Cohomology of singular Riemannian foliations

Xosé M. Masa; Ana Rodríguez-Fernández

doi:10.1016/j.crma.2006.02.022

Topology

Cohomology of singular Riemannian foliations

Presented by: Alain Connes

Xosé M. Masa ¹; Ana Rodríguez-Fernández ¹

¹ Departamento de Xeometría e Topoloxía, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain

Comptes Rendus. Mathématique, Volume 342 (2006) no. 8, pp. 601-604.

Abstract
Résumé

Associated to a smooth foliation $(M, F)$ are defined the basic and the foliated cohomologies. These cohomologies are related to the de Rham cohomology by the de Rham spectral sequence of $F$ , $E_{r, d R}^{s, t} (M)$ , constructed by filtering the de Rham complex of the manifold. For a Riemannian foliation on a compact manifold the second term of this spectral sequence, $E_{2, d R}^{s, t} (M)$ , is finite dimensional and a topological invariant.

In this Note we prove these two results, fitness dimension and topological invariance, for the cohomology of singular Riemannian foliations. The proof uses the previous theorems for the regular case and the structure of singular Riemannian foliations described by P. Molino. For the basic cohomology these results have been proved by R. Wolak.

La filtration du complexe de De Rham d'une variété feuilletée $(M, F)$ définit une suite spectrale $E_{r, d R}^{s, t} (M)$ reliant les cohomologies basique et feuilletée du feuilletage $F$ à la cohomologie de De Rham de la variété ambiance. Pour un feuilletage riemannien d'une variété compacte le deuxième terme de cette suite spectrale est un invariant topologique de dimension finie.

Dans cette Note, nous prouvons la finitude et l'invariance topologique de ce terme pour les feuilletages riemanniens singuliers. La preuve combine les résultats du cas régulier avec la description de la structure des feuilletages riemanniens singuliers faite par P. Molino. Pour la cohomologie basique, ces résultats ont été prouvés par R. Wolak.

Received: 2004-01-13
Accepted: 2006-02-21
Published online: 2006-03-20

DOI: 10.1016/j.crma.2006.02.022

Author's affiliations:

Xosé M. Masa ¹; Ana Rodríguez-Fernández ¹

¹ Departamento de Xeometría e Topoloxía, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain

@article{CRMATH_2006__342_8_601_0,
     author = {Xos\'e M. Masa and Ana Rodr{\'\i}guez-Fern\'andez},
     title = {Cohomology of singular {Riemannian} foliations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {601--604},
     publisher = {Elsevier},
     volume = {342},
     number = {8},
     year = {2006},
     doi = {10.1016/j.crma.2006.02.022},
     language = {en},
}

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Xosé M. Masa; Ana Rodríguez-Fernández. Cohomology of singular Riemannian foliations. Comptes Rendus. Mathématique, Volume 342 (2006) no. 8, pp. 601-604. doi : 10.1016/j.crma.2006.02.022. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.02.022/

References
Cited by

[1] J.A. Álvarez López A finiteness theorem for the spectral sequence of a Riemannian foliation, Illinois J. Math., Volume 33 (1989), pp. 79-92

[2] X.M. Masa Alexander–Spanier cohomology of foliated manifolds, Illinois J. Math., Volume 46 (2002), pp. 979-998

[3] P. Molino Riemannian Foliations, Progr. Math., Birkhäuser, 1988

[4] R. Wolak Basic cohomology for singular Riemannian foliations, Monatsh. Math., Volume 128 (1999), pp. 159-163

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