A Lefschetz duality for intersection homology

Valette, Guillaume

doi:10.1007/s10711-013-9856-z

Cited by 25 publications

(8 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let us consider the following commutative diagram We deduce now from Theorem C the behaviour of the two intersection homologies in the case of a refinement. As recalled in the introduction, refinements have been treated by G. Valette in the PL framework [23] and by G. Friedman in [7]. We answer this question here for CS sets.…”

Section: Corollary 53 With the Hypotheses And The Notations Of Propmentioning

confidence: 91%

Intersection homology: General perversities and topological invariance

Chataur¹,

Saralegi-Aranguren²,

Tanré³

2019

Illinois J. Math.

View full text Add to dashboard Cite

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 p(1) = 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 p * (X) ∼ = H p * (X * ) for any perversity p with the same growth conditions verifying p(1) ≥ 0.In this work, we prove a certain topological invariance within the framework of strata-dependent perversities, p, which corresponds to the classical topological invariance if 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. Definition 1.3. A stratified space is a filtered space such that any pair of strata, S andDefinition 1.4. Let X be a stratified space. The depth of X is the greater integer m for which it exists a chain of strata, S 0 ≺ S 1 ≺ · · · ≺ S m . We denote depth X = m.Example 1.5. If X is stratified, each construction of Example 1.2 is a stratified space.Definition 1.6. A stratified map, f : X → Y , is a continuous map between stratified spaces such that, for each stratum S ∈ S X , there exists a unique stratumObserve that a continuous map f : X → Y is stratified if, and only if, the pull-back of a stratum S ′ ∈ S Y is empty or a union f −1 (S ′ ) = ⊔ i∈I S i , with codim S ′ ≤ codim S i for each i ∈ I. Therefore, a stratified map sends a regular stratum in a regular one but the image of a singular stratum can be included in a regular one. Example 1.7. Let X be a stratified space. The canonical projection, pr : M × X → X, the maps ι t : X →cX with x → [x, t], ι m : X → M × X with x → (m, x) and the canonical injection of an open subset U ֒→ X, are stratified for the structures described in Example 1.2. Let us recall some properties of stratified maps from [4, Section A.2]. Proposition 1.8 ([4, Proposition A.23]). A stratified map, f : X → Y , induces the order preserving map (S X , ) → (S Y , ), defined by S → S f .Let us introduce the notion of homotopy between stratified maps. Here, the product X × [0, 1] is endowed with the product filtration. Definition 1.9. Two stratified maps f, g : X → Y are homotopic if there exists a stratified map, ϕ : X × [0, 1] → Y , such that ϕ(−, 0) = f and ϕ(−, 1) = g. Homotopy is an equivalence relation and produces the notion of homotopy equivalence between stratified spaces.The following notion of locally cone-like stratified space has been introduced by Siebenman, [20]. Definition 1.10. A CS set of dimension n is a filtered space, ∅ ⊂ X 0 ⊆ X 1 ⊆ · · · ⊆ X n−2 ...

show abstract

Section: Corollary 53 With the Hypotheses And The Notations Of Propmentioning

confidence: 91%

Intersection homology: General perversities and topological invariance

Chataur¹,

Saralegi-Aranguren²,

Tanré³

2019

Illinois J. Math.

View full text Add to dashboard Cite

show abstract

“…Condition (K2) means that the restriction of p to the Sstratification lying on each stratum T P T is in fact a classical perversity (excepted the condition pp0q " 0). On the other hand, property (K1) is in fact a growing condition of the type (24), even weaker. Although it is not completely exact, we can think a K-perversity as a perversity whose restriction to any stratum T P T is a King perversity.…”

Section: Refinement Invariance For Cs-setsmentioning

confidence: 99%

Refinement invariance of intersection (co)homologies

Saralegi-Aranguren¹

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…Nonetheless, there are such results, typically comparing just two stratifications of the same space, X and X, with X refining X (or, equivalently, X coarsening X). In [23], Valette works with piecewise linear intersection homology on piecewise linear pseudomanifolds and arbitrary perversities p : {singular strata} → N satisfying p(S) ≤ codim(S) − 2 for each singular stratum S. He shows, in our notation, that if X refines X and if their respective perversities p and Ô satisfy Ô(S) ≤ p(S) ≤ Ô(S) + codim(S) − codim(S) whenever S is a singular stratum of X contained in the singular stratum S of X then the intersection homology groups agree, i.e. I pH * (X) ∼ = I ÔH * (X).…”

Section: Introductionmentioning

confidence: 99%

“…In Section 3 we define what we call E-compatibility between ts-perversities Ô and p on a CS set X and its refinement X; here E is a ts-coefficient system common to X and X. This condition generalizes that of Valette [23], which itself stems from the Goresky-MacPherson growth condition, by incorporating the torsion information and also allowing p 1 (S) > codim(S) − 2. The central result of the paper is Theorem 3.5, which shows that the ts-Deligne sheaves from E-compatibility ts-perversities are quasi-isomorphic.…”

Section: Introductionmentioning

confidence: 99%

Topological invariance of torsion sensitive intersection homology

Friedman

2019

Preprint

View full text Add to dashboard Cite

Torsion sensitive intersection homology was introduced to unify several versions of Poincaré duality for stratified spaces into a single theorem. This unified duality theorem holds with ground coefficients in an arbitrary PID and with no local cohomology conditions on the underlying space. In this paper we consider for torsion sensitive intersection homology analogues of another important property of classical intersection homology: topological invariance. In other words, we consider to what extent the defining sheaf complexes of the theory are independent (up to quasi-isomorphism) of choice of stratification. In addition to providing torsion sensitive versions of the existing invariance theorems for classical intersection homology, our techniques provide some new results even in the classical setting.1. H 1 (E) is a locally constant sheaf of finitely generated ℘-torsion modules, 2. H 0 (E) is a locally constant sheaf of finitely generated ℘-torsion free modules, and

show abstract

A Lefschetz duality for intersection homology

Abstract: We prove a Lefschetz duality theorem for intersection homology. Usually, this result applies to pseudomanifolds with boundary which are assumed to have a "collared neighborhood of their boundary". Our duality does not need this assumption and is a generalization of the classical one.

Cited by 25 publications

References 12 publications

Intersection homology: General perversities and topological invariance

Intersection homology: General perversities and topological invariance

Refinement invariance of intersection (co)homologies

Topological invariance of torsion sensitive intersection homology

Contact Info

Product

Resources

About