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

Volume 53 / 2005

José Ignacio Royo Prieto, Martintxo Saralegi-Aranguren, Robert Wolak Bulletin Polish Acad. Sci. Math. 53 (2005), 429-440 MSC: 53C12, 53R30, 58A35, 55N30. DOI: 10.4064/ba53-4-8


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.


  • José Ignacio Royo PrietoDepartamento de Matemática Aplicada
    Universidad del País Vasco
    Alameda de Urquijo s/n, 48013 Bilbao, Spain
  • Martintxo Saralegi-ArangurenFédération 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
  • Robert WolakInstitute of Mathematics
    Jagiellonian University
    Reymonta 4
    30-059 Kraków, Poland

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