Topological invariance of torsion sensitive intersection homology
aa r X i v : . [ m a t h . G T ] A ug Topological invariance oftorsion sensitive intersection homology
Greg FriedmanTexas Christian [email protected] 30, 2019
Contents > Primary: 55N33, 55N30, 57N80, 55M051 eywords: intersection homology, intersection cohomology, Deligne sheaf, CSset, perversity, stratification
Abstract
Torsion sensitive intersection homology was introduced to unify several versions ofPoincar´e duality for stratified spaces into a single theorem. This unified duality theo-rem holds with ground coefficients in an arbitrary PID and with no local cohomologyconditions on the underlying space. In this paper we consider for torsion sensitiveintersection homology analogues of another important property of classical intersec-tion homology: topological invariance. In other words, we consider to what extent thedefining sheaf complexes of the theory are independent (up to quasi-isomorphism) ofchoice of stratification. In addition to providing torsion sensitive versions of the ex-isting invariance theorems for classical intersection homology, our techniques providesome new results even in the classical setting.
In [6] we introduced categories of torsion sensitive perverse sheaves (more briefly ts-perversesheaves ) and studied their duality properties. In the classical category of perverse sheaveson a stratified pseudomanifold [1], the intermediate extensions of the coefficient systems arethe “Deligne sheaves” whose hypercohomology groups are the intersection homology groupsof Goresky and MacPherson. The primary motivation in [6] was to create a generalizationof these Deligne sheaves for which the various intersection homology duality theorems ofGoresky-MacPherson [13, 14], Goresky-Siegel [15], and Cappell-Shaneson [4] all arise asspecial cases of a single more general duality theorem that incorporates certain torsionphenomena into the sheaf complexes but does not require the special local cohomologicalconditions on spaces that are needed for some of the original theorems. Indeed, the ts-Deligne sheaves of [6], which are the intermediate extensions of ts-coefficient systems , fulfillthat goal, and furthermore they can be characterized by a simple set of axioms generalizingthe Deligne sheaf axioms of Goresky and MacPherson.After providing a generalization of Poincar´e duality for singular spaces, the next mostimportant property of intersection homology is its topological invariance: while the inter-section homology groups are defined in terms of a stratification of the space, the resultingintersection homology groups are independent of the choice of stratification, at least assumingcertain restrictions on the perversity parameters. In this paper we consider the topologicalinvariance of the ts-Deligne sheaves up to quasi-isomorphism, including confirming a con-jecture made in [6]. In addition to extending versions of past topological invariance resultsto the torsion sensitive category, our techniques specialize to improve the previous knownresults about ordinary intersection homology. To explain, we outline some of the history.
History.
The original intersection homology groups of Goresky and MacPherson [14] aredefined on stratified pseudomanifolds and depend on perversity parameters ¯ p : Z ≥ → Z satisfying the original Goresky-MacPherson conditions: ¯ p (2) = 0 and ¯ p ( k ) ≤ ¯ p ( k + 1) ≤ p ( k ) + 1. If X is an n -dimensional stratified pseudomanifold, so in particular a filtered space X = X n ⊃ X n − ⊃ · · · ⊃ X with each X m − X m − an m -manifold (possibly empty), thenthe Deligne sheaf is constructed beginning with a local system E on X − X n − and thenperforming a sequence of pushforwards and truncations over strata of increasing codimension.The perversity value ¯ p ( k ) determines the truncation degree following the pushforward to thecodimension k strata. In [14], Goresky and MacPherson showed that for a fixed perversityand local system the resulting sheaves are independent (up to quasi-isomorphism) of theprecise choice of pseudomanifold stratification; a more detailed exposition was provided byBorel in [2, Section V.4].King [18] later gave a proof of the topological invariance of intersection homology with-out using sheaves and requiring only that ¯ p be nonnegative as well as the growth condition.Furthermore, King worked in the broader category of CS sets and allowed strata of codi-mension one. However, it should be noted that when ¯ p has values such that ¯ p ( k ) > k − p ( k ) ≤ k − k , as isthe case in [14], then these theories agree. A sheaf theoretic approach to GM intersectionhomology and its topological invariance can be found in Habegger and Saper [16], while asingular chain approach to non-GM intersection homology has been developed in [10, 20, 7].Topological invariance of non-GM intersection homology is considered in [9], where it isshown that topological invariance holds with ¯ p (1) > p ( k ) ≤ ¯ p ( k + 1) ≤ ¯ p ( k ) + 1 so longas all changes to the stratification occur within a fixed choice of X n − , but not in generalotherwise.It has since become apparent that it is useful to utilize perversities that depend not juston codimension but on the strata themselves so that we define ¯ p : { singular strata } → Z (recall that if X is an n -dimensional CS set we call the n -dimensional strata regular andthe lower dimensional strata singular ). A version of Deligne sheaves suited to such generalperversities is defined in [11], and the corresponding singular chain non-GM intersectionhomology is studied in this generality in [7]. Clearly in this generality topological invariancebecomes a more subtle issue. Nonetheless, there are such results, typically comparing just twostratifications 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 linearpseudomanifolds and arbitrary perversities ¯ p : { singular strata } → N satisfying ¯ p ( S ) ≤ codim( S ) − S . He shows, in our notation, that if X refines X andif their respective perversities ¯ p and ¯ p satisfy ¯ p ( S ) ≤ ¯ p ( S ) ≤ ¯ p ( S ) + codim( S ) − codim( S )whenever S is a singular stratum of X contained in the singular stratum S of X then theintersection homology groups agree, i.e. I ¯ p H ∗ ( X ) ∼ = I ¯ p H ∗ ( X ). Note that with Valette’sassumptions the GM and non-GM intersection homologies automatically agree.More recently in [5], Chataur, Saralegi-Aranguren, and Tanr´e consider what they call K ∗ -perversities and show that a K ∗ -perversity on a CS set X can be pushed forward to aperversity on the intrinsic coarsest stratification X ∗ and that the two resulting intersectionhomology groups are isomorphic. This theorem holds for non-GM intersection homology3which is called “tame intersection homology” in [5]), and there is also a version for GM-intersection homology with fewer conditions on the perversities. They also show that it issimilarly possibly to pull a K ∗ -perversity back to any refinement of X and obtain isomorphicintersection homology groups. Our results below include the non-GM (tame) intersectionhomology versions of these theorems as well as those of Valette as special cases. Results.
We now outline our results, mostly in order of presentation below. We workthroughout from the sheaf-theoretic point of view, which has the benefit of easily allowing fortwisted coefficient systems and also in that quasi-isomorphism of sheaves implies isomorphismof the hypercohomology groups with any system of supports. Thus, in particular, our sheafquasi-isomorphisms imply isomorphisms of intersection homology groups both with compactsupports and with closed supports, the latter corresponding to intersection homology oflocally-finite singular chains [10, 11].In Section 2 we review background material, including definitions of ts-perversities, ts-coefficient systems, and ts-Deligne sheaves, all of which generalize the standard versions. Inparticular, a ts-perversity is a function ~p : { singular strata } → Z × P ( R ) , where P ( R ) is theset of primes (up to unit) of our ground PID R and 2 P ( R ) is its power set. We write ~p ( S ) =( ~p ( S ) , ~p ( S )). Using this additional information about primes, torsion data is incorporatedinto the definition of the ts-Deligne sheaf utilizing the “torsion-tipped truncation” functorconstructed in [6] in place of the standard truncation. If ~p ( S ) = ∅ for all S and the ts-coefficient system is just a local system in degree 0 then the ts-Deligne sheaf reduces to theclassical Deligne sheaf [14, 11]. Subsection 2.1 discusses some further natural assumptionsabout coefficient systems that will be utilized in our broadest topological invariance results.In Section 3 we define what we call E -compatibility between ts-perversities ~ p 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 ( S ) > codim( S ) −
2. The central result of the paper is Theorem 3.5, which shows thatthe ts-Deligne sheaves from E -compatibility ts-perversities are quasi-isomorphic. In Sections3.1.1 and 3.1.2 we apply Theorem 3.5 to pullback and pushforward perversities, recoveringgeneralizations of the theorems of Chataur-Saralegi-Tanr´e [5]. In particular, we discusspushforwards to arbitrary coarsenings, not just the intrinsic coarsening.In Section 3.2 we consider quasi-isomorphisms of ts-Deligne sheaves arising from twoCS set stratifications of a space without the assumption that one refines the other. Thisrequires restricting ourselves to ts-perversities that depend only on codimension, so func-tions ~p : Z ≥ → Z × P ( R ) , such that ~p satisfies the Goresky-MacPherson growth conditionand ~p satisfies certain growth conditions on sets of primes. These ts-perversities are called constrained or weakly constrained depending on our requirements for the value of ~p (2). Forconstrained ts-perversities that are also adapted to the ts-coefficient systems E (a conditionrelating torsion information about E with ~p (2)), we show in Theorem 3.20 that any twostratifications yield quasi-isomorphic ts-Deligne sheaves so long as the closures of their codi-mension one strata agree. In particular this theorem holds with the classical assumptionthat codimension one strata are forbidden. We also show in the same theorem that we can4eaken the hypotheses to weakly constrained perversities and no compatibility requirementbetween ~p and E so long as the the two stratifications have the same regular strata (or,equivalently, the same codimension one skeleta). These two results generalize the classicalGoresky-MacPherson topological invariance in [14] and that for “superperversities” in [9].The key idea is to apply Theorem 3.5 using appropriate common coarsenings of the two strat-ifications. Such intrinsic stratifications, relative to coefficient systems and fixed subspaces,are constructed in Section 6, generalizing those in [18] and [16].Section 4 concerns the extent to which the conditions for E -compatibility between ts-perversities are necessary in order to obtain quasi-isomorphic ts-Deligne sheaves. We showthat the conditions on singular strata of X contained in regular strata of the coarsening X arestrictly necessary: if they fail for any stratum the sheaves cannot be quasi-isomorphic. Bycontrast, the conditions on singular strata of X contained in singular strata of the coarsening X are only “necessary in general,” meaning that we can construct examples in which failureof the conditions implies failure of quasi-isomorphism. However, these conditions may notbe necessary in special cases, for example if certain stalk cohomology groups vanish due tothe specific topology of some space; see Section 4.3 for further details. One of our maintools in this section will be a formula for computing the ts-Deligne sheaf hypercohomologyfor a join S k ∗ X in terms of the ts-Deligne sheaf hypercohomology of X ; see Corollary 4.5.This formula is obtained by first computing the hypercohomology for the suspension Σ X inProposition 4.4, which is illuminating in its own right, and then making a nice applicationof Theorem 3.5 to the iterated suspension.The original Goresky-MacPherson proof of topological invariance involved support and“cosupport” axioms concerning the dimensions on which H i ( f ∗ x P ∗ ) and H i ( f ! x P ∗ ) may failto vanish, P ∗ being the Deligne sheaf and f x the inclusion of the point x into X . Our ar-guments to this point do not involve these axioms and so are fundamentally different fromthose in [14]. In Section 5 we develop versions of these support and cosupport axioms forts-Deligne sheaves with strongly or weakly constrained ts-perversities. Strongly constrainedts-perversities require ~p (2) = 0 while simply “constrained” is a bit weaker; the stronger con-straint in this section is not strictly necessary but simplifies the discussion. In the stronglyconstrained case we provide criteria to recognize ts-Deligne sheaves without reference to anyspecific stratification, leading to Theorem 5.9, a statement of topological invariance moreanalogous to the original Goresky-MacPherson invariance theorem of [14, Uniqueness Theo-rem] or [2, Theorem 4.15]. The weakly constrained version, Theorem 5.13, again requires afixed choice of the regular strata and is more analogous to the main theorem of [9].Lastly, Section 6 concerns the details about relative intrinsic stratifications. Remarks.
When ~p ( S ) = ∅ for all S and E is a local system concentrated in degree 0,our ts-Deligne sheaves reduce to the Deligne sheaves of [14, 11]. With this assumption,many, though not all, of our results reduce to some previously-known theorems, as outlinedabove. However, we believe that even in these cases our proofs are quite different, as ourmain invariance results in Section 3 do not require analogues of the Goresky-MacPhersonsupport and cosupport axioms. For the reader interested only in the classical Deligne sheavesand Goresky-MacPherson perversities, we have extracted a simplified version of this new5rgument and presented it in [8] together with a very short second proof of the topologicalinvariance of classical intersection homology that does use support and cosupport axioms.We thank J¨org Sch¨urmann for pointing out some very helpful references and Scott Nolletfor many useful conversations. We also thank David Chataur, Martin Saralegi-Aranguren,and Daniel Tanr´e for both ongoing stimulating mathematical discussion and their generoushospitality. Basics and Notation.
Our spaces will be paracompact dimensionally homogeneous CSsets. Defined by Siebenmann [22], an n -dimensional CS set X is a Hausdorff space equippedwith a filtration X = X n ⊃ X n − ⊃ · · · X ⊃ X − = ∅ such that X k := X k − X k − is a k -manifold (possibly empty) and for x ∈ X k there is an openneighborhood U of x in X k , an open neighborhood N of x in X , a compact filtered space L (which may be empty), and, letting cL denote the open cone on L , a homeomorphism h : U × cL → N such that h ( U × c ( L j )) = X k + j +1 ∩ N for all j . The space L is calleda link of x and N is called a distinguished neighborhood of x . Note that if L = ∅ then cL = ( cL ) is a point. Dimensional homogeneity means that we assume X − X n − is dense.Such spaces are locally compact [7, Lemma 2.3.15], metrizable [5, Proposition 1.11], and offinite cohomological dimension ([7, Lemma 6.3.46] and [3, Theorem II.16.8]). See [7, Section2.3] for more details about CS sets in general. All CS sets in this paper will be assumedparacompact and dimensionally homogeneous without further mention. We also assume X is n -dimensional unless specified otherwise.Following Borel [2, Section V.2], we let U k = X − X n − k , and noting that U k +1 is thedisjoint union of U k and X n − k , we also take i k : U k ֒ → U k +1 and j k : X n − k ֒ → U k +1 . For any x ∈ X , we write f x : { x } ֒ → X . The connected components of X k are the k -dimensional strata . Strata in X n = X n − X n − are regular strata and strata in X k for k ≤ n − singular strata . Note that strata may have codimension one, which is sometimes forbiddenin other contexts.We often abuse notation and use X to refer both to the underlying space and to the spaceequipped with the stratification; when we wish to emphasize the underlying space or do notyet want to specify the stratification we also write | X | . If X and X are two stratificationsof the same space | X | , we say that X coarsens X , or that X is a refinement of X , if eachstratum of X is a union of strata of X . Our standard notation will be X for a CS set, X for acoarsening of X , X for some unrelated stratification of | X | , and X for a common coarseningof X and X or for the intrinsic stratifications constructed in Section 6.Algebraically, we fix a PID R as our ground ring throughout, and we let P ( R ) be the setof primes of R up to unit. This means that the elements of P ( R ) are technically equivalenceclasses such that p ∼ q if p = uq for some unit u , though we will abuse notation by letting aprime stand for its equivalence class; cf. [6, Section 2]. The following is Definition 2.1 of [6].6 efinition 2.1. If A is a finitely-generated R -module and ℘ ⊂ P ( R ), we define the ℘ -torsionsubmodule of A to be T ℘ A = ( x ∈ A | nx = 0 for some product n = s Y i =1 p m i i such that p i ∈ ℘ and m i , s ∈ Z ≥ ) , i.e. T ℘ A is the submodule annihilated by products of powers of primes in ℘ . If T ℘ A = A ,we say that A is ℘ -torsion . If T ℘ A = 0, we say that A is ℘ -torsion free. We take the emptyproduct to be 1, so in particular if ℘ = ∅ then T ℘ A = 0 and every A is ∅ -torsion free. If p ∈ P ( R ) is a single element, we abuse notation and write T p A instead of T { p } A .In Section 5 it will also be useful to define P + ( R ) = P ( R ) ∪ { f } , where f is a formallyadjoined element to P ( R ). This lets us define the following: Definition 2.2. If A is a finitely-generated R -module and f ∈ ℘ ⊂ P + ( R ), let T ℘ A denote A/T ℘ c , where ℘ c is the complement of ℘ in P + ( R ).In particular, T f A = A/T P ( R ) A is the torsion free quotient of A . If A is a finitelygenerated R -module then A ∼ = T ℘ A ⊕ T ℘ c A for any ℘ ⊂ P + ( R ) thanks to the structuretheorem for finitely-generated modules over a PID. ts-Deligne sheaves. We now recall some material from [6], leading to the definition of ts-Deligne sheaves. All sheaves are sheaves of R -modules, and we think of ourselves as workingin the derived category so that ∼ = denotes quasi-isomorphism. If S ∗ is a sheaf complex, then H i ( S ∗ ) denotes the derived cohomology sheaf and H i ( X ; S ∗ ) denotes hypercohomology.We begin with ts-perversities [6, Definitions 4.1 and 4.18]: Definition 2.3.
For a PID R , let P ( R ) be the set of primes of R (up to unit), and let 2 P ( R ) be its power set. A torsion-sensitive perversity (or simply ts-perversity ) on a CS set X isa function ~p : { singular strata of X } → Z × P ( R ) . We denote the components of ~p ( S ) by ~p ( S ) = ( ~p ( S ) , ~p ( S )).The complementary ts-perversity , or dual ts-perversity , D~p is defined by
D~p ( S ) = (codim( S ) − − ~p ( S ) , P ( R ) − ~p ( S )), i.e. the first component is the complementary perversity to ¯ p inthe Goresky-MacPherson sense and the second component is the set of primes in R comple-mentary to ~p ( S ).We also recall the notion of a ℘ -coefficient system, slightly generalizing [6, Definition 4.2].These objects are in the hearts of the natural t-structures constructed in [6, Definition 5.1]: Definition 2.4.
Let ℘ ⊂ P ( R ) be a set of primes of the PID R . We will call a complex ofsheaves E on a space M a ℘ -coefficient system if1. H ( E ) is a locally constant sheaf of finitely generated ℘ -torsion modules,2. H ( E ) is a locally constant sheaf of finitely generated ℘ -torsion free modules, and Even though E is a complex of sheaves, we do not write E ∗ in order to emphasize the role of E ascoefficients. H i ( E ) = 0 for i = 0 , M is a disjoint union of spaces, we call E a ts-coefficient system if itrestricts on each component of M to a ℘ -coefficient system for some ℘ (which may vary bycomponent).If X is a CS set and E is a ts-coefficient system defined over a subset U ⊂ X , we saythat the stratification of X is adapted to E if X − X n − ⊂ U , i.e. if E is defined on (at least)the regular strata of X ; cf. [2, Section V.4.12]. We will call U the domain of E . Remark . As noted in [2, Remark V.4.14.c], even if we restrict our coefficients to localsystems concentrated in a single degree and defined on dense open submanifolds of pseudo-manifolds, there can exist E for which there does not exist a CS set stratification adaptedto E . For example, define E to be the local system on R − (0 , ∪ { (1 /n, | n ∈ Z ≥ } withstalk Z and nontrivial monodromy on a small loop around each (1 /n, ~p and a ts-coefficient system E to which X is adapted, the associatedts-Deligne sheaf is defined for pseudomanifolds in [6, Definition 4.4]. The details all hold aswell for CS sets, and the ts-Deligne sheaf P ∗ X,~p, E , often written simply as P ∗ , is defined as P ∗ X,~p, E = t X ≤ ~p Ri n ∗ . . . t X n − ≤ ~p Ri ∗ E . Here each t X k ≤ ~p is a locally torsion-tipped truncation functor as defined in [6, Section 3]. Werefer the reader there for more details but note that for S ∗ defined on U k +1 we have1. (cid:16) t X n − k ≤ ~p S ∗ (cid:17) x = S ∗ x if x ∈ U k ,2. if x ∈ S ⊂ S n − k for a singular stratum S then H i (cid:16)(cid:16) t X n − k ≤ ~p S ∗ (cid:17) x (cid:17) ∼ = , i > ~p ( S ) + 1 ,T ~p ( S ) H i ( S ∗ x ) , i = ~p ( S ) + 1 ,H i ( S ∗ x ) , i ≤ ~p ( S ) . If E is a local system (i.e. a locally constant sheaf of finitely generated R -modules)concentrated in degree 0, if ~p satisfies the Goresky-MacPherson conditions, and if ~p ( S ) = ∅ for all S , then this is just the classical Deligne sheaf of Goresky and MacPherson from [14].Analogously to the Goresky-MacPherson Deligne sheaves, the ts-Deligne sheaves can becharacterized by axioms. Here is the first set of axioms from [6, Definition 4.7], generalizedfor CS sets. We write S ∗ k for S ∗ | U k . Definition 2.6.
Let X be an n -dimensional CS set, and let E be a ts-coefficient system on U . We say that the sheaf complex S ∗ on X satisfies the Axioms TAx1 ( X, ~p, E ) ifa. S ∗ is quasi-isomorphic to a complex that is bounded and that is 0 in negative degrees;b. S ∗ | U ∼ = E | U ; 8. if x ∈ S ⊂ X n − k , where S is a singular stratum, then H i ( S x ) = 0 for i > ~p ( S ) + 1and H ~p ( S )+1 ( S x ) is ~p ( S )-torsion;d. if x ∈ S ⊂ X n − k , where S is a singular stratum, then the attachment map α k : S k +1 → Ri k ∗ S k induces stalkwise cohomology isomorphisms at x in degrees ≤ ~p ( S ) and it in-duces stalkwise cohomology isomorphisms H ~p ( S )+1 ( S k +1 ,x ) → T ~p ( S ) H ~p ( S )+1 (( Ri k ∗ S k ) x ).Theorem 4.8 of [6], which also works for CS sets, shows that the ts-Deligne sheaf com-plex P ∗ X,~p, E satisfies the axioms TAx1( X, ~p, E ), and conversely any sheaf complex satisfyingTAx1( X, ~p, E ) is quasi-isomorphic to P ∗ X,~p, E . It is also observed in [6, Theorem 4.10] thatthese sheaf complexes are X -clc, meaning that each sheaf H i ( P ∗ ) is locally constant on eachstratum. This also continues to hold for CS sets, which have the property that if j is anyinclusion of a locally closed subset that is a union of strata then j ∗ , j ! , j ! , and Rj ∗ all preserve constructibility by [21, Proposition 4.0.2.3] (see also [21, Proposition 4.2.1.2.b]).As in [14, 2, 6], we can reformulate some of these axioms. Definition 2.7.
We say S ∗ satisfies the Axioms TAx1’ ( X, ~p, E ) ifa. S ∗ is X -clc and it is quasi-isomorphic to a complex that is bounded and that is 0 innegative degrees;b. S ∗ | U ∼ = E | U ;c. if x ∈ S ⊂ X n − k , where S is a singular stratum, then H i ( S ∗ x ) = 0 for i > ~p ( S ) + 1and H ~p ( S )+1 ( S ∗ x ) is ~p ( S )-torsion;d. if x ∈ S ⊂ X n − k , where S is a singular stratum, and f x : x ֒ → X is the inclusion, then(a) H i ( f ! x S ) = 0 for i ≤ ~p ( S ) + n − k + 1(b) H ~p ( S )+ n − k +2 ( f ! x S ) is ~p ( S )-torsion free.The following theorem is a slight generalization of [6, Theorem 4.13]: Theorem 2.8.
On a CS set, the axioms TAx1’ ( X, ~p, E ) are equivalent to the axioms TAx1 ( X, ~p, E ) and so any sheaf complex satisfying TAx1’ ( X, ~p, E ) is quasi-isomorphic to P ∗ X,~p, E . The proof is the same as that of [6, Theorem 4.13], replacing the theorems about con-structibility invoked from Borel (e.g. [2, Lemma V.3.10.d]) with Sch¨urmann’s [21, Proposition4.0.2.3].
Many of our theorems below compare ts-Deligne sheaves on two stratifications of a singleCS set, one coarsening another. For this it suffices to have a ts-coefficient system E definedon the regular strata of the coarser of the two stratifications for then it restricts also to ats-coefficient system on the regular strata of the finer stratification. However, we will also9e interested in theorems concerning arbitrary stratifications, and in these cases we willneed to construct common coarsenings that remain adapted to E . The full details will beprovided in Section 6, though we discuss here some notions about coefficient systems thatwill be necessary at that point as these will also be needed in some of our earlier theoremstatements. In particular, to construct these common coarsenings we will need to make someminor assumptions about the domains of our ts-coefficient systems.To motivate our restrictions, we recall that for classical intersection homology theory ona stratified pseudomanifold X it is observed by Borel in [2, Section V.4] that if E is a localsystem (i.e. a locally constant sheaf of finitely-generated R -modules) defined on a dense opensubmanifold of X whose complement has codimension ≥ X to which E can be extended uniquely up to isomorphism [2, LemmaV.4.11] (though this submanifold may not necessarily be the largest n -dimensional manifoldcontained in X due to monodromy). Since it is not clear that such a statement holds forthe more general ts-coefficient systems, we instead build a maximality assumption into ourcoefficients when necessary. Since local systems have unique such maximal extensions, wecan convince ourselves that we therefore do not lose much generality. Alternatively, if welimit ourselves to E composed of local systems, then Proposition 2.10 below shows thatmaximality can be guaranteed. Definition 2.9.
Let X be an n -dimensional CS set. We will call a sheaf complex E on X a maximal ts-coefficient system if1. the domain of definition of E includes an open n -dimensional submanifold U E of X whose complement has codimension ≥ E is a ts-coefficient system over U E (see Definition 2.4), and3. there is no larger submanifold of X to which E extends.Clearly ts-coefficient systems composed of constant sheaves defined on all of X are max-imal. The following lemma shows that ts-coefficient systems composed of locally constantsheaves (on open submanifolds of codimension at least 2) can be made maximal. Proposition 2.10.
Suppose E is a ts-coefficient system defined on an open dense subman-ifold whose complement has codimension at least 2. If E is bounded (i.e. E i = 0 for suffi-ciently large | i | ) and each E i is a local system (a locally constant sheaf of finitely-generated R -modules), then E has a maximal extension that is unique of to isomorphism. Furthermore,if X is adapted to E then X remains adapted to the extension.Proof. By assumption, each E i is defined on an open dense submanifold U ⊂ X whosecomplement has codimension at least 2, and so by [2, Lemma V.4.11] each E i has a unique This is no longer true if the local system is only defined on a dense open set whose complement hascodimension 1. For example let X = S with stratification S ⊃ { pt } . Suppose E is the constant sheaf withstalk Z on S − { pt } . Then there are two non-isomorphic extensions of E to S , namely the constant sheafwith stalk Z and the twisted sheaf with stalks Z such that a generator of π ( S ) acts by multiplication by − E i to a maximal open subset U i . Let W = ∩ i U i , which remainsopen and dense since all but finitely many of the U i will be the largest open submanifold of X . Since U ⊂ U i , we also have U ⊂ W , and we let ¯ E i = ˜ E i | W . Also by [2, Lemma V.4.11],each boundary map E i → E i +1 extends uniquely to a map ¯ E i → ¯ E i +1 . This gives us a unique(up to isomorphisms) complex ¯ E ∗ on W that cannot be extended to a larger submanifold of X . The last statement of the lemma is trivial.Another nice property of local systems is that if E is a maximal local system, X is adaptedto E with no codimension one strata, and U E is the maximal submanifold over which E isdefined, then U E is a union of strata of X . This is shown at the bottom of [2, page 92]. Wewill also need a property like this to define our common coarsenings, which motivates thefollowing definition. Once again we will then show that this condition is automatic when E consists of local systems and there are no codimension one strata. Definition 2.11.
Suppose E is a maximal ts-coefficient system on X and that U E is thelargest open submanifold on which E is defined. We say that the stratification of X is fullyadapted to E if1. X − X n − ⊂ U E , and2. U E is a union of strata of X . Proposition 2.12.
Suppose E is a maximal ts-coefficient system on X such that each E i is a local system. If X has no codimension one strata and is adapted to E then it is fullyadapted.Proof. The proof is essentially the same as the argument on [2, page 92]: We will proceed bycontradiction. Let S be a stratum of X of minimal codimension so that S intersects U E butis not contained in it. Since X is adapted to E we must have codim( S ) ≥
1. Suppose x ∈ S has a distinguished neighborhood N ∼ = B × cL such that there is some point y ∈ B × { v } (with v the cone vertex) such that y ∈ U E . We claim that then x ∈ U E .First, since x and y are in the same stratum of X and as y must have a Euclideanneighborhood in X (since y ∈ U E ), [7, Lemma 2.10.4] implies that x also has a Euclideanneighborhood; thus both x and y (and similarly all points of B × { v } ) are contained in themaximal submanifold of X . Furthermore, by assumption U E must contain N − ( N ∩ S ) = B × ( cL − { v } ), and if π : B × cL → cL is the projection then π ∗ ( E | { y }× cL ) is a local systemon N whose restriction to N − S is isomorphic to E | N − S . Since extension of local systems isunique when there are no codimension one strata by [2, Lemma V.4.11], π ∗ ( E | { y }× cL ) mustagree with E where they overlap, and so we must have N ⊂ U E or else the maximality of U E would be contradicted.Now, since U E is open in X , we have U E ∩ S open in S . The above argument shows thatif x is in the closure of U E ∩ S in S then x ∈ U E ∩ S . So U E ∩ S is open and closed in theconnected set S and is thus all of S .The preceding proposition can fail if there are codimension one strata:11 xample . Let E be the local system on R −{ } with Z stalks and nontrivial monodromyaround the origin. Let X = R filtered as R ⊃ x -axis. Then E is maximal and X is adaptedto E , but it is not fully adapted, though it can be refined to be so.As a more dramatic example, consider the example from Remark 2.5 of a maximal localsystem E that is defined on the complement in R of (0 , ∪ { (0 , /n ) | n ∈ Z ≥ } . If we againfilter X = R as R ⊃ x -axis then again X is adapted to E , but there is no fully adaptedrefinement. In this section we prove our main topological invariance theorems. These are mostly suffi-ciency statements, demonstrating that if certain conditions hold between different perver-sities on different stratifications of the same space, as well as certain relations between theperversity on the more refined stratification and the ts-coefficient system, then the two cor-responding ts-Deligne sheaves are quasi-isomorphic. We will consider necessity in Section 4.In particular, we’ll show that the conditions involving a singular stratum that is containedin a regular stratum of a coarsening are strictly necessary. However the conditions involvinga singular stratum that is contained in a singular stratum of a coarsening are only “neces-sary in general,” meaning that there are examples in which the failure of the conditions willprevent quasi-isomorphism of ts-Deligne sheaves but that the conditions may not be neces-sary if further assumptions are made about cohomology of links. The singular-in-singularconditions are also not necessary if the first components of the perversities take on extremevalues; see Remark 3.4 just below.We begin with our sufficient conditions.
Definition 3.1.
Suppose that X and X denote two CS set stratifications of the same under-lying space with X coarsening X . Let ~p and ~ p be respective ts-perversities on X and X , andlet E be a ts-coefficient system to which X (and hence also X ) is adapted. We will say that ~p and ~ p are E -compatible ts-perversities if the following conditions hold whenever a singularstratum S of X is contained in a (singular or regular) stratum S of X :1. If S is singular then ~ p ( S ) ≤ ~p ( S ) ≤ ~ p ( S ) + codim( S ) − codim( S ), and furthermore(a) if ~p ( S ) = ~ p ( S ) then ~p ( S ) ⊃ ~ p ( S ),(b) if ~p ( S ) = ~ p ( S ) + codim( S ) − codim( S ), then ~p ( S ) ⊂ ~ p ( S ).2. If S is regular then − ≤ ~p ( S ) ≤ codim( S ) −
1, and furthermore(a) if ~p ( S ) = − H ( E x ) = 0 and H ( E x ) is ~p ( S )-torsion for all x ∈ S ,(b) if ~p ( S ) = 0 then H ( E x ) is ~p ( S )-torsion for all x ∈ S ,(c) if ~p ( S ) = codim( S ) − H ( E x ) is ~p ( S )-torsion free for all x ∈ S ,(d) if ~p ( S ) = codim( S ) − H ( E x ) = 0 and H ( E x ) is ~p ( S )-torsion free for all x ∈ S . 12 emark . Since E is clc on the regular strata by assumption, the torsion conditions ofproperty 2 hold for all x ∈ S if and only if they hold for some x ∈ S . Remark . We notice that if S is singular then compatibility as just defined places noabsolute constraints on the values of ~p ( S ) and ~ p ( S ) but only relative constraints on how thesemust relate to each other. By contrast, if S is regular then there are absolute constraints onthe values of ~p ( S ) and also, for the extreme values of ~p ( S ), constraints on how ~p ( S ) relatesto E . We will see such a dichotomy throughout. One consequence is that these conditionsforbid any codimension one stratum of X being contained in a regular stratum of X (unless E is trivial on that regular stratum) as this would require either ~p ( S ) = − S ) − ~p ( S ) = 0 = codim( S ) −
1, and in either case the combined torsion assumptions implyeach H ∗ ( E x ) = 0 so that E is trivial. Remark . One situation in which the conditions of Definition 3.1 are not necessary fortopological invariance is when perversity values are so extreme that their specific valuesbecome irrelevant. For example, if ~p ( S ) < − P ∗ x = 0 for any x ∈ S regardless of theactual value of ~p ( S ). At the other extreme, [6, Theorem 4.15] implies that if X is a stratifiedpseudomanifold, P ∗ a ts-Deligne sheaf on X , and x ∈ X n − k then H i ( P ∗ x ) = 0 for i > k .The main technical tool in the proof of that theorem is [6, Lemma 4.14], but the argumentfor this lemma applies to any manifold stratified space (cf. [2, Lemma V.9.5]). The proof of[6, Theorem 4.15] therefore generalizes to CS sets, using Lemma 4.1 below in the argumentinstead of the citation to [2, Lemma V.3.8.b]. Consequently, if ~p ( S ) ≥ k then again thespecific values don’t matter. Therefore, for the purposes of Theorem 3.5 we could add to theconditions of Definition 3.1 the possibility that if S ⊂ S then either both ~p ( S ) and ~ p ( S ) are < − efficient , i.e. that − ≤ ~p ( S ) ≤ codim( S ) for all singular strata S . If ~p is not efficient, it can always be replaced by a ts-perversity that is efficient withoutaltering P ∗ . In any case, Theorem 3.5 does not require such assumptions.We now come to the main theorem of the paper, which will be the basis for the resultsin the rest of Section 3. Theorem 3.5.
Suppose that X and X denote two CS set stratifications of the same under-lying space with X coarsening X . Let E be a ts-coefficient system such that X is adaptedto E (and hence so is X ). Let ~p and ~ p be respective ts-perversities on X and X that are E -compatible, and let P ∗ and P ∗ be the respective ts-Deligne sheaves with coefficients E . Then P ∗ is quasi-isomorphic to P ∗ .Proof. By Theorem 2.8, P ∗ is characterized uniquely up to quasi-isomorphism by the axiomsTAx1’( X, ~p, E ). Therefore, to prove the proposition, it is sufficient to show that P ∗ alsosatisfies these axioms. We will use that P ∗ already satisfies the axioms TAx1’( X , ~ p , E ). Axiom a.
Since P ∗ is X -clc, it is also X -clc since X refines X . Furthermore, P ∗ is 0for ∗ < Axiom b.
We know P ∗ | X − X n − ∼ = E | X − X n − , but X − X n − ⊂ X − X n − by assumption, soalso P ∗ | X − X n − ∼ = E | X − X n − . 13 xiom c. Suppose x ∈ S ⊂ X n − k for k ≥
1. We must show that H i ( P ∗ x ) = 0 for i > ~p ( S ) + 1 and that H ~p ( S )+1 ( P ∗ x ) is ~p ( S )-torsion.First suppose that x is contained in a regular stratum of X . Then P ∗ x ∼ = E x , so H i ( P ∗ x ) ∼ = H i ( E x ). We recall that H i ( E x ) is automatically 0 if i = 0 , − ≤ ~p ( S ) ≤ codim( S ) − ~p ( S ) ≥
1, then ~p ( S )+1 ≥ H i ( E x ) =0 for i ≥ ~p ( S ) + 1. If ~p ( S ) = 0 then H i ( E x ) = 0 for i > ~p ( S ) + 1 = 1 while H ~p ( S )+1 ( E x ) = H ( E x ) is ~p ( S )-torsion in this case by the E -compatibility assumptions. Finally, if ~p ( S ) = −
1, the compatibility assumptions imply that H ( E x ) = 0, and so H i ( E x ) = 0 for i > H ( E x ) is ~p ( S )-torsion .Now suppose that x is contained in the singular stratum S of X . Then we know H i ( P ∗ x ) =0 for i > ~ p ( S ) + 1 and H ~ p ( S )+1 ( P ∗ x ) is ~ p ( S )-torsion. But by assumption ~ p ( S ) ≤ ~p ( S ), andif ~ p ( S ) = ~p ( S ) then ~ p ( S ) ⊂ ~p ( S ). It follows that P ∗ satisfies Axiom c of TAx1’( X, ~p, E ). Axiom d.
Again suppose x ∈ S ⊂ X n − k for k ≥
1, and let f x : { x } ֒ → X be the inclusion.We must show that H i ( f ! x P ∗ ) = 0 for i ≤ ~p ( S ) + n − k + 1 and that it is ~p ( S )-torsion freewhen i = ~p ( S ) + n − k + 2.Once again we first suppose that x is contained in a regular stratum of X . Then wehave f ! x P ∗ ∼ = f ∗ x P ∗ [ − n ] ∼ = E x [ − n ] by [2, Proposition V.3.7.b] and by assumption, and so H i ( f ! x P ∗ ) ∼ = H i ( E x [ − n ]) ∼ = H i − n ( E x ). So we must show H j ( E x ) is 0 for j ≤ ~p ( S ) − k + 1 andthat it is ~p ( S )-torsion free when j = ~p ( S ) − k + 2. Recall that H j ( E x ) = 0 automaticallyfor j = 0 ,
1. Now, as ~p and ~ p are E -compatible, we have that ~p ( S ) ≤ codim( S ) − k − ~p ( S ) ≤ k − ~p ( S ) − k + 2 ≤ − H j ( E x ) = 0 for j ≤ ~p ( S ) − k + 2. If ~p ( S ) = k −
2, then ~p ( S ) − k + 2 = 0 so similarly H j ( E x ) = 0 for j < ~p ( S ) − k + 2 while H ~p ( S ) − k +2 ( E x ) = H ( E x ) is ~p ( S )-torsion free in this case by the compatibility assumptions.Finally, if ~p ( S ) = k − ~p ( S ) − k + 2 = 1, we have that H ( E x ) = 0 and H ( E x ) is ~p ( S )-torsion free again by the compatibility conditions .Next we suppose that x ∈ S ⊂ X n − k and that S ⊂ S for S ⊂ X n − ℓ a singular stratumof X . Let U ∼ = R n − ℓ × cL be a distinguished neighborhood of x in the X stratification. Byabuse of notation, we can identify U with R n − ℓ × cL , letting P ∗ also denote its pullback tothis product neighborhood, which remains X -clc. Let x = ( y, v ) with f y : { y } ֒ → R n − ℓ and f v : { v } ֒ → cL the vertex inclusion. Let π : R n − ℓ × cL → R n − ℓ and π : R n − ℓ × cL → cL be the projections, and let s : cL ֒ → { y } × cL be the inclusion. By [17, Proposition 2.7.8](letting the Y n there be close balls in R n − ℓ ), we have P ∗ ∼ = π ∗ Rπ ∗ P ∗ . Let R A denote theconstant sheaf on the space A with stalks in our ground ring R . Then P ∗ ∼ = R R n − ℓ × cL L ⊗ P ∗ ∼ = π ∗ R R n − ℓ L ⊗ π ∗ Rπ ∗ P ∗ . By [2, Remark V.10.20.c], whose hypotheses are satisfied due to the constructibility (see [21,Proposition 4.0.2.2]), f ! x P ∗ ∼ = f ! y R R n − ℓ L ⊗ f ! v Rπ ∗ P ∗ . We see in this argument why ~p ( S ) ≥ − We see in this argument why ~p ( S ) ≤ codim( S ) − f ! y R R n − ℓ ∼ = f ∗ y R R n − ℓ [ − ( n − ℓ )] = R [ − ( n − ℓ )], and so f ! x P ∗ ∼ = f ! v Rπ ∗ P ∗ [ − ( n − ℓ )] . To compute H i (cid:0) f ! v Rπ ∗ P ∗ (cid:1) , consider the long exact sequence [2, Section V.1.8] → H i (cid:0) f ! v Rπ ∗ P ∗ (cid:1) → H i ( cL ; Rπ ∗ P ∗ ) α −→ H i (cid:0) cL − { v } ;¯ i ∗ Rπ ∗ P ∗ (cid:1) → , where ¯ i : cL − { v } → cL fits into the Cartesian square R n − ℓ × cL − R n − ℓ × { v } ¯ π ✲ cL − { v } R n − ℓ × cL i ❄ ∩ π ✲ cL. ¯ i ❄ ∩ We can see that ¯ i ∗ Rπ ∗ P ∗ = R ¯ π ∗ i ∗ P ∗ by replacing P ∗ with an injective resolution and thenconsidering sections over open sets. It follows that α is isomorphic to the attaching map H i ( R n − ℓ × cL ; P ∗ ) → H i ( R n − ℓ × cL ; R i ∗ i ∗ P ∗ ) = H i ( R n − ℓ × ( cL − { v } ); i ∗ P ∗ ). As P ∗ and R i ∗ i ∗ P ∗ are X -clc, [21, Proposition 4.0.2] implies that restriction to smaller distinguishedneighborhoods of x in X yields a constant map of constant direct systems, and so thishypercohomology attaching map is isomorphic to the map it induces stalkwise in the directlimit. And by the axioms TAx1( X , ~ p , E ), which P ∗ satisfies, this attaching map inducesstalk-wise cohomology isomorphisms for i ≤ ~ p ( S ) and an isomorphism onto the ~ p ( S )-torsion module for i = ~ p ( S ) + 1. So H i (cid:0) f ! v Rπ ∗ P ∗ (cid:1) = 0 for i ≤ ~ p ( S ) + 1. We alsoknow that H ~ p ( S )+2 ( cL ; Rπ ∗ P ∗ ) ∼ = H ~ p ( S )+2 (cid:0) R n − ℓ × cL ; P ∗ (cid:1) ∼ = H ~ p ( S )+2 ( P ∗ x ), the latter by[21, Proposition 4.0.2.2] again, and this is 0 by axiom TAx1’c for P ∗ . Since we have alreadynoted that in degree ~ p ( S ) + 1 the map α is an isomorphism onto the ~ p ( S )-torsion moduleof H ~ p ( S )+1 (cid:0) cL − { v } ;¯ i ∗ Rπ ∗ P ∗ (cid:1) , it follows that H ~ p ( S )+2 (cid:0) f ! v Rπ ∗ P ∗ (cid:1) is ~ p ( S )-torsion free.Returning now to H i ( f ! x P ∗ ) ∼ = H i ( f ! v Rπ ∗ P ∗ [ − ( n − ℓ )]) = H i − n + ℓ ( f ! v Rπ ∗ P ∗ ), we concludethat H i ( f ! x P ∗ ) = 0 when i − n + ℓ ≤ ~ p ( S ) + 1, i.e. when i ≤ ~ p ( S ) + n − ℓ + 1, andthat H ~ p ( S )+ n − ℓ +2 ( f ! x P ∗ ) is ~ p ( S )-torsion free. By assumption, ~p ( S ) ≤ ~ p ( S ) + codim( S ) − codim( S ) = ~ p ( S ) + k − ℓ , so ~ p ( S ) + n − ℓ + 1 ≥ ~p ( S ) − k + ℓ + n − ℓ + 1 = ~p ( S ) − k + n + 1.So H i ( f ! x P ∗ ) = 0 for i ≤ ~p ( S ) − k + n + 1 as desired. Furthermore, H ~p ( S ) − k + n +2 ( f ! x P ∗ ) willalso be 0 unless the above inequalities are equalities, in which case H ~p ( S ) − k + n +2 ( f ! x P ∗ ) = H ~ p ( S ) − ℓ + n +2 ( f ! x P ∗ ) is ~ p ( S )-torsion free. But we have assumed for this scenario that ~ p ( S ) ⊃ ~p ( S ) so that this module is also ~p ( S )-torsion free.We have now demonstrated all the axioms, completing the proof. In [5], Chataur, Saralegi, and Tanr´e consider the invariance of intersection homology underrefinement/coarsening when the perversity on the finer stratification is pulled back from a15erversity on the coarser stratification or when the perversity on the coarser stratificationis pushed forward from the finer stratification. In the following two subsections we considersuch constructions for ts-perversities. We will see that pullback perversities can alwaysbe constructed and always result in quasi-isomorphic ts-Deligne sheaves, generalizing [5,Corollary 6.13]. By contrast, pushforward perversities require certain conditions to be definedand then further conditions to provide quasi-isomorphic ts-Deligne sheaves. Our resultsabout pushforwards generalize [5, Theorem C].In this section we assume the following situation: We suppose that X and X denote twoCS set stratifications of the same underlying space with X coarsening X . Let ν : X → X denote the identity map, which is a stratified map; we sometimes refer to ν as a coarseningmap. If S is a stratum of X , let S ν denote the stratum of X containing it. We first define pullback perversities.
Definition 3.6.
Let ν : X → X be a coarsening map. Suppose X is adapted to a ts-coefficient system E , and let ~ p be a ts-perversity on X . We define the E -compatible pullbackperversity ν ∗E ~ p on the refinement X of X by:1. if S ν is singular then ν ∗E ~ p ( S ) = ~ p ( S ν ),2. if S ν is regular then ν ∗E ~ p ( S ) = (0 , ℘ ), where ℘ is the smallest subset of P ( R ) such that E is a ℘ -coefficient system on S . Remark . Since X refines X and E is adapted to X , for each singular stratum S of X contained in a regular stratum of X the ℘ in the second condition always exists. In fact, wecould use in this condition any choice of ℘ ⊂ P ( R ) such that E is a ℘ -coefficient system on S for the purposes of the following theorem; we choose the smallest such ℘ just for definiteness. Theorem 3.8. If X has no stratum of codimension one contained in a regular stratum of X then ~ p and ν ∗E ~ p are E -compatible. Consequently P ∗ ν ∗E ~ p ∼ = P ∗ ~ p .Proof. The first statement is immediate from the definitions. The second follows by Theorem3.5.
Remark . If we assume in Theorem 3.8 that E is a local system concentrated in degree0 (and so, in particular, a ∅ -coefficient system) and if ~p ( S ) = ∅ for all S , then the theorembecomes a statement about ordinary intersection homology that generalizes [5, Corollary6.12]. Chataur, Saralegi, and Tanr´e also introduce pushforward perversities in [5, Section 6]. Morespecifically, they establish conditions under which a perversity can be pushed forward froma stratification of a CS set to its intrinsic stratification and for which the correspondingintersection homology groups are isomorphic. We first generalize pushforwards and placethem in our context: 16 efinition 3.10.
We will say that the ts-perversity ~p on X can be pushed to X if ~p ( Y ) = ~p ( Z )for the two singular strata Y and Z of X whenever all the following conditions hold:1. Y ν = Z ν ,2. Y ν is singular in X , and3. dim( Y ) = dim( Z ) = dim( Y ν ).If this property holds then we define the pushforward ν ∗ ~p by ν ∗ ~p ( S ) = ~p ( S ) if S is astratum of X such that S = S ν and dim( S ) = dim( S ). Every singular stratum of X mustcontain such a stratum S , so ν ∗ ~p is well defined and without any ambiguity due to ourassumptions. Remark . If ~p is a ts-perversity that depends only on codimension, i.e. ~p ( S ) = ~p ( T )whenever codim( S ) = codim( T ), then ~p can be pushed forward to any coarsening. In thiscase we may abuse notation and also write ν ∗ ~p simply as ~p . As a further abuse, we can alsotreat Z ≥ as the domain of ~p writing ~p (codim( S )) = ~p ( S ).Unfortunately, in contrast to Theorem 3.8, a perversity and its pushforward are notnecessarily E -compatible, even when the pushforward is defined. For example, we need onlylet X be a trivially filtered manifold and let X be a refinement with a stratum S on which ~p ( S ) < − E -compatibilityis our most general criterion for topological invariance, but for comparison with earlierresults, especially [5, Theorem C], it is useful to delineate in terms of ~p exactly when ~p and ν ∗ ~p will be E -compatible. Inspection of the definitions yields the following (cf. the definitionof K ∗ -perversities in [5, Definition 6.8]): Proposition 3.12.
Let ν : X → X be a coarsening map, and suppose X is adapted tothe ts-coefficient system E . Suppose ~p is a perversity on X that can be pushed to X . Forany singular stratum S ⊂ X , let ˜ S denote any stratum of X such that ( ˜ S ) ν = S ν and dim( ˜ S ) = dim( S ν ) . Then ~p and ν ∗ ~p are E -compatible if the following conditions hold on ~p :1. If ˜ S is singular then ~p ( ˜ S ) ≤ ~p ( S ) ≤ ~p ( ˜ S ) + codim ( S ) − codim ( ˜ S ) , and furthermore(a) if ~p ( ˜ S ) = ~p ( S ) then ~p ( ˜ S ) ⊂ ~p ( S ) ,(b) if ~p ( S ) = ~p ( ˜ S ) + codim ( S ) − codim ( ˜ S ) , then ~p ( ˜ S ) ⊃ ~p ( S ) .2. If ˜ S is regular then − ≤ ~p ( S ) ≤ codim ( S ) − , and furthermore(a) If ~p ( S ) = − then H ( E x ) = 0 and H ( E x ) is ~p ( S ) -torsion for all x ∈ S ,(b) if ~p ( S ) = 0 then H ( E x ) is p ( S ) -torsion for all x ∈ S These ˜ S are called source strata in [5]. Note that the precise choice of ˜ S does not matter in the conditionsthat follow as the assumption that ~p can be pushed forward assures us that any two choices would give thesame perversity conditions. c) if ~p ( S ) = codim ( S ) − , then H ( E x ) is p ( S ) -torsion free for all x ∈ S ,(d) if ~p ( S ) = codim ( S ) − then H ( E x ) = 0 and H ( E x ) is ~p ( S ) -torsion free for all x ∈ S .Remark . Condition 1 on ~p is essentially Condition B of [5, Definition 6.8], while Con-dition 2 on ~p corresponds there to Conditions A and D. Condition C of [5] is built into ourassumption that ~p can be pushed forward. Note, however, that in [5] only pushforwards tothe intrinsic stratification are considered. So far we have considered only the situation in which we have two stratifications, X and X , one refining the other. In this case we have a point of comparison between any twots-perversities ~p and ~ p on these stratifications since any stratum S of X is contained in astratum S of X , allowing us to compare ~p ( S ) with ~ p ( S ). If we are given instead two arbitrarystratifications, then our best hope for relating them seems to be if we can find a commonrefinement or coarsening, in which case we can perhaps apply Theorem 3.5 twice, connectingeach given stratification with our common intermediary. More generally, we can attemptto compare all stratifications of a given space by finding a universal common refinement orcoarsening. We will see that such common refinements do not always exist, even for justtwo stratifications (Remark 3.24), but intrinsic common coarsenings do exist, as we’ll showin Section 6.Of course we also need to be able to assign sufficiently compatible perversities to all thesestratifications. The simplest way to proceed when faced with all possible stratifications seemsto be to revert more closely to the original definition of Goresky and MacPherson [13] inwhich perversities were assumed to be functions only of codimension. This allows one todefine perversities without any reference to a specific stratification: the perversity simplyassigns the same predetermined value to all strata of the same codimension. In such asetting, Goresky and MacPherson proved a topological invariance statement of the form,“if a perversity satisfies certain conditions then all stratifications (without codimension onestrata) of the same space with that perversity and a fixed coefficient system yield quasi-isomorphic Deligne sheaves” [14, Theorem 4.1]. This theorem was slightly refined by Borel[2, Section V.4]. In both cases, the proofs utilize common coarsenings, with the assumptionsabout the perversities being strong enough to imply (in our language) E -compatibility amongthe manifestations of the “same” perversity on all stratifications.Analogously, our goal in this section is to construct ts-perversities that depend only oncodimension and such that all stratifications yield quasi-isomorphic ts-Deligne sheaves fromthese ts-perversities. Of course we must also take into account adaptability to coefficientsystems and place certain limitations on the behavior of codimension one strata. The resultwill be a sequence of theorems each with a pair of results. One of each pair places stricterconditions on the perversities but allows for any stratifications up to some limitations oncompatibility of codimension one strata; the other removes some of the restrictions on theperversities but forces us to fix the regular strata of the stratifications.18e start with the following definitions: Definition 3.14.
We call a ts-perversity ~p constrained if it satisfies all of the following:1. ~p depends only on the codimension of strata (so we can write ~p = ( ~p , ~p ) : Z ≥ → Z × P ( R ) ),2. ~p satisfies the Goresky-MacPherson growth condition ~p ( k ) ≤ ~p ( k + 1) ≤ ~p ( k ) + 1for k ≥ ~p (2) ∈ {− , , } ,4. if ~p ( k + 1) = ~p ( k ) then ~p ( k + 1) ⊃ ~p ( k ),5. if ~p ( k + 1) = ~p ( k ) + 1 then ~p ( k + 1) ⊂ ~p ( k ).If ~p satisfies all these conditions except (3) then we say that ~p is weakly constrained . If ~p (2) = 0, we say that ~p is strongly constrained .We say that ~p is adapted to the ts-coefficient system E in the following cases:1. if ~p (2) = − H ( E x ) = 0 and H ( E x ) is ~p (2)-torsion for all x in the domain of E ,2. if ~p (2) = 0 then H ( E x ) is ~p (2)-torsion and H ( E x ) is ~p (2)-torsion free for all x inthe domain of E ,3. if ~p (2) = 1 then H ( E x ) = 0 and H ( E x ) is ~p (2)-torsion free for all x in the domainof E . Remark . Note that there are a few conditions specific to codimension 2, even though ~p is also defined for codimension one. This is because these conditions will govern whathappens when a singular stratum of X is contained in a regular stratum of X , but we knowthat for topological invariance we must preclude this for codimension one strata of X anywayand so these conditions only come into play starting at codimension 2. Remark . We will not utilize strongly constrained ts-perversities until Section 5, wherethey will be convenient.We begin by considering when a constrained ~p is E -compatible with itself across twostratificaitons. Proposition 3.17.
Suppose X is a coarsening of X and that X is adapted to a ts-coefficientsystem E . Suppose either1. no codimension one stratum of X is contained in a regular stratum of X , and ~p isconstrained and adapted to E , or2. X n − = X n − and ~p is weakly constrained. hen ~p on X and ~p on X are E -compatible and so the ts-perversity ~p ts-Deligne sheaves P ∗ on X and P ∗ on X are quasi-isomorphic.Remark . The dichotomy here is a reflection of that of Remark 3.3. If we want toallow singular strata of X in regular strata of X then we need an assumption that will forcethe perversities on such strata into the absolute bounds − ≤ ~p ( S ) ≤ codim( S ) − ~p (2) ∈ {− , , } , together with the Goresky-MacPhersongrowth condition (which is in general necessary anyway for the singular-stratum-in-singular-stratum cases — see Section 4) accomplishes this and is the weakest possible such requirementonce we have eliminated codimension one strata of X in regular strata of X (see Remark3.3). If we wish to dispense with this additional constraint on ~p (2) then we can insteadask that no singular stratum of X be contained in a regular one of X , leading to the secondalternative of the proposition.Of course we could also dabble in many more specific cases — for example we might allow ~p (2) = − ~p (3) = − X from appearingin regular strata of X (plus conditions involving E ). Such scenarios are in any case stillcaptured by Theorem 3.5, so we leave the reader to formulate his or her own variants andinstead consider just these two broad situations that are closest in keeping to the previoustheorems of [14, 9, 5]. Proof of Proposition 3.17.
In each case we check the conditions of Definition 3.1. As notedin Remark 3.11, in this case we simply write ~p for both perversities, as well as writing ~p (codim( S )) = ~p ( S ) when convenient.Assuming the first hypotheses, suppose S is a singular stratum of X in the singularstratum S of X . Then codim( S ) ≤ codim( S ) so the first condition of Definition 3.1 followsby iterating Condition 2 of Definition 3.14. Furthermore, since ~p is non-decreasing, ~p ( S ) = ~p ( S ) only if ~p ( k ) is constant from codim( S ) to codim( S ), in which case Condition 4 ofDefinition 3.14 inductively implies that ~p ( S ) ⊃ ~p ( S ) as needed. Similarly, if ~p ( S ) = ~p ( S ) + codim( S ) − codim( S ) then ~p ( k + 1) = ~p ( k ) + 1 for codim( S ) ≤ k < codim( S ) andso Condition 5 of Definition 3.14 implies ~p ( S ) ⊂ ~p ( S ).Now suppose S is regular. By assumption codim( S ) ≥ ~p , together with ~p (2) ∈ {− , , } , ensures that − ≤ ~p ( S ) ≤ codim( S ) − • If ~p ( S ) = −
1, the growth condition implies that ~p ( k ) = − ≤ k ≤ codim( S ).Since ~p is adapted to E , we have by definition that H ( E x ) = 0 and H ( E x ) is ~p (2)-torsion for all x in the domain of E , so in particular for x ∈ S . Now Condition 4 ofDefinition 3.14 implies that ~p ( S ) ⊃ ~p (2), so H ( E x ) is also ~p ( S )-torsion for all x ∈ S . • If ~p ( S ) = codim( S ) −
1, the growth condition implies that ~p ( k ) = k − ≤ k ≤ codim( S ) and in particular that ~p (2) = 1. Since ~p is adapted to E , we have bydefinition that H ( E x ) = 0 and H ( E x ) is ~p (2)-torsion free for all x in the domain of E , so in particular for x ∈ S . Condition 5 of Definition 3.14 implies that ~p ( S ) ⊂ ~p (2),so H ( E x ) is also ~p ( S )-torsion free for all x ∈ S .20 If ~p ( S ) = 0 the growth condition implies that ~p (2) ∈ {− , } . If ~p (2) = − ~p is adapted to E , we have H ( E x ) = 0 for all x in the domain of E , in whichcase certainly H ( E x ) is ~p ( S )-torsion for all x ∈ S . If ~p (2) = 0 the growth conditionimplies that ~p ( k ) = 0 for 2 ≤ k ≤ codim( S ). Since ~p is adapted to E , H ( E x ) is ~p (2)-torsion. Condition 4 of Definition 3.14 implies that ~p ( S ) ⊃ ~p (2), so H ( E x ) isalso ~p ( S )-torsion for all x ∈ S . • If ~p ( S ) = codim( S ) − ~p (2) ∈ { , } . If ~p (2) = 1then since ~p is adapted to E , we have H ( E x ) = 0 for all x in the domain of E , inwhich case certainly H ( E x ) is ~p ( S )-torsion free for all x ∈ S . If ~p (2) = 0 the growthcondition implies that ~p ( k ) = k − ≤ k ≤ codim( S ). Since ~p is adapted to E , H ( E x ) is ~p (2)-torsion free. Condition 5 of Definition 3.14 implies that ~p ( S ) ⊂ ~p (2),so H ( E x ) is also ~p ( S )-torsion free for all x ∈ S .In the second situation of the lemma, no singular stratum of X is contained in a regularstratum of X . The argument is therefore exactly as above except that we don’t need to verifyany of Condition 2 of Definition 3.1. Therefore, we don’t need Condition 3 of Definition 3.14nor the assumption that ~p is adapted to E .The final statement of the proposition follows from Theorem 3.5. Corollary 3.19.
Suppose that X and X are any two CS set stratifications of the same space,both of which are adapted to the ts-coefficient system E . Suppose either1. ~p is a constrained ts-perversity, ~p is adapted to E , and X and X possess a commoncoarsening X that is adapted to E such that no codimension one stratum of X or X iscontained in a regular stratum of X , or2. ~p is a weakly constrained ts-perversity and X and X possess a common coarsening X that is adapted to E and such that X n − = X n − = X n − .Then X and X have quasi-isomorphic ts-perversity ~p ts-Deligne sheaves, i.e. P ∗ X,~p, E ∼ = P ∗X ,~p, E .Proof. For each case apply Proposition 3.17 twice.The preceding corollary shows that we can compare Deligne sheaves from different strat-ifications provided appropriate common coarsenings exist. With minor hypotheses, suchcommon coarsenings are constructed in Section 6 with the following result. Recall the def-initions of maximal ts-coefficient systems from Definition 2.9 and of “fully adapted” fromDefinition 2.11.
Theorem 3.20.
Suppose that X and X are any two CS set stratifications of the same space,both fully adapted to the maximal ts-coefficient system E . Suppose either1. ~p is a constrained ts-perversity adapted to E and the closure of the union of the codi-mension one strata of X (which may be empty) is equal to the closure of the union ofthe codimension one strata of X , or . ~p is a weakly constrained ts-perversity and X n − = X n − .Then X and X have quasi-isomorphic ts-perversity ~p ts-Deligne sheaves, i.e. P ∗ X,~p, E ∼ = P ∗X ,~p, E .Proof. Let C denote the closure of the union of the codimension one strata of X . In thefirst case, Proposition 6.4 yields an intrinsic stratification X E ,C that is a common coarseningof X and X that is adapted to E and such that no codimension one stratum of X or X iscontained in a regular stratum of X (in fact the closures of the unions of the codimensionone strata of X , X , and X are all the same set). In the second case, Proposition 6.4 yieldsan intrinsic stratification X E ,X n − = X E , X n − that is a common coarsening of X and X that is adapted to E and such that X n − E ,X n − = X n − = X n − . This last statement holdsbecause Property 4 of Proposition 6.4 shows that if U E is the domain of E then in this case X E ,X n − − X n − E ,X n − = U E − X n − , which must be X − X n − as X is adapted to E . Thus X and X E ,X n − have the same regular strata and hence the same n − X .The theorem now follows from the preceding corollary. Remark . If we take E to be a constant sheaf concentrated in degree 0, set ~p ( k ) = ∅ for all k , and consider only pseudomanifold stratifications with no codimension one strata,we recover from the first case of the theorem the original topological invariance result ofGoresky-MacPherson [14, Theorem 4.1], though with a fairly different proof (see also [8]).Similarly, with the same assumptions on E and ~p and with ~p (2) >
0, the second case ofthe theorem recovers the main result of [9].
Remark . In general, if we are given two CS set stratifications X and X of the samespace without any conditions on their codimension one strata there is not necessarily acommon coarsening X such that no codimension one stratum of X or X is contained in aregular stratum of X . For example, just let X be the plane R with its trivial stratificationand let X be any other stratification with a codimension one stratum. The only commoncoarsening is X itself in which a regular stratum of X already contains a codimension stratumof X . Even if we rule out starting with such a bad situation, consider letting X and X bestratifications of R , one with the x -axis as singular stratum and one with the y -axis assingular stratum. Again, there is no appropriate common coarsening.We have a similar result to Corollary 3.19 concerning common refinements instead ofcommon coarsenings, though this is generally less useful as common refinements do not existin general, even with restrictions on codimension one strata; see Remark 3.24. Nonetheless,we include this corollary as it may occasionally be useful when working with spaces for whichthe hypotheses of the preceding corollaries do not apply. Corollary 3.23.
Suppose that ~p is any ts-perversity depending only on codimension and that X and X are any two CS set stratifications of the same space, both of which are adaptedto the ts-coefficient system E . If X and X posses a common refinement X such that nocodimension one stratum of X is contained in a regular stratum of X or X , then the ts-perversity ~p ts-Deligne sheaves on X and X are quasi-isomorphic.Proof. This follows from Theorem 3.8 using X as an intermediary.22 emark . Unfortunately, unlike the case of coarsenings for which any two stratificationshave a common coarsening (given appropriate assumptions on coefficients and codimensionone strata), common refinements do not exist in general. For example, let X be R n , n ≥ R n ⊃ { x -axis } , and let X be R n stratified by R n ⊃ Z , where Z is the imageof f : R → R n given by f ( t ) = ( t, t sin(1 /t ) , , . . . ,
0) for t = 0 and f (0) = ~
0. Then Z isa C submanifold and so possesses a tubular neighborhood, making X a CS set. But theintersection of Z and the x -axis is the union of the origin with the points (cid:8) nπ , , . . . , (cid:9) andso any common refinement X of the stratifications X and X would require X to have anaccumulation point, which is not possible for CS sets. In this section we consider the necessity of the E -compatibility conditions on ts-perversities(Definition 3.1) in order for our main topological invariance result (Theorem 3.5) to hold.We will show that the “singular-in-regular” conditions are strictly necessary, using that thelinks of strata in such situations are limited to homology spheres. For the “singular-in-singular” conditions, there are many more possibilities and so we only have necessity ingeneral, meaning that we will construct examples that show that failure of the conditionsof Definition 3.1 can result in non-quasi-isomorphic ts-Deligne sheaves unless certain stalkcohomology modules (or some of their torsion submodules or torsion-free quotient modules)vanish. As noted in Remark 3.4, such vanishing is assured when the first component ofa ts-perversity has values less than − S ) −
1, but for efficient ts-perversities (those for which − ≤ ~p ( S ) ≤ codim( S ) − S ) suchvanishing depends on the particulars of X , E , and ~p . There are certainly stronger conditionsthat can be imposed on ( X, X , E ) that would allow some weakening of Definition 3.1. Asan extreme example we could take E = 0 in which case P ∗ = P ∗ = 0 regardless of thestratifications or perversities. Another less trivial, though still somewhat artificial, exampleis noted in Remark 3.18. In fact all such extra conditions are likely to be somewhat artificialin our current general context.For historical context, we recall that if E is concentrated in degree 0, if ~p and ~ p dependonly on codimension, and if ~p ( k ) = ~ p ( k ) = ∅ for all k ≥ P ∗ and P ∗ will be the original Deligne sheaves of Goresky and MacPherson[14]. In this case the hypercohomology groups with compact supports will be the classicalintersection homology groups. If ¯ p ( k ) ≤ k − k , as in [14], then these will agree withthe singular intersection homology groups of King [18], called “GM intersection homology”in [7]. For GM intersection homology, the Goresky-MacPherson growth condition ¯ p ( k ) ≤ ¯ p ( k + 1) ≤ ¯ p ( k ) + 1 is known to be necessary in general for topological invariance; see King[18, Section 2]. The main idea is to compare c (Σ X ) with R × cX , where cX is the open coneon X and Σ X is the suspension. For compact stratified X these spaces are topologicallyhomeomorphic but with different natural stratifications coming from that on X . Goreskyand MacPherson also assume ¯ p (2) = 0 (and no codimension one strata), though King showsthat this assumption is not necessary for the topological invariance of singular chain GMintersection homology, only that ¯ p (1) ≥
0. 23f ¯ p ( k ) > k − k the hypercohomology of the Deligne sheaf is called “non-GMintersection homology” in [7]. Non-GM intersection homology can only be obtained usingsingular chains after making some additional modifications; see [7, Chapter 6]. We showed in[9, Section 3] that if ¯ p (2) > X n − = X n − . The general necessity ofthe growth condition for non-GM intersection homology can be argued identically as in theGM case, and we will use essentially the same basic argument below for the case of singularstrata of X in singular strata of X .In the remainder of this section, we first develop some computational machinery in Section4.1. We then consider the “singular-in-regular” situation in Section 4.2 and the “singular-in-singular” situation in Section 4.3. Throughout we continue our notation from the precedingsections, namely X coarsens X with corresponding ts-perversities ~ p and ~p and ts-Delignesheaves P ∗ and P ∗ . We also assume X adapted to a ts-coefficient system E . We need some tools for computation. Our first lemma is standard:
Lemma 4.1.
Suppose Y is a filtered space, and let X = R k × Y have the product filtration X j = R k × X j − k . If S ∗ is X -clc then H i ( X ; S ∗ ) ∼ = H i ( Y ; S ∗ | Y ) , identifying Y with any { z } × Y ⊂ R k × Y .Proof. Let π : R k × Y → Y be the projection and let s : { z } × Y ֒ → R k × Y be theinclusion. Since S ∗ is X -clc, in particular H j ( S ∗ ) is constant along each R k × { y } , andso S ∗ = π ∗ Rπ ∗ S ∗ by [17, Proposition 2.7.8] (letting the Y n there be close balls in R k ).Then s ∗ S ∗ ∼ = s ∗ π ∗ Rπ ∗ S ∗ = Rπ ∗ S ∗ , so H i ( R k × Y ; S ∗ ) ∼ = H i ( Y ; Rπ ∗ S ∗ ) ∼ = H i ( Y ; s ∗ S ∗ ) = H i ( Y ; S ∗ | Y ). Lemma 4.2.
Let L be a compact filtered set such that X = R k × cL is a CS set. Let x = ( s, v ) ∈ R k × cL with s ∈ R k arbitrary and v ∈ cL the cone vertex. If P ∗ is a ts-Delignesheaf on R k × cL , then identifying L with some copy ( z, t, L ) ⊂ R k × (0 , × L ⊂ R k × cL ,we have H i ( P ∗ x ) ∼ = , i > ~p ( S ) + 1 ,T ~p ( S ) H i ( L ; P ∗ | L ) , i = ~p ( S ) + 1 , H i ( L ; P ∗ | L ) , i ≤ ~p ( S ) . Proof.
Let S be the stratum R k × { v } . Let W = X − S = R k × ( cL − { v } ) ∼ = R k +1 × L , let i : W ֒ → X , and let P ∗ W = P ∗ | W . From the definition of the ts-Deligne sheaf and the torsiontipped truncation, we have H i ( P ∗ x ) ∼ = , i > ~p ( S ) + 1 ,T ~p ( S ) H ~p ( S )+1 (( Ri ∗ P ∗ W ) x ) , i = ~p ( S ) + 1 ,H i (( Ri ∗ P ∗ W ) x ) , i ≤ ~p ( S ) . P ∗ W is W -clc, Ri ∗ P ∗ W is X -clc by [21, Proposition 4.0.2.3], and so H i (( Ri ∗ P ∗ W ) x ) ∼ = H i ( X ; Ri ∗ P ∗ W ) by [21, Proposition 4.0.2.2]. But then H i ( X ; Ri ∗ P ∗ W ) ∼ = H i ( W ; P ∗ W ) ∼ = H i ( R k +1 × L ; P ∗ W ) ∼ = H i ( L ; P ∗ | L ) , using Lemma 4.1.If L is itself a CS set, then P ∗ | L will itself be a ts-Deligne sheaf, as we show in thenext lemma. To establish notation, suppose X is a CS set and ~p, E , P ∗ are a ts-perversity,ts-coefficient system, and ts-Deligne sheaf on X . If Y ⊂ X is a CS set with the inducedstratification (in the sense of the statement of the lemma below) we will denote by ~p Y thets-perversity on Y such that ~p Y ( S ) = ~p ( Z ) if the singular stratum S of Y is contained in thesingular stratum Z of X . We also write E Y for the restriction of E to the intersection of itsdomain with Y , and we will write P ∗ Y for the Deligne sheaf on Y with respect to ~p Y , E Y . Lemma 4.3.
Suppose that either1. Y is an open subset of the CS set X stratified by Y j = Y ∩ X j or2. Y is a CS set and X = R m × Y with X j = R m × X j − m and we identify Y with { z } × Y for some z ∈ R m .Let ~p, E be a ts-perversity and ts-coefficient system on X . Then P ∗ | Y ∼ = P ∗ Y , i.e. the restric-tion of the ts-Deligne sheaf on X to Y is quasi-isomorphic to the ~p Y , E Y ts-Deligne sheaf on Y .Proof. Note that in both cases both X and Y are CS sets by [7, Lemmas 2.3.13 and 2.11.4].We know that ts-Deligne sheaves are characterized up to quasi-isomorphic by the axiomsTAx1’. That P ∗ satisfies these axioms on X implies that P ∗ | Y satisfies the axioms on Y .The only axiom that is not obvious is the last axiom when X = R m × Y . In this casesuppose X is n -dimensional so that Y is n − m dimensional. Let x ∈ Y ( n − m ) − k ⊂ X n − k . Let f x : { x } ֒ → X and g x : { x } ֒ → Y for x in a singular stratum. Now, exactly as in the proofof Theorem 3.5, if π : R m × Y → Y is the projection we have f ! x P ∗ ∼ = g ! x Rπ ∗ P ∗ [ − m ] , while Rπ ∗ P ∗ ∼ = P ∗ | Y by the proof of Lemma 4.1. Thus H i ( f ! x P ∗ ) ∼ = H i − m ( g ! x P ∗ | Y ) . The axiom for P ∗ | Y now follows from the axiom for P ∗ .In what follows we shall abuse notation and write the restrictions ~p Y , E Y , and P ∗ Y assimply ~p , E , and P ∗ if it is clear what is meant from context. We let Σ X denote the(unreduced) suspension, stratified by (Σ X ) i = Σ( X i − ) with (Σ X ) = { n , s } , the union ofthe two suspension points. Proposition 4.4.
Let X n − be a compact CS set with suspension Σ X . Let ~p be a ts -perversity on Σ X such that ~p ( n ) = ~p ( s ) , and denote the common value ( p, ℘ ) . Let E be a s-coefficient system to which Σ X is adapted, and let P ∗ be the associated ts-Deligne sheaf.Then H i (Σ X ; P ∗ ) ∼ = H i − ( X ; P ∗ ) , i ≥ p + 3 , H p +1 ( X ; P ∗ ) /T ℘ H p +1 ( X ; P ∗ ) , i = p + 2 ,T ℘ H p +1 ( X ; P ∗ ) , i = p + 1 , H i ( X ; P ∗ ) , i ≤ p. Proof.
Let U = Σ X − { s } ∼ = cX and U = Σ X − { n } ∼ = cX . We consider the Mayer-Vietorissequence [17, Remark 2.6.10] → H i (Σ X ; P ∗ ) → H i ( U ; P ∗ ) ⊕ H i ( U ; P ∗ ) → H i ( U ∩ U ; P ∗ ) → . Note that U ∩ U ∼ = (0 , × X so we have H i ( U ∩ U ; P ∗ ) ∼ = H i ( X ; P ∗ ) by Lemma 4.1.Now consider H i ( U ; P ∗ ). Since P ∗ is Σ X -clc and since U is a distinguished neighborhoodof n , we have H i ( U ; P ∗ ) ∼ = H i ( P ∗ n ) by [21, Proposition 4.0.2.2]. Since this is 0 for i > p + 1,and similarly for U , we have H i (Σ X ; P ∗ ) ∼ = H i − ( U ∩ U ; P ∗ ) ∼ = H i − ( X ; P ∗ ) for i > p + 2.For i ≤ p , using [21, Proposition 4.0.2.2] again implies that the hypercohomology mapfrom U to U −{ n } = U ∩ U is isomorphic to the attaching map, which is an isomorphism byaxiom TAx1d, and similarly for U . It follows that in this range H i (Σ X ; P ∗ ) ∼ = H i ( U j ; P ∗ ) ∼ = H i ( U ; P ∗ ) ∼ = H i ( X ; P ∗ ) for j = 1 , ✲ H p +1 (Σ X ; P ∗ ) ✲ H p +1 ( U ; P ∗ ) ⊕ H p +1 ( U ; P ∗ ) ✲ H p +1 ( U ∩ U ; P ∗ ) ✲ H p +2 (Σ X ; P ∗ ) ✲ . Once again we have that H p +1 ( U ; P ∗ ) ∼ = H p +1 ( P ∗ n ), which in this case is T ℘ H p +1 ( U ∩ U ; P ∗ ) ∼ = T ℘ H p +1 ( X ; P ∗ ) by Lemma 4.2. Each of the maps H p +1 ( U j ; P ∗ ) → H p +1 ( U ∩ U ; P ∗ ) thus corresponds to the inclusion (up to sign) of the ℘ -torsion subgroup. So H p +1 (Σ X ; P ∗ ) ∼ = T ℘ H p +1 ( X ; P ∗ ) and H p +2 (Σ X ; P ∗ ) ∼ = H p +1 ( X ; P ∗ ) /T ℘ H p +1 ( X ; P ∗ ).As a corollary, and as a nice example of an application of Theorem 3.5, we compute H i ( S k ∗ X ; P ∗ ) for k >
0, where X is a compact CS set, S k is the k -sphere with trivialstratification, and S k ∗ X is the join. Rather than use the join stratification of [7, Section2.11], however, it will be more natural for us below to use the following stratification. Recallthat we can decompose S k ∗ X into cS k × X and S k × cX (see [7, Section 2.11]). We give S k × cX ⊂ S k ∗ X the stratification it obtains from the cone and product stratifications,while we stratify cS k × X as D k +1 × X , where D k +1 is the interior of the unit disk with thetrivial stratification. These two stratifications agree on the overlap S k × (0 , × X and sopatch together to give a CS set stratification of S k ∗ X . Letting v denote the cone vertex,we identify S k with the stratum S k × { v } ⊂ S k × cX ⊂ S k ∗ X . Corollary 4.5.
Let X be a compact CS set, let S k , k > , be the k -sphere with trivialstratification, and let S k ∗ X be the join with the stratification as above. Let ~p be a ts-perversity on S k ∗ X , and let E be a ts-coefficient system to which S k ∗ X is adapted. Then i ( S k ∗ X ; P ∗ ) ∼ = H i − k − ( X ; P ∗ ) , i > ~p ( S k ) + k + 2 , H ~p ( S k )+1 ( X ; P ∗ ) /T ~p ( S k ) H ~p ( S k )+1 ( X ; P ∗ ) , i = ~p ( S k ) + k + 2 , , ~p ( S k ) + 1 < i < ~p ( S k ) + k + 2 ,T ~p ( S k ) H ~p ( S k )+1 ( X ; P ∗ ) , i = ~p ( S k ) + 1 , H i ( X ; P ∗ ) , i ≤ ~p ( S k ) . Proof.
Let Σ k +1 X be the k + 1 times iterated suspension of X . Topologically (ignoringstratifications) Σ k +1 X ∼ = S k ∗ X . Furthermore, the stratification of Σ k +1 ( X ) as an iteratedsuspension refines the stratification of S k ∗ X . In fact, the stratifications are identical on S k ∗ X − S k , which is D k +1 × X with the product stratification. In particular, no singularstratum of Σ k +1 X is contained in a regular stratum of S k ∗ X . Let ~q be the ts-perversityon Σ k X such that ~q ( Z ) = ~p ( Z ) for strata S shared by Σ k +1 X and S k ∗ X and such that ~q ( Z ) = ~p ( S k ) if Z is a stratum of Σ k +1 X contained in S k . Then ~p and ~q are E -compatiblefor any E , and so by Theorem 3.5 the ~p ts-Deligne sheaf on S k ∗ X and the ~q ts-Deligne sheafon Σ k +1 X are quasi-isomorphic. The computation now follows by applying Proposition 4.4iteratively. In this section we will show that if S is a singular stratum of X contained in a regularstratum S of X then the conditions of Definition 3.1 for such strata are strictly necessary inorder to have P ∗ ∼ = P ∗ over S . On S we have P ∗ ∼ = E so we will assume also that P ∗ ∼ = E over S and see that contradictions occur if any of the conditions of Definition 3.1 for thisscenario fail.Suppose x ∈ S . Then x has a distinguished neighborhood R n − k × cL , k >
0. Topologicallythis is homeomorphic to cS n − k − × cL ∼ = c ( S n − k − ∗ L ). Furthermore, since x is in a regularstratum of X , we have ( c ( S n − k − ∗ L ) , x ) ∼ = ( R n , x ) by [7, Corollary 2.10.2] and its proof.Thus H ∗ ( S n − k − ∗ L ) ∼ = H ∗ ( S n − ), and so L must be a k − L as L k − to remind us of this.Returning to our ts-Deligne sheaves, we have H ∗ ( P ∗ x ) = H ∗ ( E x ). Since x has a Euclideanneighborhood, E is clc on our neighborhood of x and so each derived cohomology sheaf H i ( E )is constant on this neighborhood. By assumption there is some ℘ ⊂ P ( R ) such that H ( E x )is ℘ -torsion while H ( E x ) is ℘ -torsion free. All other H i ( E x ) are 0.On the other hand, by Lemma 4.2, H i ( P ∗ x ) ∼ = , i > ~p ( S ) + 1 ,T ~p ( S ) H i ( L k − ; E ) , i = ~p ( S ) + 1 , H i ( L k − ; E ) , i ≤ ~p ( S ) . Since there is some ℘ such that H ( E x ) is ℘ -torsion and H ( E x ) is ℘ -torsion free, any map H ( L k − , H ( E x )) → H k − ( L k − , H ( E x )) must be trivial, so the hypercohomology spectral27equence for H ∗ ( L k − ; E ) degenerates (using also that L k − has the cohomology of a k − k = 1 we have H i ( L , E ) ∼ = H i ( E x ) ⊕ H i ( E x ), and if k ≥ H i ( L k − ; E ) ∼ = H ( E x ) , i = k,H ( E x ) , i = k − ,H ( E x ) , i = 1 ,H ( E x ) , i = 0 , , otherwise.Similarly if k = 2, the only possibly nontrivial groups are for i = 0 , , H ( L ; E ) ∼ = H ( L ; H ( E )) ∼ = H ( E x ) and H ( L ; E ) ∼ = H ( L ; H ( E )) ∼ = H ( E x ). But the only easyinformation we have about H ( L ; E ) is that it must fit in the extension problem0 → H ( E x ) → H ( L ; E ) q −→ H ( E x ) → . (1)Given these preliminaries, we can now consider the necessity of the conditions of Defini-tion 3.1, separately in the cases k = 1, k = 2, and k ≥
3, by comparing H ∗ ( E x ) with H ∗ ( P ∗ x )as obtained from the preceding computations. k = 1 . In this case H i ( P x ) comes by torsion-tipped truncating H i ( L ; E ) ∼ = H i ( E x ) ⊕ H i ( E x ).Thus there is no way that H i ( P x ) can equal H i ( P ∗ x ) ∼ = H i ( E x ), regardless of perversity unless E is trivial. This shows why we must always rule out codimension one strata of X in regularstrata of X . k ≥ . Given the computations above, we can readily see that if E is not trivial then H ∗ ( P x ) ∼ = H ∗ ( E x ) (and hence H ∗ ( P x ) ∼ = H ∗ ( P x )) if and only one of the following scenariosholds:1. ~p ( S ) = − H ( E x ) is ~p ( S )-torsion, H ( E x ) = 0,2. ~p ( S ) = 0, H ( E x ) is ~p ( S )-torsion, H ( E x ) arbitrary,3. 1 ≤ ~p ( S ) ≤ k − H ∗ ( E x ) arbitrary,4. ~p ( S ) = k − H ( E x ) is ~p ( S )-torsion free, H ( E x ) is arbitrary,5. ~p ( S ) = k − H ( E x ) = 0, H ( E x ) is ~p ( S )-torsion free.Of course here “arbitrary” still means within the limitations of E being a ℘ -coefficient systemfor some ℘ ⊂ P ( R ), and we have recovered precisely the conditions from Definition 3.1.28 = 2 . This case is a bit more delicate as we can’t pin down H ( L ; E ) in general. However,we see again that to have H ∗ ( P x ) ∼ = H ∗ ( P ∗ x ) ∼ = H ∗ ( E x ) with E nontrivial, it is first of allnecessary to have ~p ( S ) ≥ −
1, and also by the same arguments if ~p ( S ) = −
1, then we musthave H ( E x ) = 0, and H ( E x ) must be ~p ( S ) torsion.If ~p ( S ) = 1 = k −
1, then to have H ( P x ) ∼ = H ( E x ) requires H ( L ; E ) ∼ = H ( E x ), whichfrom the short exact sequence above requires H ( E x ) = 0. And looking at degree 2 we mustagain have that H ( E x ) is ~p ( S )-torsion free.If ~p ( S ) ≥
2, then we must have H ( E x ) = 0 for degree 2 to work but again we also need H ( L ; E ) ∼ = H ( E x ) and so H ( E x ) = 0, which forces E to be trivial.Finally, suppose ~p ( S ) = 0 = k −
2. Then to have H ∗ ( P x ) ∼ = H ∗ ( P ∗ x ) we must have T ~p ( S ) H ( L ; E ) ∼ = H ( E x ). In particular, H ( E x ) must be ~p ( S )-torsion. Let k : H ( E x ) ֒ → H ( L ; E ) take H ( E x ) isomorphically onto T ~p ( S ) H ( S ; E ), and consider the composition qk , where q is the quotient map in (1). If z ∈ H ( E x ), z = 0, then z is ℘ -torsion forsome ℘ such that E is a ℘ -coefficient system, so qk ( z ) = 0 or else there would be a ℘ -torsion element in ker( q ) ∼ = H ( E z ), violating that E is a ℘ -coefficient system. Thus qk isinjective. Since H ( E x ) is a finitely-generated torsion module over a PID, it is Artinian andso every injective endomorphism is an isomorphism [19, Lemma II.4. α ]. Precomposing k with the inverse of this isomorphism provides a splitting of (1). Consequently we also have H ( E x ) ∼ = H ( L ; E ) /T ~p ( S ) H ( L ; E ), and so H ( E x ) must be ~p ( S )-torsion free. In this case we consider the necessity of the conditions of Definition 3.1 for a singular stratum S of X contained in singular stratum S of X . As noted above, these conditions won’t alwaysbe necessary in the strictest sense since weaker conditions might suffice depending on thelocal cohomology computations resulting from specific choices of spaces, perversities, andcoefficient systems. Instead, we show the conditions to be “necessary in general,” meaningthat we will demonstrate the existence of examples where P ∗ is not quasi-isomorphic to P ∗ because the conditions fail. Accordingly, we can choose to work in relatively simple settings.We first discuss necessity in the codimension 0 setting, followed by the codimension > S ⊂ S and dim( S ) = dim( S ). In this case, a point x ∈ S may have distinguishedneighborhoods in X and X that are filtered homeomorphic and so have the same link L . If P ∗ ∼ = P ∗ then in particular H i ( L ; P ∗ ) ∼ = H i ( L ; P ∗ ). It is then clear from Lemma 4.2 that wewill need to have ~p ( S ) = ~ p ( S ) in order to have H ∗ ( P ∗ x ) ∼ = H ∗ ( P ∗ x ) unless there are furtherrestrictions on H i ( L ; P ∗ ). In fact, since H ( E x ) is torsion, we can treat it as a module over R/ Ann( H ( E x )). If H ( E x ) = 0 thenAnn( H ( E x )) = 0, and this is an Artinian ring since R is a PID. If H ( E x ) = 0, it is clearly Artinian. .3.2 Codimension > L to be a trivially-stratified n − k − H i ( L ; E ) is nontrivial in any dimension we like or has ℘ -torsion in anydimension 0 to n − k − L and E . We then consider X = R k +1 × cL for k ≥
0. Topologically, R k +1 × cL ∼ = cS k × cL ∼ = c ( S k ∗ L ). If we instead stratify this spaceas the cone on S k ∗ L , using the stratification of S k ∗ L described just before Corollary 4.5we obtain a refinement X of X . In fact, X differs from X only by the addition of a singlezero-dimensional stratum, namely the vertex V of c ( S k ∗ L ), which we can identify with(0 , v ) ∈ R k +1 × cL if v is the cone vertex of cL .By such a construction, we obtain an X and X such that X has a singular stratum S = { V } contained in the singular stratum S = R k +1 × { v } of X and such that codim( S ) − codim( S ) = k + 1. The actual values of codim( S ) and codim( S ) will of course depend ondim( L ). If we want examples with higher dimensional strata we can consider instead R m × X and R m × X , though Lemma 4.1 shows that the cohomology computations will be the same.We will tend to use V when referring to the point and S when thinking of the stratum S = { V } .We suppose that the ts-Deligne sheaves P ∗ and P ∗ are quasi-isomorphic when restricted tothe complement of V and consider what would be necessary for them to be quasi-isomorphicat V .Based on Lemma 4.2, and our assumption that P ∗ ∼ = P ∗ off of V , we have H i ( P ∗ V ) ∼ = , i > ~ p ( S ) + 1 ,T ~ p ( S ) H i ( L ; P ∗ | L ) , i = ~ p ( S ) + 1 , H i ( L ; P ∗ | L ) , i ≤ ~ p ( S )and H i ( P ∗ V ) ∼ = , i > ~p ( S ) + 1 ,T ~p ( S ) H i ( S k ∗ L ; P ∗ | S k ∗ L ) , i = ~p ( S ) + 1 , H i ( S k ∗ L ; P ∗ | S k ∗ L ) , i ≤ ~p ( S ) . We can compute H i ( S k ∗ L ; P ∗ | S k ∗ L ) in terms of H ∗ ( L ; P ∗ | L ) using Corollary 4.5: H i ( S k ∗ L ; P ∗ | S k ∗ L ) ∼ = H i − k − ( L ; P ∗ ) , i > ~ p ( S ) + k + 2 , H ~ p ( S )+1 ( L ; P ∗ ) /T ~p ( S ) H ~p ( S )+1 ( L ; P ∗ ) , i = ~ p ( S ) + k + 2 , , ~ p ( S ) + 1 < i < ~ p ( S ) + k + 2 ,T ~ p ( S ) H ~ p ( S )+1 ( L ; P ∗ ) , i = ~ p ( S ) + 1 , H i ( L ; P ∗ ) , i ≤ ~ p ( S ) . (2)From these equations, we can see what constraints are necessary in this case and why:In order to have H i ( P ∗ V ) ∼ = H i ( P V ) we must truncate H i ( S k ∗ L ; P ∗ | S k ∗ L ) in such a waythat the result agrees with H i ( P ∗ V ). If ~p ( S ) < ~ p ( S ) then in general we will be forcing30 i ( P ∗ V ) to be 0 in some degrees ≤ ~ p ( S ) + 1 in which H i ( P ∗ V ) will not generally be 0without further vanishing assumptions. Furthermore, even if ~p ( S ) = ~ p ( S ) we must have ~p ( S ) ⊃ ~ p ( S ) in order to make sure we get all of T ~ p ( S ) H i ( L ; P ∗ | L ) in degree ~ p ( S ) + 1.Similarly, if ~p ( S ) ≥ ~ p ( S ) + k + 2, i.e. if ~p ( S ) > ~ p ( S ) + codim( S ) − codim( S ), thenthe term H ~ p ( S )+1 ( L ; P ∗ ) /T ~p ( S ) H ~p ( S )+1 ( L ; P ∗ ) (as well as possibly some of those above it in(2)) will appear in H ∗ ( P ∗ V ) even though it does not appear in H i ( P ∗ V ), but these are notnecessarily trivial. If ~p ( S ) = ~ p ( S ) + k + 1, i.e. if ~p ( S ) = ~ p ( S ) + codim( S ) − codim( S ), theonly problematic degree is H ~ p ( S )+ k +2 ( P ∗ V ) ∼ = T ~p ( S ) (cid:0) H ~ p ( S )+1 ( L ; P ∗ ) /T ~p ( S ) H ~p ( S )+1 ( L ; P ∗ ) (cid:1) .For this to vanish in general we need ~p ( S ) ⊂ ~ p ( S ). In addition to the original Ax1 axioms of [14, Section 3.3] (and the slight modificationAx1’), the Deligne sheaves of Goresky-MacPherson can also be characterized by a verydifferent set of axioms that were used in the original proofs of topological invariance ofintersection homology in [14, Section 4]. Called Ax2, these are phrased in terms of thesupport and “cosupport” dimensions of the Deligne sheaves, i.e. the dimensions of the sets onwhich H i ( P x ) and H i ( f ! x P ∗ ) are non-zero for the various i . This perspective has historicallybeen very useful, to the extent that these axioms are sometimes used to define intersectionhomology, e.g. see [16]. In this section we formulate a version of these axioms for ts-Delignesheaves and show that they are equivalent to the TAx1 axioms discussed above. We culminatewith Theorem 5.9, which is another formulation of our topological invariance results moreattuned to this context. Unfortunately, support and cosupport axioms can only characterize ts-Deligne sheaves if welimit ourselves to constrained ts-perversities. In this subsection we will see why that is. Thebasic issue is that we know our ts-Deligne sheaves are characterized by the axioms TAx1’ andso would like to see when it is possible to recover the information content of those axiomsfrom support and cosupport information. We will further assume below that our constrainedts-perversities satisfy ~p (2) = 0, i.e. that they are strongly constrained. While not strictlynecessary, this stronger condition allows us to avoid dealing with a plethora of case analysesand strong restrictions on ts-coefficients.We first note that knowing that some property holds on some k -dimensional union ofstrata is not enough by itself to tell us whether or not that property holds on all k -dimensionalstrata. So in order to convert (co)support information into information about behavior on allstrata of a given dimension, all strata of the same dimension need to be treated equivalently,i.e. we need to consider ts-perversities that are functions of codimension alone.To see why ~p must be nondecreasing, let us simplify and consider field coefficients, inwhich case our ts-Deligne sheaves are simply the usual Deligne sheaves. The axioms TAx1’tell us that the key information in this case for characterizing Deligne sheaves is knowingfor each stratum S the degrees for which H i ( P ∗ x ) and H i ( f ! x P ∗ ) are 0 for x ∈ S . Note that31onstructibility assumptions will tell us that these modules vanish for some x ∈ S if andonly if they vanish for all x ∈ S . Let us focus on H i ( P ∗ x ) and consider how TAx1’ translatesinto support information. The diagram (3) below serves as a good model (though focus onlyon the * entries for now) with codimension k increasing to the right, degree i increasingupward, the heights of the columns given by the values of ~p ( k ), and so each ∗ representinga (possibly) non-zero H i ( P ∗ x ), x ∈ X n − k .Suppose now that for a specific i we know that the dimension of the support of H i ( P ∗ x )is n − k . This tells us that H i ( P ∗ x ) = 0 for x in strata of codimensions < k , correspondingin the diagram to no ∗ entries at height i for columns left of k . It also tells us that ~p ( k )must be ≥ i . Now if ~p is nondecreasing then as k increases the columns get taller and as i increases the support dimensions of the H i ( P ∗ x ) get smaller. Suppose, however, that weallow ~p to decrease at some point; e.g. suppose we change the ~p shown in the diagram sothat ~p (9) = 0. Since the diagram suggests that the support dimension of H ( P ∗ x ) is n − ~p (9) to force H ( P ∗ x ) = 0 on a set of dimension n − H ( P ∗ x ). Conversely, if our only information is support dimensions andwe are trying to recover values of ~p ( k ), the vanishing of H ( P ∗ x ) on an n − n − ~p (9). Roughly said: if ~p is nondecreasing, then a diagramsuch as (3) allows us to recover the column heights from the row depths and vice versa; thisis essentially the content of the support axiom. However, if ~p is allowed to decrease, this isno longer possible.A similar consideration implies that we need the dual perversity D~p to be nondecreasingin its first component (recall Definition 2.3). To see this, let D denote Verdier duality; then[6, Theorem 4.19] says that DP ∗ ~p [ − n ] ∼ = P ∗ D~p , where P ∗ D~p is also taken with respect to thedual coefficient system DE . The key observation for our purposes is that f ! x P ∗ ~p ∼ = f ! x DDP ∗ ~p ∼ = D f ∗ x DP ∗ ~p ∼ = D f ∗ x P ∗ D~p [ n ] . So continuing to assume field coefficients and letting X be a pseudomanifold for simplic-ity, we have H i ( f ! x P ∗ ~p ) ∼ = H n − i ( f ∗ x P ∗ D~p ), using the Universal Coefficient Theorem for Verdierduality [2, Section V.7.7] and the finite generation implied by the constructibility of thesheaves [6, Theorem 4.10]. Consequently, information about H i ( f ! x P ∗ ~p ) is equivalent to infor-mation about H n − i ( f ∗ x P ∗ D~p ), and so the same argument above applies to say that obtainingfull TAx1’ information from cosupport dimensions relies on the presupposition that
D~p isnondecreasing.Since ~p ( k ) + D~p ( k ) = k −
2, we can only have ~p and D~p both nondecreasing if ~p satisfies the Goresky-MacPherson condition ~p ( k ) ≤ ~p ( k + 1) ≤ ~p ( k ) + 1. Furthermore,if we want to allow the full range of possible ts-coefficients so that H i ( E x ) may be nonzerofor both i = 0 , P ∗ as if it is also truncated over the regular stratausing ~p (0) ≥
0. The nondecreasing requirements on ~p and D~p then imply that we musthave both ~p ( k ) and D~p ( k ) always ≥
0. This is only possible if there are no codimensionone strata and ~p (2) = 0. Thus we see that ~p must be constrained with ~p (2) = 0, andcodimension one strata must be disallowed. Remark . This last choice of ~p (2) = 0 is a bit artificial. If we allow either H ( E ) = 032r H ( E ) = 0 then we could again consider any constrained ts-perversity adapted to thets-coefficient system. As noted above, however, we will leave these more general cases to theinterested reader.Similar considerations imply that if ~p ( k ) stays constant over some range of k , the sets ~p ( k ) must be nondecreasing. The same restriction on D~p implies that in a range where ~p ( k )is strictly increasing, the sets ~p ( k ) must be nonincreasing. Altogether, we have now arguedthat we should limit ourselves to strongly constrained ts-perversities (or at least constrainedones).Lastly, there is one other way in which we must constrain our data. Returning to PIDcoefficients, suppose that T p H ( E x ) = 0 for some p ∈ P ( R ). Then dim { supp( T p H ( P ∗ x )) } = n , and analogously to the above arguments, support information would be insufficient to tellus about T p H ( P ∗ x ) on higher codimension strata. To remedy this, we must assume that if T p H ( E x ) = 0 then p ∈ ~p ( k ) for all k such that ~p ( k ) = 0, so that p torsion is always allowedin degree 1. In particular, we must have that H ( E x ) is ~p (2)-torsion. Analogously, using that ~p (2) = 0, TAx1’ also says that we will need ts-Deligne sheaves to have T ~p (2) H n ( f ! x P ∗ ) = 0.But on the manifold U , we have H n ( f ! x P ∗ ) ∼ = H n ( f ! x E ) ∼ = H ( E x ), so we will not be ableto detect T ~p (2) H n ( f ! x P ∗ ) = 0 on X n − if T ~p (2) H ( E x ) is ever non-zero, as this would implydim { x | T ~p (2) H ( E x ) = 0 } = n . So if T p H ( E x ) = 0 for some x then we need to have p / ∈ ~p (2). Alternatively, if p ∈ ~p (2), then we must have T ~p (2) H ( E x ) = 0 for all x . Buttogether these conditions are equivalent to ~p being adapted to E .Therefore, we limit ourselves in this section primarily to the case where ~p is a stronglyconstrained ts-perversities (Definition 3.14) that is adapted to E , though we will see inSection 5.4 that we can also consider weakly constrained ts-perversities if we allow ourselvessome additional information. Classically, one can visual perversities satisfying the Goresky-MacPherson condition as “sub-step” functions. Similarly, strongly constrained ts-perversities can be visualized in diagramssuch as the following in which the ground ring is Z :4 { } { , , } ∗ { } ∗ ∗ ∗ { , } ∗ ∗ ∗ ∗ { } { , } ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ~p ( k ) while the setsof primes give ~p ( k ). The diagram is meant to evoke the cut-off degrees for the truncationsdetermined by ~p with the primes in each ~p ( k ) surviving for an extra degree. In particular,the displayed ts-perversity is given by 33 p (2) = (0 , ∅ ) ~p (3) = (0 , { } ) ~p (4) = (0 , { , } ) ~p (5) = (1 , ∅ ) ~p (6) = (1 , ∅ ) ~p (7) = (1 , { , } ) ~p (8) = (2 , { } ) ~p (9) = (3 , { } ) ~p (10) = (3 , { , , } ) ~p (11) = (4 , ∅ )Note that ~p satisfies the Goresky-MacPherson conditions while ~p grows in each row butshrinks with each step up.Now let ~p be a strongly constrained ts-perversity, let m ∈ Z ≥ , and let p ∈ P + ( R ),recalling from Section 2 that P + ( R ) = P ( R ) ∪ { f } . If m > ~p ( n ) + 1 or if m = ~p ( n ) and p / ∈ ~p ( n ), set ~p − ( m, p ) = ∞ . Otherwise, generalizing the definition in [14, Section 4.1], wedefine ~p − ( m, p ) = ( min { c ≥ | ~p ( c ) = m − , p ∈ ~p ( c ) } , if such a c exists , min { c ≥ | ~p ( c ) = m } , otherwise . In terms of our diagram above, ~p − ( m, p ) is the column number of the leftmost entryin the m th row containing p . If p is not listed explicitly in the m th row, then this is thecolumn of the leftmost ∗ . In our example above, ~p − (1 ,
2) = 3 while ~p − (1 ,
5) = 5. Note thatsince ~p ( c ) = − c , we have ~p − (0 , p ) = 2 for any p . Also, if p = f then p ∈ ~p ( c ) isimpossible and so ~p − ( m, f ) = min { c ≥ | ~p ( c ) = m } as in [14]. Remark . If ~p ( k ) = ∅ for all k ≥
2, then ~p − reduces to the ~p − of [14, Section 4.1] (in[2, Section V.4.6], Borel writes ≥ instead of = in the definition, but under the Goresky-MacPherson perversity restriction, min { c | ~p ( c ) = m } = min { c | ~p ( c ) ≥ m } since ~p musttake all values between 0 and ~p ( n )).On the other hand, if ~p ( k ) = P ( R ) for all k ≥ ~p − ( m, p ) = min { c ≥ | ~p ( c ) = m − } = ~p − ( m − , f ) for all p ∈ P ( R ) and all m > Remark . Thinking in terms of diagrams as above and using that ~p is strongly constrained,we see that for a fixed p ∈ P + ( R ) we have k ≥ ~p − ( m, p ) if and only if either1. ~p ( k ) ≥ m or2. ~p ( k ) = m − p ∈ ~p ( k ).In particular if p = f then k ≥ ~p − ( m, f ) if and only ~p ( k ) ≥ m . If m = 0 this says that k ≥ ~p − (0 , p ) = 2 if and only if ~p ( k ) ≥
0, though this is tautological as both statements arealways true.This observation does not require the last property of constrained ts-perversities, onlythe first four. However, below we will need these statements for both ~p and its dual D~p .The last property is needed for the dual to also be strongly constrained.
Lemma 5.4. ~p is a strongly constrained ts-perversity if and only if its dual D~p is alsostrongly constrained. roof. For ease of notation, let ~q = D~p . As
DD~p = ~p , it suffices to show that if ~p is stronglyconstrained then so is ~q . Clearly ~p is a function of codimension if and only if ~q is. The growthcondition only concerns ~p and ~q and is true of classical Goresky-MacPherson perversities;it follows from ~p ( k ) + ~q ( k ) = k −
2. Similarly ~p (2) = 0 if and only if ~q (2) = 0. Next,note that ~p ( k + 1) = ~p ( k ) + 1 if and only if ~q ( k + 1) = ~q ( k ) and ~p ( k + 1) = ~p ( k )if and only if ~q ( k + 1) = ~q ( k ) + 1. Further, ~p ( k ) and ~q ( k ) are complementary sets ofprimes. So if ~p ( k + 1) ⊃ ~p ( k ) whenever ~p ( k + 1) = ~p ( k ) then ~q ( k + 1) ⊂ ~q ( k ) whenever ~q ( k + 1) = ~q ( k ) + 1. Similarly if ~p ( k + 1) ⊂ ~p ( k ) whenever ~p ( k + 1) = ~p ( k ) + 1 then ~q ( k + 1) ⊃ ~q ( k ) whenever ~q ( k + 1) = ~q ( k ). We can now formulate a version of the Goresky-MacPherson axioms Ax2. We follow moreclosely the exposition in [2, Section V.4], which is more detailed than [14]. In the followingdefinition we assume X to be a CS set of dimension n with no codimension one strata, that ~p is a strongly constrained ts-perversity, that ~q = D~p , and that X and ~p are adapted to thets-coefficient system E . Definition 5.5.
We say the sheaf complex S ∗ satisfies the Axioms TAx2 ( X, ~p, E ) ifa. S ∗ is X -clc and it is quasi-isomorphic to a complex that is bounded and that is 0 innegative degrees;b. S ∗ | U ∼ = E | U ;c. (a) If j > { x ∈ X | T p H j ( S ∗ x ) = 0 } ≤ n − ~p − ( j, p ) for all p ∈ P + ( R ).(b) dim { x ∈ X | T p H ( S ∗ x ) = 0 } ≤ n − ~p − (1 , p ) for all p ∈ P + ( R ) such that p / ∈ ~p (2).d. (a) If j < n then dim { x ∈ X | T p H j ( f ! x S ∗ ) = 0 } ≤ n − ~q − ( n − j + 1 , p ) for all p ∈ P ( R ) and dim { x ∈ X | T f H j ( f ! x S ∗ ) = 0 } ≤ n − ~q − ( n − j, f ).(b) dim { x ∈ X | T p H n ( f ! x S ∗ ) = 0 } ≤ n − ~q − (1 , p ) for all p ∈ ~p (2).Note that in axiom (c) we may have p = f , but in axiom (d) the only appearance of f isexplicit since f / ∈ ~p (2). If ~p ( k ) = ∅ for all k then these axioms reduce to those of Borel in[2, Section 4.7]. Proposition 5.6.
The sheaf complex S ∗ satisfies TAx1’ ( X, ~p, E ) if and only it satisfiesTAx2 ( X, ~p, E ) .Proof. The proof emulates that of [2, Proposition V.4.9], though it is a bit more complicatedsince we must consider the torsion effects and also take more care with some special caseswhen the degree j is near 0 or n . We label codimension by k and assume 2 ≤ k ≤ n throughout, since X has no codimension one strata by assumption. The first two axiomsof TAx1’ and TAx2 agree, so we will show that the two third axioms and the two fourthaxioms are equivalent given the hypotheses. 351 ′ c ⇒ c ) . First suppose S ∗ satisfies TAx1’c. We first observe that it is possible to havedim { x ∈ X | T p H j ( S ∗ x ) = 0 } = n if j = 0 or if j = 1 and p ∈ ~p (2) since S ∗ | U ∼ = E and these properties are true of E . However, also thanks to the properties of E , these arethe only cases for which dim { x ∈ X | T p H j ( S ∗ x ) = 0 } = n . In all other cases, T p H j ( S ∗ x )is supported in X n − . So consider these other cases, i.e. either j = 1 and p / ∈ ~p (2) or j >
1. If x ∈ X n − k for k ≥ T p H j ( S ∗ x ) = 0 then by TAx1’c either j ≤ ~p ( k ) orwe have j = ~p ( k ) + 1 and p ∈ ~p ( k ). By Remark 5.3, this implies k ≥ ~p − ( j, p ). Thendim X n − k ≤ n − k ≤ n − p − ( j, p ). This yields TAx2c.(1 ′ c ⇐ c ) . Conversely, suppose TAx2c holds. Now fix k ≥ x ∈ X n − k . We mustshow that T p H j ( S ∗ x ) = 0 if j > ~p ( k ) + 1 or if we have j = ~p ( k ) + 1 and p ∈ P + ( R ) − ~p ( k ).So first suppose j ≥ ~p ( k ) + 2 ≥
2, the last inequality by the assumptions on ~p . Then ~p − ( j, p ) > k using Remark 5.3. Similarly Remark 5.3 implies that if j = ~p ( k ) + 1 ≥ p / ∈ ~p ( k ) then ~p − ( j, p ) > k . We also note that since ~p (2) ⊂ ~p ( k ) for all k such that ~p ( k ) = 0, if ~p ( k ) = 0 and p / ∈ ~p ( k ), then p / ∈ ~p (2). So if j ≥ ~p ( k ) + 2 or if we have j = ~p ( k ) + 1 ≥ p / ∈ ~p ( k ) then either j ≥ j = 1 with p / ∈ ~p (2). In either case theassumptions say that dim { x ∈ X | T p H j ( S ∗ x ) = 0 } ≤ n − ~p − ( j, p ) < n − k . Since S ∗ is X -clc, if T p H j ( S ∗ x ) = 0 then also T p H j ( S ∗ y ) = 0 for all y in the n − k dimensional stratumcontaining x . Hence we must have in these cases T p H j ( S ∗ x ) = 0. This implies TAx1’c.(1 ′ d ⇒ d ) . Next suppose S ∗ satisfies TAx1’d. Since S ∗ | U ∼ = E and since f ! x = f ∗ [ − n ]on U , we have for x ∈ U that T p H j ( f ! x S ∗ ) = 0 only for j = n, n + 1 and furthermorethat T p H n ( f ! x S ∗ ) = 0 if p ∈ ~p (2). So for j < n or for j = n and p ∈ ~p (2), the dimensiondim { x ∈ X | T p H j ( f ! x S ∗ ) = 0 } is determined entirely by points in X n − .So suppose x ∈ X n − k for k ≥ j < n or that j = n and p ∈ ~p (2). If T p H j ( f ! x S ∗ ) = 0 then by assumption either1. j ≥ ~p ( k ) + n − k + 3 = n − ~q ( k ) + 1, or2. j = ~p ( k ) + n − k + 2 = n − ~q ( k ) and p ∈ ~q ( k ) ∪ { f } .In the first scenario we can conclude by Remark 5.3 that k ≥ ~q − ( n − j +1 , p ) while the secondscenario gives us k ≥ ~q − ( n − j + 1 , p ) if p ∈ ~q ( k ) and k ≥ ~q − ( n − j, f ) if p = f . Note thatin either case if p = f then we conclude k ≥ ~q − ( n − j + 1 , p ), though of course the value of q − ( n − j + 1 , p ) can depend on p . So if p = f , then dim( X n − k ) ≤ n − k ≤ n − ~q − ( n − j + 1 , p )and so dim { x ∈ X | T p H j ( f ! x S ∗ ) = 0 } ≤ n − ~q − ( n − j + 1 , p ). Similarly, if p = f we obtaindim { x ∈ X | T f H j ( f ! x S ∗ ) = 0 } ≤ n − ~q − ( n − j, f ). So we have TAx2d.(1 ′ d ⇐ d ) . Finally, suppose TAx2c holds. If n ≤ j ≤ ~p ( k ) + n − k + 1, then k − ≤ ~p ( k ),which is impossible. So in considering the first condition of TAx1’d we may assume j < n .Similarly, n ≤ j = ~p ( k ) + n − k + 2 implies k − ≤ ~p ( k ), which is possible only when we We remark that if ~p ( k ) = ∅ for all k , which corresponds to S ∗ satisfying the original Goresky-MacPherson axioms, then ~q ( k ) = P ( R ) for all k . In this case q − ( n − j + 1 , p ) = q − ( n − j, f ) by Remark5.2, which is again consistent with the expectation from the classical case. j = n . Thus for the second condition of TAx1’d we may consider only j ≤ n .First suppose j < n . Since S ∗ is X -clc (by either set of axioms), j ! k S ∗ is clc by[21, Proposition 4.0.2.3], so if T p H j ( f ! x S ∗ ) = 0 for some x ∈ X n − k then the same is truefor all other points X n − k . Thus for p ∈ P ( R ), if x ∈ X n − k and T p H j ( f ! x S ∗ ) = 0 then n − k ≤ n − ~q − ( n − j + 1 , p ), so k ≥ ~q − ( n − j + 1 , p ). Thus by Remark 5.3 either ~q ( k ) ≥ n − j + 1 or ~q ( k ) = n − j and p ∈ ~q ( k ). This translates to j ≥ ~p ( k ) + n − k + 3or j = ~p ( k ) + n − k + 2 and p / ∈ ~p ( k ). Similarly, if p = f , the assumptions imply n − k ≤ n − ~q − ( n − j, f ) or k ≥ ~q − ( n − j, f ), which means that ~q ( k ) ≥ n − j . This translatesto j ≥ ~p ( k ) + n − k + 2. So, altogether, if T p H j ( f ! x S ∗ ) = 0 then j ≥ ~p ( k ) + n − k + 2 andif j = ~p ( k ) + n − k + 2 then p / ∈ ~p ( k ). This is TAx1’d.Now suppose x ∈ X n − k , j = n = ~p ( k ) + n − k + 2, and p ∈ ~p ( k ). We must showthat T p H n ( f ! x S ∗ ) = 0. In this case ~p ( k ) = k − ~p ( k ) = k − k . So ~q must be ¯0 up through k . In this case the hypotheses onconstrained ts-perversities imply that ~p ( k ) ⊂ ~p ( c ) ⊂ ~p (2) for all 2 ≤ c ≤ k , so in particularwe may use the second condition of TAx2d. Further, since p ∈ ~p ( c ) for all 2 ≤ c ≤ k then p / ∈ ~q ( c ) for all 2 ≤ c ≤ k and our hypotheses imply n − j + 1 = 1. Consequently, ~q − ( n − j + 1 , p ) = ~q − (1 , p ) > k . So TAx2d implies that dim { T p H n ( f ! x S ∗ ) = 0 } < n − k .So T p H n ( f ! x S ∗ ) = 0 as needed.Altogether this shows TAx1.d. Corollary 5.7.
Suppose ~p is a strongly constrained ts-perversity, X is a CS set withoutcodimension one strata, and X and ~p are adapted to the ts-coefficient system E . Then S ∗ satisfies TAx1 ( X, ~p, E ) if and only if it satisfies TAx1’ ( X, ~p, E ) if and only if it satisfiesTAx2 ( X, ~p, E ) . Any of these axioms characterize S ∗ uniquely up to isomorphism as thets-Deligne sheaf P ∗ X,~p, E .Proof. This follows directly from the preceding proposition, Theorem 2.8, and [6, Theorem4.8].
Definition 5.8.
Let | X | be a space, let E be a maximal ts-coefficient system on | X | , andlet ~p be a strongly constrained perversity adapted to E . We say S ∗ satisfies the AxiomsTAx2’ ( ~p, E ) ifa. S ∗ is quasi-isomorphic to a complex that is bounded and that is 0 in negative degrees;b. S ∗ is X -clc for some CS set stratification X of | X | without codimension one stratathat is adapted to E , and S ∗ | U ∼ = E | U ;c. (a) If j > { x ∈ | X | | T p H j ( S ∗ x ) = 0 } ≤ n − ~p − ( j, p ) for all p ∈ P + ( R ).(b) dim { x ∈ | X | | T p H ( S ∗ x ) = 0 } ≤ n − ~p − (1 , p ) for all p ∈ P + ( R ) such that p / ∈ ~p (2).d. (a) If j < n then dim { x ∈ | X | | T p H j ( f ! x S ∗ ) = 0 } ≤ n − ~q − ( n − j + 1 , p ) for all p ∈ P ( R ) and dim { x ∈ | X | | T f H j ( f ! x S ∗ ) = 0 } ≤ n − ~q − ( n − j, f ).37b) dim { x ∈ | X | | T p H n ( f ! x S ∗ ) = 0 } ≤ n − ~q − (1 , p ) for all p ∈ ~p (2).Our Axioms TAx2’( ~p, E ) are slightly different from the Axioms (Ax2) E of [2, Section4.13] even beyond the incorporation of torsion information and the generalization to CSsets. As observed in [2, Remark V.4.14.b], the axioms there do not assume any relationbetween the stratification of X and the coefficient system E as we have done in the secondaxiom. However, it is also observed in this remark that (in that setting), a sheaf complexsatisfies (Ax2) E if and only if it satisfies (Ax2) X, E for some stratification (in Borel’s case apseudomanifold stratification) adapted to E . Our Axioms TAx2’( ~p, E ) are therefore a bitless general than Borel’s Axioms (Ax2) E in this sense, though as in Section 2.1 we can adaptBorel’s remark if each E i is a local system and E i = 0 for sufficiently large | i | . In this casewe need not assume X adapted to E in the second axiom.In either case, the upshot is the same: a sheaf complex satisfies our TAx2’( ~p, E ) if andonly if it satisfies TAx2( X, ~p, E ) for some stratification X of | X | (adapted to E ); furthermorethe axioms TAx2’( ~p, E ) are stratification independent. Putting this together with our priorresults we obtain a torsion sensitive analogue of [2, Theorem V.4.15]: Theorem 5.9.
Suppose E is a maximal ts-coefficient system with domain U E on a space | X | and that ~p is a strongly constrained perversity adapted to E . Suppose X has a CS setstratification with no codimension one strata that is fully adapted to E . Then there is a sheafcomplex P ∗ satisfying TAx2’ ( ~p, E ) with P ∗ | U E ∼ = E and such that P ∗ satisfies TAx2 ( X, ~p, E ) for every CS set stratification X of | X | without codimension one strata that is fully adaptedto E .Proof. As X has a CS set stratification with no codimension one strata that is fully adaptedto E , there is an intrinsic stratification X fully adapted to E by Proposition 6.4. Let P ∗ bethe ts-Deligne sheaf with respect to X . Then P ∗ | U E ∼ = E since X − X n − = X − X n − = U E byProposition 6.4. Proposition 3.17 implies P ∗ is quasi-isomorphic to the ts-Deligne sheavescoming from any of the other stratifications, and we know these satisfy the axioms byCorollary 5.7. Analogously to the Goresky-MacPherson axioms Ax2, our Axioms TAx2 depend only veryweakly on the stratification: TAx2 only mentions a particular stratification to specify thatit is adapted to the coefficients, that P ∗ is X -clc, and that P ∗ ∼ = E over the regular strata.TAx2’ only asks for this with respect to some stratification. Consequently we obtain ourversion of topological invariance in Theorem 5.9.In Section 5.1 we argued that in order for the support and cosupport axioms to implyour earlier TAx1’ axioms it is necessary to use constrained perversities that are adapted to E and to forbid codimension one strata. However, it is possible to avoid all of these constraintsexcept for the Goresky-MacPherson growth condition at the expense of modifying the TAx2axioms to depend more heavily on the stratification. In fact, we can obtain Theorem 5.13,38elow, which generalizes the main theorem of [9]. The proofs are all analogous to thoseabove, though in fact simpler since special care no longer needs to be taken in extremedegrees.For the following, we let ~p be a weakly constrained perversity with domain Z ≥ . We canthen extend ~p − to be a function Z × P + ( R ) → Z ≥ by declaring that if m < ~p (1) then ~p − ( m, p ) = 1. Definition 5.10.
Let X be a CS set (possibly with codimension one strata) adapted toa ts-coefficient system E . Let ~p be a weakly constrained ts-perversity. We say the sheafcomplex S ∗ satisfies the Axioms TAx2 ( X, ~p, E , X n − ) ifa. S ∗ is X -clc and it is quasi-isomorphic to a complex that is bounded and that is 0 innegative degrees;b. S ∗ | U ∼ = E | U ;c. dim { x ∈ X n − | T p H j ( S ∗ x ) = 0 } ≤ n − ~p − ( j, p ) for all p ∈ P + ( R ).d. dim { x ∈ X n − | T p H j ( f ! x S ∗ ) = 0 } ≤ n − ~q − ( n − j + 1 , p ) for all p ∈ P ( R ) anddim { x ∈ X n − | T f H j ( f ! x S ∗ ) = 0 } ≤ n − ~q − ( n − j, f ). Proposition 5.11.
Let ~p be a weakly constrained perversity, and suppose X is a CS set,possibly with codimension one strata, adapted to the ts-coefficient system E . Then the sheafcomplex S ∗ satisfies TAx1’ ( X, ~p, E ) if and only it satisfies TAx2 ( X, ~p, E , X n − ) . Definition 5.12.
Let | X | be a space, Σ a closed subspace, ~p a weakly constrained per-versity, and E any maximal ts-coefficient system on | X | . We say S ∗ satisfies the AxiomsTAx2’ ( ~p, E , Σ) ifa. S ∗ is quasi-isomorphic to a complex that is bounded and that is 0 in negative degrees;b. S ∗ is X -clc for some CS set stratification X of | X | (possibly with codimension onestrata) such that X n − = Σ and X that is adapted to E , and S ∗ | U ∼ = E | U ;c. dim { x ∈ Σ | T p H j ( S ∗ x ) = 0 } ≤ n − ~p − ( j, p ) for all p ∈ P + ( R ).d. dim { x ∈ Σ | T p H j ( f ! x S ∗ ) = 0 } ≤ n − ~q − ( n − j + 1 , p ) for all p ∈ P ( R ) and dim { x ∈ Σ | T f H j ( f ! x S ∗ ) = 0 } ≤ n − ~q − ( n − j, f ). Theorem 5.13.
Suppose ~p is a weakly constrained perversity and that E is a maximal ts-coefficient system with domain U E on a space | X | with closed subspace Σ . Suppose | X | hasa CS set stratification X that is fully adapted to E and such that X n − = Σ . Then there is asheaf complex P ∗ satisfying TAx2’ ( ~p, E , Σ) with P ∗ | U E − Σ ∼ = E | U E − Σ and such that P ∗ satisfiesTAx2 ( X, ~p, E , Σ) for every CS set stratification X with X n − = Σ that is fully adapted to E . Intrinsic stratifications
In this section we consider common coarsenings of CS sets. In general, each CS set possessesan intrinsic stratification that coarsens all others; this is due to King and Sullivan [18] and athorough discussion can be found in [7, Section 2.10]. However, since not all stratificationsare adapted to a given coefficient system, it is necessary to refine the construction to take thecoefficients into account. A version of such a construction for fairly general sheaf complexescan be found in Habegger-Saper [16, Section 3]. Furthermore, we have seen in Section 3that we may wish to only consider coarsenings that preserve some subspace, without lettingstrata in the subspace “merge” with other strata not in the subspace. So this is a furtheringredient we will consider for our common coarsenings. Many of the basic ideas are thesame as in the above references, but in order to account for the additional ingredients weprovide most of the details.Recall that the usual intrinsic stratification of a CS set X is determined by an equivalencerelation so that x , x ∈ X are equivalent, denoted x ∼ x , if they possess neighborhoods U , U such that ( U , x ) ∼ = ( U , x ) as topological space pairs (i.e. ignoring the filtrations)[7, Definition 2.10.3]. If x , x are both in the same stratum of X , then x ∼ x [7, Lemma2.10.4]. Furthermore, if we let X i be the union of the equivalence classes that only containstrata of X of dimension ≤ i , then the X i filter X as a CS set that does not depend onthe initial filtration of X as a CS set and that coarsens all other CS set stratifications [7,Proposition 2.10.5]. This provides an intrinsic coarsest CS set stratification of X .To account for subspaces, the Frontier Condition [7, Definition 2.2.16., Lemma 2.3.7]implies that if we have a stratum S that we don’t wish to merge with some other stratum T of lower codimension then points in the closure of S also cannot merge with T . Consequently,it makes sense for our fixed subspaces to be closed unions of strata. We therefore make thefollowing definition. The assumption that X be fully adapted to a maximal ts-coefficientsystem E will be critical in the following arguments; see Section 2.1 for those definitions. Definition 6.1.
Let X be a CS set fully adapted to the maximal ts-coefficient system E withdomain the open n -manifold U E , and let C be a closed union of strata of X . We say x ∼ E ,C y for points x, y ∈ X if there is a homeomorphism of space pairs (ignoring the stratifications) h : ( U x , x ) → ( U y , y ) so that1. h ( U x ∩ C ) = U y ∩ C ,2. h ( U x ∩ U E ) = U y ∩ U E , and3. h ∗ ( E | U y ∩ U E ) is quasi-isomorphic to E | U x ∩ U E , i.e. h ∗ ( E | U y ∩ U E ) ∼ = E | U x ∩ U E in the derivedcategory.For the rest of our discussion we fix E and C and so simply write ∼ for our relation. Lemma 6.2. ∼ is an equivalence relation.2. If x, y ∈ X are in the same stratum of X then x ∼ y . roof. The relation is clearly reflexive. For symmetry and transitivity, the only parts thatare not obvious are the behavior of E . Suppose x ∼ y and y ∼ z with homeomorphisms h : U x → U y and g : U y → U z . Then h ∗ E | U y ∩ U E ∼ = E | U x ∩ U E and g ∗ E | U z ∩ U E ∼ = E | U y ∩ U E , so( gh ) ∗ E | U z ∩ U E ∼ = h ∗ g ∗ E | U z ∩ U E ∼ = h ∗ E | U y ∩ U E ∼ = E | U x ∩ U E demonstrating transitivity. Similarly,if x ∼ y then ( h − ) ∗ E | U x ∩ U E ∼ = ( h − ) ∗ h ∗ E | U y ∩ U E ∼ = ( hh − ) ∗ E | U y ∩ U E ∼ = E | U y ∩ U E . So ∼ is anequivalence relation.Now suppose U x is a distinguished neighborhood of x by the filtered homeomorphism g : R k × cL ֒ → X . Let y ∈ U x be contained in the same stratum as x , in which case U x is alsoa distinguished neighborhood of y . Then g − ( x ) , g − ( y ) ⊂ R k × { v } = R k , where v is thecone point. Let f be a homeomorphism of R k that takes g − ( x ) to g − ( y ), and let h = f × idon R k × cL . Then ghg − is a homeomorphism U x → U y = U x that takes x to y . Since ghg − preserves strata and since X is fully adapted to E and C is a union of strata, the map ghg − restricts to a homeomorphism from U x ∩ U E to itself and also from U x ∩ C to itself.Furthermore, since E is clc on its domain of definition and since X is fully adapted to E , g ∗ E will be clc on g − ( U x ∩ U E ), which will be a set of the form R k × V . Let π : R k × cL → cL bethe projection. By [17, Proposition 2.7.8], g ∗ E ∼ = π ∗ Rπ ∗ g ∗ E on its domain. So since πh = π we have( ghg − ) ∗ E = ( g − ) ∗ h ∗ g ∗ E ∼ = ( g − ) ∗ h ∗ π ∗ Rπ ∗ g ∗ E ∼ = ( g − ) ∗ π ∗ Rπ ∗ g ∗ ∼ = ( g − ) ∗ g ∗ E ∼ = E on its domain. So x ∼ y .Now suppose x is in the stratum S of x and let W be the set of points in S equivalentto x . By the above argument, W is an open subset of S . On the other hand, if y ∈ S is inthe closure of W , then any distinguished neighborhood of y must intersect W , so y ∈ W bythe above. Thus W is closed. Since strata are connected, W must be all of S .By the lemma, the equivalence classes under ∼ are unions of strata of X . Let X i E ,C bethe union of the equivalence classes made up only of strata of dimension ≤ i , and let X E ,C be the stratification of | X | with these skeleta. Definition 6.3.
We call X E ,C the intrinsic stratification of X rel ( E , C ).The following proposition contains the properties of X E ,C , including that this is a CS setand that it provides a common coarsening of all CS set stratifications of | X | that are fullyadapted to E and for which C is a closed union of strata (Property 7). We will only needthe case k = 1 of the last statement, Property 8, which concerns the closure of the unionof strata of codimension one. However, the proof is equivalent for any k so we provide themore general version. Proposition 6.4.
Let X be an n -dimensional CS set fully adapted to the maximal ts-coefficient system E with domain the open n -manifold U E , and let C be a closed union ofstrata of X of codimension ≥ . Then:1. The sets X i E ,C filter | X | as a CS set.2. If x and y are in the same stratum of X E ,C then x ∼ y . . C is a union of strata of X E ,C .4. X − X n − = U E − C .5. X E ,C is fully adapted to E .6. Suppose X is another CS set stratification of | X | that is fully adapted to E and suchthat C is also a closed union of strata of X . Then starting with X results in the same X i E ,C , i.e. the intrinsic stratification of X rel ( E , C ) is also X E ,C .7. Suppose X is another CS set stratification of | X | that is fully adapted to E and suchthat C is also a closed union of strata of X . Then X refines X E ,C , i.e. each stratumof X E ,C is a union of strata of X . Hence, X E ,C is a common coarsening of all suchstratifications.8. Suppose C k is the closure of the union of strata of X of codimension k . Then C k isalso the closure of the union of strata of X E ,C k of codimension k .Proof. We write simply X rather than X E ,C . We take each statement in turn. The proof that the sets X i filter | X | as a CS set is essentially identically to the classicalcase [7, Proposition 2.10.5], using Lemma 6.2. We sketch the argument as we will use someof the details below.First observe that if x ∼ y by the homeomorphism h : U x → U y and if z ∈ U x then z ∼ h ( z ) letting U z = U x and U h ( z ) = U y . Now suppose x ∈ | X | − X i so that x is equivalentto a point y in a stratum of X of dimension > i . By restricting to a smaller U x if necessary,we can assume h ( U x ) is contained in a distinguished neighborhood of y in X . Then h ( U x )intersects only strata of dimension > i and so each point of U x is equivalent to a point in astratum of dimension > i . So h ( U x ) ∈ | X | − X i . Thus | X | − X i is open so X i is closed andthe X i provide a closed filtration of X , as clearly X i ⊂ X i +1 .Next suppose x ∈ X i ∩ X i . Then x has a distinguished neighborhood N ∼ = R i × cL in X . It is shown in the proof of [7, Proposition 2.10.5] that if we think of L as embedded asthe image of { } × { / } × L and refilter | L | by ℓ j − i − = | L | ∩ X j then the image of R i × cℓ becomes a distinguished neighborhood of x in X . If z ∈ X i − X i ∩ X i then z is equivalent tosome point x ∈ X i ∩ X i , and we obtain a distinguished neighborhood for z in X as the filteredhomeomorphic image of a distinguished neighborhood of x in X under the homeomorphismof the equivalence. See [7] for details. It follows from Lemma 6.2 and the construction of distinguished neighborhoods in theproof of Property 1 that each point x has a distinguished neighborhood R i × cL in X suchthat the points of R i × { v } , i.e. all the points in the neighborhood that are in the samestratum of X as x , are equivalent. Property 2 now follows from the same sort of open/closedargument as in the proof of Lemma 6.2. 42 . By the preceding property, all points in any fixed stratum of X are equivalent. Fromthe definition of the equivalence relation, this would not be possible if any stratum of X intersected both C and its complement. Hence any stratum intersecting C is contained in C , and C is a union of strata. Suppose x ∈ X n − X n − . Then by definition x is equivalent to a point z in X n − X n − .But since X is adapted to E , the point z has a Euclidean neighborhood on which E isdefined. Hence so does x . Thus X − X n − ⊂ U E . Furthermore, by construction of ∼ and theassumption that C is the closure of a union of strata of codimension ≥
1, no point in C canbe equivalent to a point in X − X n − , so X − X n − ⊂ U E − C .Next suppose x ∈ U E − C . Then x has a Euclidean neighborhood in | X | − C on which E is defined. By the argument in the proof of Lemma 6.2, x will be equivalent to every otherpoint in this neighborhood. In particular x is equivalent to a point in an n -dimensionalstratum of X and so x ∈ X n − X n − . We have already seen that X − X n − = U E − C , so in particular X − X n − ⊂ U E . Itremains to show that U E ∩ C is a union of strata of X . Let S ⊂ X i , i ≤ n −
1, be anystratum of X . Then U E ∩ S is open in S since U E is an open set. Thus it suffices to showthat U E ∩ S is also closed in S . Let x be in the closure of U E ∩ S so that every neighborhoodof x intersects U E ∩ S . By the proof of Property 1, x is equivalent to a point z ∈ X i and the homeomorphism h : U z → U x induces (possibly after restriction to a subspace)a filtered homeomorphism from a distinguished neighborhood of z in X to a distinguishedneighborhood of x in X . In particular, h takes a neighborhood B of z in X i to a neighborhoodof x in X i . Since X is fully adapted to E , either B ⊂ U E or B ∩ U E = ∅ . But since h ( B )is a neighborhood of x in S , there is some y ∈ h ( B ) ∩ U E , and hence h − ( y ) ∈ B ∩ U E bydefinition of ∼ . Thus z ∈ U E and so is x . Here we modify the proof of [7, Proposition 2.10.5]. Let X and X be two CS setstratifications of | X | fully adapted to E and such that C is a closed union of strata of X .Let X and ˆ X be the resulting coarsenings. The equivalence relation ∼ does not depend onthe stratifications, and so the equivalence relations used to define X and ˆ X are the same andwe will use the same symbol for both. However, the definitions of the skeleta X i and ˆ X i do a priori depend on the stratifications, so this is what we must consider.Clearly X n = | X | = ˆ X n , and by our preceding arguments X − X n − = U E − C = ˆ X − ˆ X n − so that X n − = ˆ X n − . Now let x ∈ X i for some i < n −
1. Then x cannot be equivalentto any point in X j with j > i by definition. Suppose x ∈ ˆ X j for some j > i , and let S bethe stratum of ˆ X j containing x . By Property 2, the points of S are all equivalent to x . Butnow dimension considerations show that there must be points arbitrary close to x that areequivalent to x but not contained in X i , and in particular there is therefore some stratumof X of dimension > i containing points equivalent to x , a contradiction. So x is not in anyˆ X j with j > i and so x ∈ ˆ X i . Thus X i ⊂ ˆ X i , and the same argument shows the converse. So X i = ˆ X i for all i . 43 . The statement follows from the preceding one and Lemma 6.2. Let C k denote the closure of the strata of X of codimension k . By the Frontier Condition, C k ⊂ X n − k . So if x ∈ X n − k then x cannot be equivalent to any point in X − X n − k as suchpoints have neighborhoods that do not intersect C k . So x ∈ X n − k . But also x clearly cannotbe equivalent to a point that is only equivalent to points in strata of X of dimension < n − k ,so x ∈ X n − k − X n − k − . Thus X n − k − X n − k − ⊂ X n − k − X n − k − . Taking closures, C k ⊂ C k .Next, suppose x ∈ X n − k − X n − k − . By definition x is equivalent to a point in a stratum of X of dimension n − k . If x / ∈ C k then x has a neighborhood that does not intersect C k and so x is not equivalent to a point in C k , a contradiction. So x ∈ C k . Thus X n − k − X n − k − ⊂ C k ,so, taking closures, C k ⊂ C k . References [1] A. A. Be˘ılinson, J. Bernstein, and P. Deligne,
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