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

Martintxo Saralegi-Aranguren¹ ; Robert Wolak²

¹ Laboratoire 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
² Institute of Mathematics Jagiellonian University Reymonta 4 30-059 Kraków, Poland

Annales Polonici Mathematici, Tome 89 (2006) no. 3, pp. 203-246.

Résumé

% 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).

Détail

DOI : 10.4064/ap89-3-1

Keywords: study cohomology properties singular foliation cal determined action mit phi colon times where abelian lie group preserves riemannian metric compact manifold precisely prove basic intersection cohomology mathbb * overline mathcal finite dimensional satisfies poincar duality duality includes known situations bullet poincar duality basic cohomology action mit phi almost bullet poincar duality intersection cohomology group compact connected