The BIC of a singular foliation defined by an abelian group of isometries

Volume 89 / 2006

Martintxo Saralegi-Aranguren, Robert Wolak Annales Polonici Mathematici 89 (2006), 203-246 MSC: 53C12, 57R30, 55N33, 58A35, 22Fxx. DOI: 10.4064/ap89-3-1


% We study the cohomology properties of the singular foliation $\cal F$ determined by an action ${\mit\Phi} \colon G \times M\to M$ where the abelian Lie group $G$ preserves a riemannian metric on the compact manifold $M$. More precisely, we prove that the basic intersection cohomology $\mathbb H^{*}_{\overline{p}}{(M/\mathcal F)}$ is finite-dimensional and satisfies the Poincaré duality. This duality includes two well known situations:

$\bullet$ Poincaré duality for basic cohomology (the action ${\mit\Phi}$ is almost free).

$\bullet$ Poincaré duality for intersection cohomology (the group $G$ is compact and connected).


  • Martintxo Saralegi-ArangurenLaboratoire de Mathématiques de Lens EA 2462
    Fédération CNRS Nord-Pas-de-Calais FR 2956
    Faculté des Sciences Jean Perrin
    Université d'Artois
    Rue Jean Souvraz S.P. 18
    62 307 Lens Cedex, France
  • Robert WolakInstitute of Mathematics
    Jagiellonian University
    Reymonta 4
    30-059 Kraków, Poland

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