Continuous cohomology of groups and classifying spaces
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- Bull. Amer. Math. Soc. 84 (1978), 513-530
References
- 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
- Daniel Baker, Differential characters and Borel cohomology, Topology 16 (1977), no. 4, 441–449. MR 646086, DOI 10.1016/0040-9383(77)90049-0
- 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
- Armand Borel, Compact Clifford-Klein forms of symmetric spaces, Topology 2 (1963), 111–122. MR 146301, DOI 10.1016/0040-9383(63)90026-0 5. A. Borel, Seminar notes IAS 1976-1977.
- 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
- 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
- Raoul Bott, Some remarks on continuous cohomology, Manifolds—Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973) Univ. Tokyo Press, Tokyo, 1975, pp. 161–170. MR 0368027
- R. Bott, H. Shulman, and J. Stasheff, On the de Rham theory of certain classifying spaces, Advances in Math. 20 (1976), no. 1, 43–56. MR 402769, DOI 10.1016/0001-8708(76)90169-9
- Lawrence G. Brown, Extensions of topological groups, Pacific J. Math. 39 (1971), 71–78. MR 307264, DOI 10.2140/pjm.1971.39.71
- Henri Cartan, Notions d’algèbre différentielle; application aux groupes de Lie et aux variétés où opère un groupe de Lie, Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège; Masson & Cie, Paris, 1951, pp. 15–27 (French). MR 0042426 12. J-L. Cathelineau, Un objet simplicial associé à un recouvrement d’un groupe topologique, Université de Poitiers (preprint).
- W. Casselman and D. Wigner, Continuous cohomology and a conjecture of Serre’s, Invent. Math. 25 (1974), 199–211. MR 352333, DOI 10.1007/BF01389727
- Jeff Cheeger and James Simons, Differential characters and geometric invariants, Geometry and topology (College Park, Md., 1983/84) Lecture Notes in Math., vol. 1167, Springer, Berlin, 1985, pp. 50–80. MR 827262, DOI 10.1007/BFb0075216
- Claude Chevalley and Samuel Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85–124. MR 24908, DOI 10.1090/S0002-9947-1948-0024908-8
- Albrecht Dold and Richard Lashof, Principal quasi-fibrations and fibre homotopy equivalence of bundles, Illinois J. Math. 3 (1959), 285–305. MR 101521
- 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, Characteristic classes of foliations, Uspehi Mat. Nauk 28 (1973), no. 2(170), 3–17 (Russian). MR 0415635
- I. M. Gel′fand and D. B. Fuks, Cohomologies of the Lie algebra of tangent vector fields of a smooth manifold, Funkcional. Anal. i Priložen. 3 (1969), no. 3, 32–52 (Russian). MR 0256411
- 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 21. C. Godbillon, Cohomologie d’algèbres de Lie de champs de vecteurs formels, Séminaire Bourbaki 421 (1972/73).
- 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
- Werner Greub, Stephen Halperin, and Ray Vanstone, Connections, curvature, and cohomology, Pure and Applied Mathematics, Vol. 47-III, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Volume III: Cohomology of principal bundles and homogeneous spaces. MR 0400275
- André Haefliger, Sur les classes caractéristiques des feuilletages, Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 412, Lecture Notes in Math., Vol. 317, Springer, Berlin, 1973, pp. 239–260 (French). MR 0420655
- A. Haefliger, Feuilletages sur les variétés ouvertes, Topology 9 (1970), 183–194 (French). MR 263104, DOI 10.1016/0040-9383(70)90040-6
- James L. Heitsch, Deformations of secondary characteristic classes, Topology 12 (1973), 381–388. MR 321106, DOI 10.1016/0040-9383(73)90030-X 27. J. Heitsch, Independent variation of secondary classes (manuscript).
- Alex Heller, Principal bundles and group extensions with applications to Hopf algebras, J. Pure Appl. Algebra 3 (1973), 219–250. MR 327871, DOI 10.1016/0022-4049(73)90011-X
- G. Hochschild and G. D. Mostow, Cohomology of Lie groups, Illinois J. Math. 6 (1962), 367–401. MR 147577, DOI 10.1215/ijm/1255632500
- Sze-tsen Hu, Cohomology rings of compact connected groups and their homogeneous spaces, Ann. of Math. (2) 55 (1952), 391–419. MR 47054, DOI 10.2307/1969641
- Sze-tsen Hu, Cohomology theory in topological groups, Michigan Math. J. 1 (1952), 11–59. MR 51244
- Franz W. Kamber and Philippe Tondeur, Characteristic invariants of foliated bundles, Manuscripta Math. 11 (1974), 51–89. MR 334237, DOI 10.1007/BF01189091 33. F. Kamber and Ph. Tondeur, On the linear independence of certain cohomology classes of BГ (preprint). 34. M. Lazard, Groupes analytiques p-adiques, Publ. Math. IHES 26 (1975). 35. S. Lie, Om en classe geometriske Transformationen, Ges. der Wiss. zu Christiania, 1870. 36. S. Lie, Theorie der Transformations-gruppen, Teubner, Leipzig, 1890.
- George W. Mackey, Les ensembles boréliens et les extensions des groupes, J. Math. Pures Appl. (9) 36 (1957), 171–178 (French). MR 89998
- Saunders MacLane, Homology, 1st ed., Die Grundlehren der mathematischen Wissenschaften, Band 114, Springer-Verlag, Berlin-New York, 1967. MR 0349792 39. S. Mac Lane, Retiring Presidential Address, Bull. Amer. Math. Soc. 82 (1976), 1-40.
- Saunders Mac Lane, The work of Samuel Eilenberg in topology, Algebra, topology, and category theory (a collection of papers in honor of Samuel Eilenberg), Academic Press, New York, 1976, pp. 133–144. MR 0414284
- R. James Milgram, The bar construction and abelian $H$-spaces, Illinois J. Math. 11 (1967), 242–250. MR 208595
- John Milnor, Construction of universal bundles. II, Ann. of Math. (2) 63 (1956), 430–436. MR 77932, DOI 10.2307/1970012
- Calvin C. Moore, Extensions and low dimensional cohomology theory of locally compact groups. I, II, Trans. Amer. Math. Soc. 113 (1964), 40–63; ibid. 113 (1964), 64–86. MR 171880, DOI 10.1090/S0002-9947-1964-0171880-5
- Calvin C. Moore, Restrictions of unitary representations to subgroups and ergodic theory: Group extensions and group cohomology, Group Representations in Math. and Phys. (Battelle Seattle 1969 Rencontres), Lecture Notes in Phys., Vol. 6, Springer, Berlin, 1970, pp. 1–35. MR 0279232 45. J. C. Moore, Algèbre homologique et homologie des espaces classifiants, Séminaire H. Cartan (1959/60).
- 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
- Paul S. Mostert, Local cross sections in locally compact groups, Proc. Amer. Math. Soc. 4 (1953), 645–649. MR 56614, DOI 10.1090/S0002-9939-1953-0056614-5
- G. D. Mostow, Cohomology of topological groups and solvmanifolds, Ann. of Math. (2) 73 (1961), 20–48. MR 125179, DOI 10.2307/1970281
- Mark Alan Mostow, Continuous cohomology of spaces with two topologies, Mem. Amer. Math. Soc. 7 (1976), no. 175, x+142. MR 413132, DOI 10.1090/memo/0175 50. M. Mostow and J. Perchik, Notes on Gel’fand-Fuks cohomology and characteristic classes (Lectures by Raoul Bott), New Mexico State Univ., 1973. 51. O. H. Rasmussen, Non-vanishing and continuous variation of exotic characteristic classes for foliations, Thesis, Princeton Univ., 1974.
- Graeme Segal, Categories and cohomology theories, Topology 13 (1974), 293–312. MR 353298, DOI 10.1016/0040-9383(74)90022-6
- Graeme Segal, Cohomology of topological groups, Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69) Academic Press, London, 1970, pp. 377–387. MR 0280572
- Jean-Pierre Serre, Cohomologie Galoisienne, Lecture Notes in Mathematics, Vol. 5, Springer-Verlag, Berlin-New York, 1973. Cours au Collège de France, Paris, 1962–1963; Avec des textes inédits de J. Tate et de Jean-Louis Verdier; Quatrième édition. MR 0404227, DOI 10.1007/978-3-662-21553-1
- Herbert Shulman, Covering dimension and characteristic classes for foliations, 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. 189–190. MR 520535
- James Stasheff, $H$-spaces from a homotopy point of view, Lecture Notes in Mathematics, Vol. 161, Springer-Verlag, Berlin-New York, 1970. MR 0270372, DOI 10.1007/BFb0065896
- James Dillon Stasheff, On extensions of $H$-spaces, Trans. Amer. Math. Soc. 105 (1962), 126–135. MR 178469, DOI 10.1090/S0002-9947-1962-0178469-0
- William Thurston, Foliations and groups of diffeomorphisms, Bull. Amer. Math. Soc. 80 (1974), 304–307. MR 339267, DOI 10.1090/S0002-9904-1974-13475-0
- William P. Thurston, The theory of foliations of codimension greater than one, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 1, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 321. MR 0375345 60. W. Thurston, Variations of the Godbillon-Vey invariant in higher dimensions (to appear).
- W. T. van Est, A generalization of the Cartan-Leray spectral sequence. I, II, Nederl. Akad. Wetensch. Proc. Ser. A 61 = Indag. Math. 20 (1958), 399–413. MR 0103467, DOI 10.1016/S1385-7258(58)50055-9
- W. T. van Est, Group cohomology and Lie algebra cohomology in Lie groups. I, II, Nederl. Akad. Wetensch. Proc. Ser. A. 56 = Indagationes Math. 15 (1953), 484–492, 493–504. MR 0059285, DOI 10.1016/S1385-7258(53)50061-7
- W. T. van Est, On the algebraic cohomology concepts in Lie groups. I, II, Nederl. Akad. Wetensch. Proc. Ser. A. 58 = Indag. Math. 17 (1955), 225–233, 286–294. MR 0070959
- W. T. van Est, Une application d’une méthode de Cartan-Leray, Nederl. Akad. Wetensch. Proc. Ser. A. 58 = Indag. Math. 17 (1955), 542–544 (French). MR 0073108 65. C. Watkiss, Thesis, Univ. of Toronto, 1975.
- David Wigner, Algebraic cohomology of topological groups, Trans. Amer. Math. Soc. 178 (1973), 83–93. MR 338132, DOI 10.1090/S0002-9947-1973-0338132-7 67. A. Haefliger, Differentiable cohomology, Cours donné au CIME (1976).
- H. Samelson, Heinz Hopf zum Gedenken. II. Zum wissenschaftlichen Werk von Heinz Hopf, Jber. Deutsch. Math.-Verein. 78 (1976/77), no. 3, 126–146. MR 427001
- 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
Additional Information
- Journal: Bull. Amer. Math. Soc. 84 (1978), 513-530
- MSC (1970): Primary 55F35, 55F40, 55B35, 55H99; Secondary 17B65, 18H25, 18H10, 57F15
- DOI: https://doi.org/10.1090/S0002-9904-1978-14488-7
- MathSciNet review: 0494071