11 research outputs found
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 where the abelian Lie group
preserves a riemannian metric on the compact manifold . 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 is almost
free).
-- Poincar\'e Duality for intersection cohomology (the group 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 . 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 where is the singular
foliation determined by an isometric action of a Lie group on the compact
manifold , is finite dimensional
Tautness for riemannian foliations on non-compact manifolds
For a riemannian foliation on a closed manifold , it is
known that is taut (i.e. the leaves are minimal submanifolds) if
and only if the (tautness) class defined by the mean curvature form
(relatively to a suitable riemannian metric ) is zero. In the
transversally orientable case, tautness is equivalent to the non-vanishing of
the top basic cohomology group , 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
, when is oriented. When 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