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

A Gysin sequence for a semifree action of S^3

International audienceIn this work we shall consider smooth semifree (i.e., free outside the fixed point set) actions of S^3 on a manifold M. We exhibit a Gysin sequence relating the cohomology of M with the intersection cohomology of the orbit space M/S^3. This generalizes the usual Gysin sequence associated with a free action of S^3.

Homological properties of stratified spaces

International audienceNous montrons un Théorème de de Rham entre l'homologie d'intersection et la cohomologie d'intersection dans le cadre des pseudovariétés strtifiées

Cohomologie d'intersection des actions toriques simples

International audienceNous décrivons le deuxième terme de la suite spectrale de Leray-Serre associée à l'action d'un tore (de profondeur 1) en termes de la cohomologie d'intersection de l'espace d'orbites

Poincar\'e duality, cap product and Borel-Moore intersection Homology

Using a cap product, we construct an explicit Poincar\'e duality isomorphism between the blown-up intersection cohomology and the Borel-Moore intersection homology, for any commutative ring of coefficients and second-countable, oriented pseudomanifolds

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