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

We study the cohomology properties of the singular foliation \F determined by an action $\Phi \colon G \times M\to M$ where the abelian Lie group $G$ preserves a riemannian metric on the compact manifold $M$. More precisely, we prove that the basic intersection cohomology \lau{\IH}{*}{\per{p}}{\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 $\Phi$ is almost free). -- Poincar\'e Duality for intersection cohomology (the group $G$ is compact and connected)

Top dimensional group of the basic intersection cohomology for singular riemannian foliations

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 Poincar\'e 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 its basic cohomology does not satisfy the Poincar\'e Duality property. We recover this property in the basic intersection cohomology. It is not by chance that the top dimensional basic intersection cohomology groups of the example are isomorphic to either 0 or $\mathbb{R}$. We prove in this Note that this holds for any singular riemannian foliation of a compact connected manifold. As a Corollary, we get that the tautness of the regular stratum of the singular riemannian foliation can be detected by the basic intersection cohomology.Comment: 11 pages. Accepted for publication in the Bulletin of the Polish Academy of Science

Finitness of the basic intersection cohomology of a Killing foliation

We prove that the basic intersection cohomology ${I H}^{^{*}}_{_{\bar{p}}}{(M/\mathcal{F})},$ where $\mathcal{F}$ is the singular foliation determined by an isometric action of a Lie group $G$ on the compact manifold $M$, is finite dimensional

Tautness for riemannian foliations on non-compact manifolds

For a riemannian foliation $\mathcal{F}$ on a closed manifold $M$, it is known that $\mathcal{F}$ is taut (i.e. the leaves are minimal submanifolds) if and only if the (tautness) class defined by the mean curvature form $\kappa_\mu$ (relatively to a suitable riemannian metric $\mu$) is zero. In the transversally orientable case, tautness is equivalent to the non-vanishing of the top basic cohomology group $H^{^{n}}(M/\mathcal{F})$, where n = \codim \mathcal{F}. By the Poincar\'e Duality, this last condition is equivalent to the non-vanishing of the basic twisted cohomology group $H^{^{0}}_{_{\kappa_\mu}}(M/\mathcal{F})$, when $M$ is oriented. When $M$ is not compact, the tautness class is not even defined in general. In this work, we recover the previous study and results for a particular case of riemannian foliations on non compact manifolds: the regular part of a singular riemannian foliation on a compact manifold (CERF).Comment: 18 page

Equivariant intersection cohomology of the circle actions

In this paper, we prove that the orbit space B and the Euler class of an action of the circle S^1 on X determine both the equivariant intersection cohomology of the pseudomanifold X and its localization. We also construct a spectral sequence converging to the equivariant intersection cohomology of X whose third term is described in terms of the intersection cohomology of B.Comment: Final version as accepted in RACSAM. The final publication is available at springerlink.com; Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 201

Cohomological tautness for Riemannian foliations

In this paper we present some new results on the tautness of Riemannian foliations in their historical context. The first part of the paper gives a short history of the problem. For a closed manifold, the tautness of a Riemannian foliation can be characterized cohomologically. We extend this cohomological characterization to a class of foliations which includes the foliated strata of any singular Riemannian foliation of a closed manifold

Degradability of cross-linked polyurethanes based on synthetic polyhydroxybutyrate and modified with polylactide

In many areas of application of conventional non-degradable cross-linked polyurethanes (PUR), there is a need for their degradation under the influence of specific environmental factors. It is practiced by incorporation of sensitive to degradation compounds (usually of natural origin) into the polyurethane structure, or by mixing them with polyurethanes. Cross-linked polyurethanes (with 10 and 30%wt amount of synthetic poly([R,S]-3-hydroxybutyrate) (R,S-PHB) in soft segments) and their physical blends with poly([d,l]-lactide) (PDLLA) were investigated and then degraded under hydrolytic (phosphate buffer solution) and oxidative (CoCl2/H2O2) conditions. The rate of degradation was monitored by changes of samples mass, morphology of surface and their thermal properties. Despite the small weight losses of samples, the changes of thermal properties of polymers and topography of their surface indicated that they were susceptible to gradual degradation under oxidative and hydrolytic conditions. Blends of PDLLA and polyurethane with 30 wt% of R,S-PHB in soft segments and PUR/PDLLA blends absorbed more water and degraded faster than polyurethane with low amount of R,S-PHB