La cohomologie basique d'un feuilletage riemannien est de dimension finie
type: Item
instance of: scholarly article
title: La cohomologie basique d'un feuilletage riemannien est de dimension finie
zbMATH Open document ID: 0536.57013
DOI: 10.1007/BF01161658
author: A. El Kacimi Alaoui
author: Vlad Sergiescu
author: Gilbert Hector
published in: Mathematische Zeitschrift
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173561
review text: Let f be a codimension n foliation on a compact connected manifold M. (i) If F is transversally parallelisable we construct a spectral sequence converging to the base-like cohomology \(H^*(M/F)\) of the foliated manifold (M,F) and show that this cohomology is finite dimensional. (ii) If F is Riemannian, the lifted foliation \(\hat F\) to the transverse orthonormal frame bundle \(SO(n)\to \hat M\to M\) is transversally parallelisable. To this fiber bundle we associate a spectral sequence connecting the base-like cohomology of (M,F), the cohomology of (\^M,\^F). Using this spectral sequence and (i) we show that \(H^*(M/F)\) is finite dimensional and \(H^ n(M/F)\) is isomorphic to 0 or \({\mathbb{R}}\).
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 55N30
zbMATH DE Number: 3852038
zbMATH Keywords: transversally parallelisable foliations
zbMATH Keywords: Riemannian foliations
zbMATH Keywords: transverse Euler class
zbMATH Keywords: baselike Euler class
zbMATH Keywords: codimension n foliation on a compact connected manifold
zbMATH Keywords: spectral sequence
zbMATH Keywords: base-like cohomology
MaRDI profile type: Publication
cites work: Q3961493
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cites work: Foliated manifolds with bundle-like metrics
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cites work: On the de Rham cohomology of the leaf space of a foliation
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description: scientific article; zbMATH DE number 3852038