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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0597668-4
PII: S 0002-9939(1981)0597668-4
Keywords: Classifying spaces, foliations, characteristic classes, minimal models
Article copyright: © Copyright 1981 American Mathematical Society