversally parallelizable foliations of codimension two, Trans. Amer. Math. Soc. 194 (1974), 79–102, erratum, ibid. 207 (1975), 406. MR 370617, DOI 10.1090/S0002-9947-1974-0370617-0 [D] T. Duchamp, Characteristic invariants of G-foliations, Ph.D. thesis, University of Illinois, Urbana, Illinois, 1976.
- Johan L. Dupont, Simplicial de Rham cohomology and characteristic classes of flat bundles, Topology 15 (1976), no. 3, 233–245. MR 413122, DOI 10.1016/0040-9383(76)90038-0
- D. B. Fuks, Finite-dimensional Lie algebras of formal vector fields and characteristic classes of homogeneous foliations, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 1, 57–64, 221 (Russian). MR 0413125
- Dimitri Fuchs, Non-trivialité des classes caractéristiques de ${\mathfrak {g}}$-structures. Application aux classes caractéristiques de feuilletages, C. R. Acad. Sci. Paris Sér. A-B 284 (1977), no. 17, A1017–A1019. MR 433469
- Dimitri Fuchs, Non-trivialité des classes caractéristiques de ${\mathfrak {g}}$-structures. Applications aux variations des classes caractéristiques de feuilletages, C. R. Acad. Sci. Paris Sér. A-B 284 (1977), no. 18, A1105–A1107. MR 433470
- I. M. Gel′fand and D. B. Fuks, Upper bounds for the cohomology of infinite-dimensional Lie algebras. , Funkcional. Anal. i Priložen. 4 (1970), no. 4, 70–71 (Russian). MR 0287589
- I. M. Gel′fand and D. B. Fuks, Cohomologies of the Lie algebra of formal vector fields, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 322–337 (Russian). MR 0266195 [GB] C. Godbillon, Cohomologies d’algèbres de Lie de champs de vecteurs, Séminaire Bourbaki, 25e année (1971/72), no. 421.
- Claude Godbillon and Jacques Vey, Un invariant des feuilletages de codimension $1$, C. R. Acad. Sci. Paris Sér. A-B 273 (1971), A92-A95 (French). MR 283816
- A. Haefliger, Feuilletages sur les variétés ouvertes, Topology 9 (1970), 183–194 (French). MR 263104, DOI 10.1016/0040-9383(70)90040-6 [H2] A. Haefliger, Sur les classes caractéristiques des feuilletages, Séminaire Bourbaki, 24e année (1971/72), n° 412.
- A. Haefliger, Sur la cohomologie de Gelfand-Fuchs, Differential topology and geometry (Proc. Colloq., Dijon, 1974) Lecture Notes in Math., Vol. 484, Springer, Berlin, 1975, pp. 121–152 (French). MR 0394698
- André Haefliger, Sur la cohomologie de l’algèbre de Lie des champs de vecteurs, Ann. Sci. École Norm. Sup. (4) 9 (1976), no. 4, 503–532 (French). MR 448367, DOI 10.24033/asens.1316 [H5] A. Haefliger, Differentiable cohomology, Cours donné au C.I.M.E., 1976.
- James L. Heitsch, Deformations of secondary characteristic classes, Topology 12 (1973), 381–388. MR 321106, DOI 10.1016/0040-9383(73)90030-X
- James L. Heitsch, Derivatives of secondary characteristic classes, J. Differential Geometry 13 (1978), no. 3, 311–339. MR 551563 [HT3] J. Heitsch, Independent variation of secondary characteristic classes (to appear).
- Jean-Louis Koszul, Homologie et cohomologie des algèbres de Lie, Bull. Soc. Math. France 78 (1950), 65–127 (French). MR 36511, DOI 10.24033/bsmf.1410
- J. L. Koszul, Sur un type d’algèbres différentielles en rapport avec la transgression, Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège; Masson & Cie, Paris, 1951, pp. 73–81 (French). MR 0042428
- J.-L. Koszul, Déformations de connexions localement plates, Ann. Inst. Fourier (Grenoble) 18 (1968), no. fasc. 1, 103–114 (French). MR 239529, DOI 10.5802/aif.279
- Franz Kamber and Philippe Tondeur, Flat manifolds, Lecture Notes in Mathematics, No. 67, Springer-Verlag, Berlin-New York, 1968. MR 0238342, DOI 10.1007/BFb0076909
- Franz W. Kamber and Philippe Tondeur, Invariant differential operators and cohomology of Lie algebra sheaves, Differentialgeometrie im Großen (Tagung, Math. Forschungsinst., Oberwolfach, 1969) Ber Math. Forschungsinst. Oberwolfach, Heft 4, Bibliographisches Inst., Mannheim, 1971, pp. 177–230. MR 0370616
- Franz W. Kamber and Philippe Tondeur, Invariant differential operators and the cohomology of Lie algebra sheaves, Memoirs of the American Mathematical Society, No. 113, American Mathematical Society, Providence, R.I., 1971. MR 0413129 [KT4] F. Kamber and Ph. Tondeur, Characteristic classes of modules over a sheaf of Lie algebras, Notices Amer. Math. Soc. 19 (1972), A-401.
- Franz W. Kamber and Philippe Tondeur, Cohomologie des algèbres de Weil relatives tronquées, C. R. Acad. Sci. Paris Sér. A-B 276 (1973), A459–A462 (French). MR 321127
- Franz W. Kamber and Philippe Tondeur, Algèbres de Weil semi-simpliciales, C. R. Acad. Sci. Paris Sér. A-B 276 (1973), A1177–A1179 (French). MR 321128
- Franz W. Kamber and Philippe Tondeur, Characteristic invariants of foliated bundles, Manuscripta Math. 11 (1974), 51–89. MR 334237, DOI 10.1007/BF01189091
- Franz W. Kamber and Philippe Tondeur, Semisimplicial Weil algebras and characteristic classes for foliated bundles in Čech cohomology, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 1, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 283–294. MR 0385886
- Franz W. Kamber and Philippe Tondeur, Foliated bundles and characteristic classes, Lecture Notes in Mathematics, Vol. 493, Springer-Verlag, Berlin-New York, 1975. MR 0402773, DOI 10.1007/BFb0081558
- Franz Kamber and Philippe Tondeur, Classes caractéristiques généralisées des fibrés feuilletés localement homogènes, C. R. Acad. Sci. Paris Sér. A 279 (1974), 847–850 (French). MR 368030
- Franz W. Kamber and Philippe Tondeur, Non-trivial characteristic invariants of homogeneous foliated bundles, Ann. Sci. École Norm. Sup. (4) 8 (1975), no. 4, 433–486. MR 394700, DOI 10.24033/asens.1298 [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, Tohoku Math. J. (2) 30 (1978), no. 3, 373–422. MR 509023, DOI 10.2748/tmj/1178229977
- Franz W. Kamber and Philippe Tondeur, Classes caractéristiques et suites spectrales d’Eilenberg-Moore, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 12, Ai, A883–A886. 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).
- H. Blaine Lawson Jr., Foliations, Bull. Amer. Math. Soc. 80 (1974), 369–418. MR 343289, DOI 10.1090/S0002-9904-1974-13432-4
- H. Blaine Lawson Jr., The quantitative theory of foliations, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 27, American Mathematical Society, Providence, R.I., 1977. Expository lectures from the CBMS Regional Conference held at Washington University, St. Louis, Mo., January 6–10, 1975. MR 0448368
- Daniel Lehmann, Classes caractéristiques exotiques et ${\cal I}$-connexité des espaces de connexions, Ann. Inst. Fourier (Grenoble) 24 (1974), no. 3, xiv, 267–306 (French, with English summary). Avec une appendice par B. Callenaere et D. Lehmann. MR 362342
- Connor Lazarov and Joel Pasternack, Secondary characteristic classes for Riemannian foliations, J. Differential Geometry 11 (1976), no. 3, 365–385. MR 445513
- Connor Lazarov and Joel Pasternack, Residues and characteristic classes for Riemannian foliations, J. Differential Geometry 11 (1976), no. 4, 599–612. MR 445514
- Pierre Molino, Classe d’Atiyah d’un feuilletage et connexions transverses projetables, C. R. Acad. Sci. Paris Sér. A-B 272 (1971), A779–A781 (French). MR 281224
- Pierre Molino, Classes caractéristiques et obstruction d’Atiyah pour les fibrés principaux feuilletés, C. R. Acad. Sci. Paris Sér. A-B 272 (1971), A1376–A1378 (French). MR 287571
- Pierre Molino, Feuilletages et classes caractéristiques, Symposia Mathematica, Vol. X (Convegno di Geometria Differenziale, INDAM, Rome, 1971) Academic Press, London, 1972, pp. 199–200 (French). MR 0358806
- Pierre Molino, Propriétés cohomologiques et propriétés topologiques des feuilletages à connexion transverse projetable, Topology 12 (1973), 317–325 (French). MR 353332, DOI 10.1016/0040-9383(73)90026-8
- Shigeyuki Morita, A remark on the continuous variation of secondary characteristic classes for foliations, J. Math. Soc. Japan 29 (1977), no. 2, 253–260. MR 442952, DOI 10.2969/jmsj/02920253 [MS] V. P. Maslov, Théorie des perturbations et méthodes asymptotiques, Gauthier-Villars, Paris, 1972.
- Yozô Matsushima, On Betti numbers of compact, locally sysmmetric Riemannian manifolds, Osaka Math. J. 14 (1962), 1–20. MR 141138
- Seiki Nishikawa and Hajime Sato, On characteristic classes of Riemannian, conformal and projective foliations, J. Math. Soc. Japan 28 (1976), no. 2, 223–241. MR 400255, DOI 10.2969/jmsj/02820223
- J. S. Pasternack, Foliations and compact Lie group actions, Comment. Math. Helv. 46 (1971), 467–477. MR 300307, DOI 10.1007/BF02566859
- Bruce L. Reinhart, Foliated manifolds with bundle-like metrics, Ann. of Math. (2) 69 (1959), 119–132. MR 107279, DOI 10.2307/1970097
- Herbert Shulman and David Tischler, Leaf invariants for foliations and the Van Est isomorphism, J. Differential Geometry 11 (1976), no. 4, 535–546. MR 451262
- William Thurston, Noncobordant foliations of $S^{3}$, Bull. Amer. Math. Soc. 78 (1972), 511–514. MR 298692, DOI 10.1090/S0002-9904-1972-12975-6
- Kenzi Yamato, Examples of foliations with non trivial exotic characteristic classes, Osaka Math. J. 12 (1975), no. 2, 401–417. MR 397749
Additional Information
- 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