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Given a perversity function in the sense of intersection homology theory, the method of intersection spaces assigns to certain oriented stratified spaces cell complexes whose ordinary reduced homology with real coefficients satisfies…

The theory of intersection spaces assigns cell complexes to certain stratified topological pseudomanifolds depending on a perversity function in the sense of intersection homology. The main property of the intersection spaces is Poincar\'e…

The method of intersection spaces associates rational Poincar\'e complexes to singular stratified spaces. For a conifold transition, the resulting cohomology theory yields the correct count of all present massless 3-branes in type IIB…

We construct an explicit de Rham isomorphism relating the cohomology rings of Banagl's de Rham and spatial approach to intersection space cohomology for stratified pseudomanifolds with isolated singularities. Intersection space…

In this paper, we mainly build up the theory of sheaf-correspondence filtered spaces and stratified de Rham complexes for studying singular spaces. We prove the finiteness of a stratified de Rham cohomology and obtain its isomorphism to…

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…

Given a compact stratified pseudomanifold with a Thom-Mather stratification and a class of riemannian metrics over its regular part, we study the relationships between the $L^{2}$ de Rham and Hodge cohomology and the intersection cohomology…

Let $X$ be any subanalytic compact pseudomanifold. We show a De Rham theorem for $L^\infty$ forms. We prove that the cohomology of $L^\infty$ forms is isomorphic to intersection cohomology in the maximal perversity.

In this article, we prove that there is a canonical Verdier self-dual intersection space sheaf complex for the middle perversity on Witt spaces that admit compatible trivializations for their link bundles, for example toric varieties. If…

Intersection cohomology is a way to enhance classical cohomology, allowing us to use a famous result called Poincar\'e duality on a large class of spaces known as stratified pseudomanifolds. There is a theoretically powerful way to arrive…

In previous works, we have introduced the blown-up intersection cohomology and used it to extend Sullivan's minimal models theory to the framework of pseudomanifolds, and to give a positive answer to a conjecture of M. Goresky and W. Pardon…

Within its traditional range of perversity parameters, intersection cohomology is a topological invariant of pseudomanifolds. This is no longer true once one allows superperversities, in which case intersection cohomology may depend on the…

We generalize the first author's construction of intersection spaces to the case of stratified pseudomanifolds of stratification depth 1 with twisted link bundles, assuming that each link possesses an equivariant Moore approximation for a…

A stratified space is a topological space together with a decomposition into strata corresponding to different types of singularities. Examples of such spaces appear everywhere in topology and geometry. The study of stratified spaces…

We develop a generalization to non-Witt spaces of the intersection homology theory of Goresky-MacPherson. The second author has described the self-dual sheaves compatible with intersection homology, and the other authors have described a…

We show that intersection homology extends Poincare duality to manifold homotopically stratified spaces (satisfying mild restrictions). This includes showing that, on such spaces, the sheaf of singular intersection chains is…

We define a simplicial category called the category of derived manifolds. It contains the category of smooth manifolds as a full discrete subcategory, and it is closed under taking arbitrary intersections in a manifold. A derived manifold…

Let (M,F) be a foliated manifold. We study the relationship between the basic cohomology Hb(M,F) of the foliation and the De Rham cohomology H(DF) of the space of leaves M/F as a quotient diffeological space. We prove that for an arbitrary…

Let X be a pseudomanifold. In this text, we use a simplicial blow-up to define a cochain complex whose cohomology with coefficients in a field, is isomorphic to the intersection cohomology of X, introduced by M. Goresky and R. MacPherson.…

We show by studying the symplectic geometry of the extended moduli space that the intersection cohomology of the representation space $Hom(\pi_1(\Sigma),G)/G$ for a simply connected compact Lie group $G$ is naturally embedded into the $G$…