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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…
A special subcomplex of the singular chain complex for a topological space, historically called oriented singular chain complex is used here with the new name "alternative" singular chain complex. It was already known that this subcomplex…
Given pure-dimensional (generalized) cycles $\mu_1$ and $\mu_2$ on a complex manifold $Y$ we introduce a product $\mu_1\diamond_{Y} \mu_2$ that is a generalized cycle whose multiplicities at each point are the local intersection numbers at…
The method of intersection spaces associates cell-complexes depending on a perversity to certain types of stratified pseudomanifolds in such a way that Poincar\'e duality holds between the ordinary rational cohomology groups of the…
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…
We introduce techniques of Suslin, Voevodsky, and others into the study of singular varieties. Our approach is modeled after Goresky-MacPherson intersection homology. We provide a formulation of perversity cycle spaces leading to perversity…
Global intersection theories for smooth algebraic varieties via products in {\it appropriate}\, Poincar\'e duality theories are obtained. We assume given a (twisted) cohomology theory $H^*$ having a cup product structure and we let consider…
We give a new residual intersection decomposition for the refined intersection products of Fulton-MacPherson. Our formula refines the celebrated residual intersection formula of Fulton, Kleiman, Laksov, and MacPherson. The new decomposition…
We determine the product structure on Hochschild cohomology of commutative algebras in low degrees, obtaining the answer in all degrees for complete intersection algebras. As applications, we consider cyclic extension algebras as well as…
Let X be a complex analytic manifold. Given a closed subspace $Y\subset X$ of pure codimension p>0, we consider the sheaf of local algebraic cohomology $H^p_{[Y]}({\cal O}_X)$, and ${\cal L}(Y,X)\subset H^p_{[Y]}({\cal O}_X)$ the…
We prove a conjecture raised by M. Goresky and W. Pardon, concerning the range of validity of the perverse degree of Steenrod squares in intersection cohomology. This answer turns out of importance for the definition of characteristic…
We study the refinement invariance of several intersection (co)homologies existing in the literature. These (co)homologies have been introduced in order to establish the Poincar\'e Duality in variousl contexts. We found the classical…
This paper is a sequel to [Ga1]. We study the semi-infinite category on the Ran version of the affine Grassmannian, and study a particular object in it that we call the semi-infinite intersection cohomology sheaf. Unlike the situation of…
Since the 1974 paper by Peskine and Szpiro, liaison theory via complete intersections, and more generally via Gorenstein varieties, has become a standard tool kit in commutative algebra and algebraic geometry, allowing to compare algebraic…
We study the multiplicative structure of orbifold Hochschild cohomology in an attempt to generalize the results of Kontsevich and Calaque-Van den Bergh relating the Hochschild and polyvector field cohomology rings of a smooth variety. We…
We prove that the pair-of-pants product on the Floer homology of the cotangent bundle of a compact manifold M corresponds to the Chas-Sullivan loop product on the singular homology of the loop space of M. We also prove related results…
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…
The string topology coproduct is often perceived as a counterpart in string topology to the Chas-Sullivan product. However, in certain aspects the string topology coproduct is much harder to understand than the Chas-Sullivan product. In…
We study homological properties of a locally complete intersection ring by importing facts from homological algebra over exterior algebras. One application is showing that the thick subcategories of the bounded derived category of a locally…
In this paper we establish duality theorems relating Bott-Chern and Aeppli cohomology, both with and without compact support, on non-compact complex manifolds under suitable pseudoconvexity assumptions. In particular, on Stein manifolds we…