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

We study the cohomology properties of the singular foliation $\text{F}$ determined by an action $\mathrm{\Phi }\text{\hspace{0.17em}⁣}:G\times M\to M$ where the abelian Lie group $G$ preserves a riemannian metric on the compact manifold $M$. More precisely, we prove that the basic intersection cohomology $\text{lau}\text{IH}\ast \text{per}p\text{mf}$ is finite dimensional and verifies the Poincar'e Duality. This duality includes two well-known situations: – Poincar'e Duality for basic cohomology (the action $\mathrm{\Phi }$ is almost free). – Poincar'e Duality for intersection cohomology (the group $G$ is compact and connected).