De Rham intersection cohomology for general perversities

For a stratified pseudomanifold $X$, we have the de Rham Theorem \lau{\IH}{*}{\per{p}}{X} = \lau{\IH}{\per{t} - \per{p}}{*}{X}, for a perversity \per{p} verifying \per{0} \leq \per{p} \leq \per{t}, where \per{t} denotes the top perversity. We extend this result to any perversity \per{p}. In the direction cohomology $\mapsto$ homology, we obtain the isomorphism \lau{\IH}{*}{\per{p}}{X} = \lau{\IH}{\per{t} -\per{p}}{*}{X,\ib{X}{\per{p}}}, where {\displaystyle \ib{X}{\per{p}} = \bigcup\_{S \preceq S\_{1} \atop \per{p} (S\_{1})< 0}S = \bigcup\_{\per{p} (S)< 0}\bar{S}.} In the direction homology $\mapsto$ cohomology, we obtain the isomorphism \lau{\IH}{\per{p}}{*}{X}=\lau{\IH}{*}{\max (\per{0},\per{t} -\per{p})}{X}. In our paper stratified pseudomanifolds with one-codimensional strata are allowed

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)

The Gysin sequence for S3{\mathbb S}^3-actions on manifolds

We construct a Gysin sequence associated to any smooth ${\mathbb S}^3$-action on a smooth manifold.Comment: Accepted for publication in Publicationes Mathematicae Debrecen, scheduled for 2014 Publicationes Mathematicae Debrecen (2014

Intersection Homology. General perversities and topological invariance

Topological invariance of the intersection homology of a pseudomanifold without codimension one strata, proven by Goresky and MacPherson, is one of the main features of this homology. This property is true for codimension-dependent perversities with some growth conditions, verifying $\overline p(1)=\overline p(2)=0$. King reproves this invariance by associating an intrinsic pseudomanifold $X^*$ to any pseudomanifold $X$. His proof consists of an isomorphism between the associated intersection homologies $H^{\overline{p}}_{*}(X) \cong H^{\overline{p}}_{*}( X^*)$ for any perversity $\overline{p}$ with the same growth conditions verifying $\overline p(1)\geq 0$. In this work, we prove a certain topological invariance within the framework of strata-dependent perversities, $\overline{p}$, which corresponds to the classical topological invariance if $\overline{p}$ is a GM-perversity. We also extend it to the tame intersection homology, a variation of the intersection homology, particularly suited for ``large'' perversities, if there is no singular strata on $X$ becoming regular in $X^*$. In particular, under the above conditions, the intersection homology and the tame intersection homology are invariant under a refinement of the stratification