Top-Dimensional Group of the Basic Intersection Cohomology for Singular Riemannian Foliations

José Ignacio Royo Prieto ¹   ; Martintxo Saralegi-Aranguren ²   ; Robert Wolak ³

¹ Departamento de Matemática Aplicada Universidad del País Vasco Alameda de Urquijo s/n, 48013 Bilbao, Spain
² Fédération CNRS Nord-Pas-de-Calais FR 2956 UPRES-EA 2462 LML Faculté Jean Perrin Université d'Artois Rue Jean Souvraz SP 18 62 307 Lens Cedex, France
³ Institute of Mathematics Jagiellonian University Reymonta 4 30-059 Kraków, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 53 (2005) no. 4, pp. 429-440

Résumé

It is known that, for a regular riemannian foliation on a compact manifold, the properties of its basic cohomology (non-vanishing of the top-dimensional group and Poincaré duality) and the tautness of the foliation are closely related. If we consider singular riemannian foliations, there is little or no relation between these properties.We present an example of a singular isometric flow for which the top-dimensional basic cohomology group is non-trivial, but the basic cohomology does not satisfy the Poincaré Duality. However, we recover the Poincaré Duality in the basic intersection cohomology.It is not accidental that the top-dimensional basic intersection cohomology groups of the example are isomorphic to either $0$ or $\mathbb R$. We prove that this holds for any singular riemannian foliation of a compact connected manifold. As a corollary, we show that the tautness of the regular stratum of the singular riemannian foliation can be detected by the basic intersection cohomology.

Détail

DOI : 10.4064/ba53-4-8

Keywords: known regular riemannian foliation compact manifold properties its basic cohomology non vanishing top dimensional group poincar duality tautness foliation closely related consider singular riemannian foliations there little relation between these properties present example singular isometric flow which top dimensional basic cohomology group non trivial basic cohomology does satisfy poincar duality however recover poincar duality basic intersection cohomology accidental top dimensional basic intersection cohomology groups example isomorphic either mathbb prove holds singular riemannian foliation compact connected manifold corollary tautness regular stratum singular riemannian foliation detected basic intersection cohomology