Intersection homology: General perversities and topological invariance

Abstract

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 ...

Blown-up intersection cochains and Deligne’s sheaves

“…• It satisfies Mayer-Vietoris property, [5,Proposition 7.10]. This comes from the fact that the classical subdivision operator Sd : C * (X; R) → C * (X; R) can also be defined on the complex C p * (X; R) and gives a quasi-isomorphism.…”

“…This comes from the fact that the classical subdivision operator Sd : C * (X; R) → C * (X; R) can also be defined on the complex C p * (X; R) and gives a quasi-isomorphism. As a consequence if we consider an open covering U of X the inclusion of U -small chains [5,Corollaire 7.13].…”

Topological invariance of torsion sensitive intersection homology

“…In [5], Chataur, Saralegi, and Tanré consider the invariance of intersection homology under refinement/coarsening when the perversity on the finer stratification is pulled back from a perversity on the coarser stratification or when the perversity on the coarser stratification is pushed forward from the finer stratification. In the following two subsections we consider such constructions for ts-perversities.…”

“…More recently in [5], Chataur, Saralegi-Aranguren, and Tanré consider what they call K * -perversities and show that a K * -perversity on a CS set X can be pushed forward to a perversity on the intrinsic coarsest stratification X * and that the two resulting intersection homology groups are isomorphic. This theorem holds for non-GM intersection homology (which is called "tame intersection homology" in [5]), and there is also a version for GMintersection homology with fewer conditions on the perversities. They also show that it is similarly possibly to pull a K * -perversity back to any refinement of X and obtain isomorphic intersection homology groups.…”

“…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. In Sections 3.1.1 and 3.1.2 we apply Theorem 3.5 to pullback and pushforward perversities, recovering generalizations of the theorems of Chataur-Saralegi-Tanré [5]. In particular, we discuss pushforwards to arbitrary coarsenings, not just the intrinsic coarsening.…”