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On the homotopy and cohomology of the classifying space of Riemannian foliations


Author: Steven Hurder
Journal: Proc. Amer. Math. Soc. 81 (1981), 485-489
MSC: Primary 57R32; Secondary 55Q35, 55R60, 57R20
DOI: https://doi.org/10.1090/S0002-9939-1981-0597668-4
MathSciNet review: 597668
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Abstract: Let $ G$ be a closed subgroup of the general linear group. Let $ B\tilde \Gamma _G^q$ be the classifying space for $ G$-foliated microbundles of rank $ q$. (The $ G$-foliation is not assumed to be integrable.) The homotopy fiber $ F\tilde \Gamma _G^q$ of the classifying map $ \nu :B\tilde \Gamma _G^q \to BG$ is shown to be $ (q - 1)$-connected. For the orthogonal group, this implies $ FR{\Gamma ^q}$ is $ (q - 1)$-connected. The indecomposable classes in $ {H^ * }(R{W_q})$ therefore are mapped to linearly independent classes in $ {H^ * }(FR{\Gamma ^q})$; the indecomposable variable classes are mapped to independently variable classes. Related results on the homotopy groups $ {\pi _ * }(FR{\Gamma ^q})$ also follow.


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

  • [1] T. Duchamp, Characteristic invariants of $ G$-foliations, Ph. D. thesis, University of Illinois, Urbana, Ill., 1976.
  • [2] A. Haefliger, Feuilletages sur les variétés ourvertes, Topology 9 (1970), 183-194. MR 0263104 (41:7709)
  • [3] -, Homotopy and integrability, Lecture Notes in Math., vol. 197, Springer-Verlag, Berlin and New York, 1971, pp. 133-163. MR 0285027 (44:2251)
  • [4] -, Whitehead products and differential forms, Lecture Notes in Math., vol. 652, SpringerVerlag, Berlin and New York, pp. 13-24. MR 505648 (80a:55012)
  • [5] S. Hurder, Dual homotopy invariants of $ G$-foliations, Ph. D. Thesis, University of Illinois, Urbana, Ill., 1980.
  • [6] F. Kamber and Ph. Tondeur, Non-trivial invariants of homogeneous foliated bundles, Ann. Sci. École Norm. Sup. 8 (1975), 433-486. MR 0394700 (52:15499)
  • [7] -, $ G$-foliations and their characteristic classes, Bull. Amer. Math. Soc. 84 (1978), 1086-1124. MR 508449 (80b:57024)
  • [8] C. Lazarov and J. Pasternack, Secondary characteristic classes for Riemannian foliations, J. Differential Geometry 11 (1976), 365-385; Residues and characteristic classes for Riemannian foliations, J. Differential Geometry 11 (1976), 599-612. MR 0445514 (56:3854)
  • [9] J. Pasternack, Foliations and compact Lie groups actions, Comment. Math. Helv. 46 (1971), 467-477. MR 0300307 (45:9353)
  • [10] D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269-331. MR 0646078 (58:31119)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0597668-4
Keywords: Classifying spaces, foliations, characteristic classes, minimal models
Article copyright: © Copyright 1981 American Mathematical Society

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