Cohomologie d’intersection modérée. Un théorème de de Rham

Cenkl, Bohumil; Hector, Gilbert; Saralegi-Aranguren, Martintxo

doi:10.2140/pjm.1995.169.235

Cited by 4 publications

(2 citation statements)

References 17 publications

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“…We define another cochain complex on K, based on a simplicial version of a blow-up, already present in [7] for differential forms on singular manifolds and for some particular simplicial complexes in [9]. To any filtered simplex, ∆ = ∆ j 0 * • • • * ∆ jn , we associate a prism, ∆ = c∆ j 0 × • • • × c∆ j n−1 × ∆ jn , where c∆ j i is the simplicial cone on ∆ j i .…”

Section: Simplicial Blow-up and Intersection-cohomologymentioning

confidence: 99%

Intersection Cohomology. Simplicial Blow-up and Rational Homotopy

Chataur¹,

Saralegi-Aranguren²,

Tanré³

2018

Memoirs of the AMS

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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 do it simplicially in the setting of a filtered version of face sets, also called simplicial sets without degeneracies, in the sense of C.P. Rourke and B.J. Sanderson. We define perverse local systems over filtered face sets and intersection cohomology with coefficients in a perverse local system. In particular, as announced above when X is a pseudomanifold, we get a perverse local system of cochains quasi-isomorphic to the intersection cochains of Goresky and MacPherson, over a field. We show also that these two complexes of cochains are quasi-isomorphic to a filtered version of Sullivan's differential forms over the field Q. In a second step, we use these forms to extend Sullivan's presentation of rational homotopy type to intersection cohomology.For that, we construct a functor from the category of filtered face sets to a category of perverse commutative differential graded Q-algebras (CDGA's) due to Hovey. We establish also the existence and unicity of a positively graded, minimal model of some perverse CDGA's, including the perverse forms over a filtered face set and their intersection cohomology. Finally, we prove the topological invariance of the minimal model of a PLpseudomanifold whose regular part is connected, and this theory creates new topological invariants. This point of view brings a definition of formality in the intersection setting and examples are given. In particular, we show that any nodal hypersurface in CP(4), is intersection-formal.

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Section: Simplicial Blow-up and Intersection-cohomologymentioning

confidence: 99%

Intersection Cohomology. Simplicial Blow-up and Rational Homotopy

Chataur¹,

Saralegi-Aranguren²,

Tanré³

2018

Memoirs of the AMS

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“…Inspirited by [1] and the notion of complexity we introduce the residual complex LpKq which allow us to adapt the previous strategy: (1) (2) the description of the relative complex pK, LpKqq as a disjoint union of pβ ˚Lβ , Bβ ˚Lβ q, where each link L β is a filtered complex of smaller complexity, (3) and the verification of the existence of isomorphisms between the singular and the simplicial intersection homologies of β ˚Lβ , by computation and induction, and of Bβ ˚Lβ " Lpβ ˚Lβ q by induction.…”

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Simlpicial intersection homology revisited

Chataur,

Saralegi-Aranguren,

Tanré

2021

Preprint

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Intersection homology is defined for simplicial, singular and PL chains. In the case of a filtered simplicial complex, it is well known that the three versions are isomorphic. This isomorphism is established by using the PL case as an intermediate between the singular and the simplicial situations. Here, we give a proof similar to the classical proof for ordinary simplicial complexes.We also study the intersection blown-up cohomology that we have previously introduced. In the case of a pseudomanifold, this cohomology owns a Poincaré isomorphism with the intersection homology, for any coefficient ring, thanks to a cap product with a fundamental class. We prove that the simplicial and the singular blown-up cohomologies of a filtered simplicial complex are isomorphic. From this result, we can now compute the blown-up intersection cohomology of a pseudomanifold from a triangulation.Finally, we introduce a blown-up intersection cohomology for PL-spaces and prove that it is isomorphic to the singular ones. We also show that the cup product in perversity 0 of a CS-set coincides with the cup product of the singular cohomology of the underlying topological space.The cohomology of a triangulated manifold can be computed in many different ways :-using the simplices of the triangulation, -using the simplicial set of singular simplices, -using sheaf theory, -using homotopy classes into Eilenberg-MacLane spaces. This paper and our previous works on blown-up cochains [4,7,8] show that we have the same picture for intersection cohomology. Furthermore, we show the existence of an isomorphism between the singular and simplicial definitions of intersection homology, without involving the PL structure and without limitation to CS sets.Let us recall some historical background of intersection homology as it was introduced by Goresky and MacPherson. In their first paper on intersection homology, they introduce it for a pseudomanifold X together with a fixed PL-structure and a parameter p, now called Goresky-MacPherson perversity. They define the complex of p-intersection, IC p ˚pX q, as a subcomplex of the complex of PL-chains and the p-intersection homology as the homology of IC p ˚pX q. Let R be a commutative ring of coefficients. We could also start with a fixed triangulation on a stratified Euclidean simplicial complex, K, and define a complex of p-intersection, C p ˚pK q, as a subcomplex of the simplicial chains. This is, for instance, the concrete case in [16], where MacPherson and Vilonen give a construction of the category of perverse constructible sheaves on a Thom-Mather stratified space with a fixed triangulation. In an appendix to this paper, Goresky and MacPherson prove that the two intersection homologies are isomorphic; i.e., the simplicial one on K and the previous PL one on the PL space X associated to K. They do that with a nice construction of a left inverse to the inclusion C p ˚pK q Ñ IC p ˚pX q, see the book of Friedman ([9, Section 3.3]) for a detailed version of this argumentation.A third step is to consider the...

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The de Rham theorem for the noncommutative complex of Cenkl and Porter

Mejias

2002

International Journal of Mathematics and Mathematical Sciences

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We use noncommutative differential forms (which were first introduced by Connes) to construct a noncommutative version of the complex of Cenkl and PorterΩ∗,∗(X)for a simplicial setX. The algebraΩ∗,∗(X)is a differential graded algebra with a filtrationΩ∗,q(X)⊂Ω∗,q+1(X), such thatΩ∗,q(X)is aℚq-module, whereℚ0=ℚ1=ℤandℚq=ℤ[1/2,…,1/q]forq>1. Then we use noncommutative versions of the Poincaré lemma and Stokes' theorem to prove the noncommutative tame de Rham theorem: ifXis a simplicial set of finite type, then for eachq≥1and anyℚq-moduleM, integration of forms induces a natural isomorphism ofℚq-modulesI:Hi(Ω∗,q(X),M)→Hi(X;M)for alli≥0. Next, we introduce a complex of noncommutative tame de Rham currentsΩ∗,∗(X)and we prove the noncommutative tame de Rham theorem for homology: ifXis a simplicial set of finite type, then for eachq≥1and anyℚq-moduleM, there is a natural isomorphism ofℚq-modulesI:Hi(X;M)→Hi(Ω∗,q(X),M)for alli≥0.

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Cohomologie d’intersection modérée. Un théorème de de Rham

Cited by 4 publications

References 17 publications

Intersection Cohomology. Simplicial Blow-up and Rational Homotopy

Intersection Cohomology. Simplicial Blow-up and Rational Homotopy

Simlpicial intersection homology revisited

The de Rham theorem for the noncommutative complex of Cenkl and Porter

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