The BIC of a singular foliation defined by an abelian group of isometries
Volume 89 / 2006
Annales Polonici Mathematici 89 (2006), 203-246
MSC: 53C12, 57R30, 55N33, 58A35, 22Fxx.
DOI: 10.4064/ap89-3-1
Abstract
% 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).
