Topological invariance of torsion sensitive intersection homology

TL;DR

This paper establishes the topological invariance of torsion sensitive intersection homology, showing that its defining sheaf complexes are independent of stratification choices, extending classical results and providing new insights.

Contribution

It proves topological invariance for torsion sensitive intersection homology and offers new results even in classical intersection homology.

Findings

Proves independence of sheaf complexes from stratification choices.

Extends classical invariance theorems to torsion sensitive intersection homology.

Provides new invariance results in classical intersection homology.

Abstract

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

Tables

Table 1. Table 1: Types of perversities and resulting topological invariance

Type	Definition	Conditions	Top. Invariance
GM perversity	${\mathbb{Z}}_{\ge 2}\to \mathbb{Z}$	$\overline{p}\left(2\right)=0$ $\overline{p}\left(k\right)\le \overline{p}\left(k+1\right)\le \overline{p}\left(k\right)+1$	PL IH [GM1] Sheaf IH [GM2]
King perversity	${\mathbb{Z}}_{\ge 1}\to \mathbb{Z}$	$\overline{p}\left(k\right)\le \overline{p}\left(k+1\right)\le \overline{p}\left(k\right)+1$	Singular IH [Ki] Partial for sheaf IH [GBF11]
General perversity	$\left\{\text{sing. strata}\right\}\to \mathbb{Z}$		With conditions [Va14, CST-inv]
ts-perversity	$\left\{\text{sing. strata}\right\}\to \mathbb{Z}\times 2^{P\left(R\right)}$		See Section 3

Equations

$$I^{\bar{p}}{H}_{\ast }(X)\cong I^{\bar{p}}{H}_{\ast }(Y).$$

$$\{\text{PL pseudomanifolds}\}\subset \{\text{topological pseudomanifolds}\}\subset \{\text{CS sets}\}.$$

$$X=X^n\supset X^{n-1}\supset \cdots X^0\supset X^{-1}=\varnothing$$

$$T^{\wp }A=\left\{x\in A\mid nx=0\text{ for some product }n=\prod _{i=1}^s{p}_i^{m_i}\text{ such that pi∈℘ and mi,s∈Z≥0}\right\},$$

$${\mathcal{P}}_{X,\vec{p},\mathcal{E}}^{\ast }={\mathfrak{t}}_{\le \vec{p}}^{X_0}R{i}_{n\ast }\dots {\mathfrak{t}}_{\le \vec{p}}^{X_{n-1}}R{i}_{1\ast }\mathcal{E}.$$

$$H^i\left({\left({\mathfrak{t}}_{\le \vec{p}}^{X_{n-k}}{\mathcal{S}}^{\ast }\right)}_x\right)\cong \{\begin{array}{ll}0,& i>{\vec{p}}_1(S)+1,\\ T^{{\vec{p}}_2(S)}{H}^i({\mathcal{S}}_x^{\ast }),& i={\vec{p}}_1(S)+1,\\ H^i({\mathcal{S}}_x^{\ast }),& i\le {\vec{p}}_1(S).\end{array}$$

$$H^i({\mathcal{P}}_x^{\ast })\cong \{\begin{array}{ll}0,& i>{\vec{p}}_1(S)+1,\\ T^{{\vec{p}}_2(S)}{\mathbb{H}}^i(L;{\mathcal{P}}^{\ast }{|}_L),& i={\vec{p}}_1(S)+1,\\ {\mathbb{H}}^i(L;{\mathcal{P}}^{\ast }{|}_L),& i\le {\vec{p}}_1(S).\end{array}$$

$${\mathbb{P}}^{\ast }\cong {\mathcal{R}}_{{\mathbb{R}}^{n-\ell }\times cL}\stackrel{L}{\otimes }{\mathbb{P}}^{\ast }\cong {\pi }_1^{\ast }{\mathcal{R}}_{{\mathbb{R}}^{n-\ell }}\stackrel{L}{\otimes }{\pi }_2^{\ast }R{\pi }_{2\ast }{\mathbb{P}}^{\ast }.$$

$$f_x^{!}{\mathbb{P}}^{\ast }\cong f_y^{!}{\mathcal{R}}_{{\mathbb{R}}^{n-\ell }}\stackrel{L}{\otimes }{f}_v^{!}R{\pi }_{2\ast }{\mathbb{P}}^{\ast }.$$

$$f_x^{!}{\mathbb{P}}^{\ast }\cong f_v^{!}R{\pi }_{2\ast }{\mathbb{P}}^{\ast }[-(n-\ell )].$$

$$\to H^i\left(f_v^{!}R{\pi }_{2\ast }{\mathbb{P}}^{\ast }\right)\to {\mathbb{H}}^i\left(cL;R{\pi }_{2\ast }{\mathbb{P}}^{\ast }\right)\stackrel{\alpha }{\to }{\mathbb{H}}^i\left(cL-\{v\};{\bar{\mathfrak{i}}}^{\ast }R{\pi }_{2\ast }{\mathbb{P}}^{\ast }\right)\to ,$$

$$H^i({\mathcal{P}}_x^{\ast })\cong \{\begin{array}{ll}0,& i>{\vec{p}}_1(S)+1,\\ T^{{\vec{p}}_2(S)}{\mathbb{H}}^i(L;{\mathcal{P}}^{\ast }{|}_L),& i={\vec{p}}_1(S)+1,\\ {\mathbb{H}}^i(L;{\mathcal{P}}^{\ast }{|}_L),& i\le {\vec{p}}_1(S).\end{array}$$

$$H^i({\mathcal{P}}_x^{\ast })\cong \{\begin{array}{ll}0,& i>{\vec{p}}_1(S)+1,\\ T^{{\vec{p}}_2(S)}{H}^{{\vec{p}}_1(S)+1}((R{i}_{\ast }{\mathcal{P}}_W^{\ast }{)}_x),& i={\vec{p}}_1(S)+1,\\ H^i((R{i}_{\ast }{\mathcal{P}}_W^{\ast }{)}_x),& i\le {\vec{p}}_1(S).\end{array}$$

$${\mathbb{H}}^i(X;R{i}_{\ast }{\mathcal{P}}_W^{\ast })\cong {\mathbb{H}}^i(W;{\mathcal{P}}_W^{\ast })\cong {\mathbb{H}}^i({\mathbb{R}}^{k+1}\times L;{\mathcal{P}}_W^{\ast })\cong {\mathbb{H}}^i(L;{\mathcal{P}}^{\ast }{|}_L),$$

$$H^i(f_x^{!}{\mathcal{P}}^{\ast })\cong H^{i-m}(g_x^{!}{\mathcal{P}}^{\ast }{|}_Y).$$

$${\mathbb{H}}^i(\Sigma X;{\mathcal{P}}^{\ast })\cong \{\begin{array}{ll}{\mathbb{H}}^{i-1}(X;{\mathcal{P}}^{\ast }),& i\ge p+3,\\ {\mathbb{H}}^{p+1}(X;{\mathcal{P}}^{\ast })/T^{\wp }{\mathbb{H}}^{p+1}(X;{\mathcal{P}}^{\ast }),& i=p+2,\\ T^{\wp }{\mathbb{H}}^{p+1}(X;{\mathcal{P}}^{\ast }),& i=p+1,\\ {\mathbb{H}}^i(X;{\mathcal{P}}^{\ast }),& i\le p.\end{array}$$

$$\to {\mathbb{H}}^i(\Sigma X;{\mathcal{P}}^{\ast })\to {\mathbb{H}}^i(U_1;{\mathcal{P}}^{\ast })\oplus {\mathbb{H}}^i(U_2;{\mathcal{P}}^{\ast })\to {\mathbb{H}}^i(U_1\cap U_2;{\mathcal{P}}^{\ast })\to .$$

$$H_i(\Sigma \mathbb{R}{P}^3)=\{\begin{array}{ll}\mathbb{Z},& i=4,\\ 0,& i=3,\\ {\mathbb{Z}}_2,& i=2,\\ 0,& i=1,\\ \mathbb{Z},& i=0,\end{array}{H}^i(\Sigma \mathbb{R}{P}^3)=\{\begin{array}{ll}\mathbb{Z},& i=4,\\ {\mathbb{Z}}_2,& i=3,\\ 0,& i=2,\\ 0,& i=1,\\ \mathbb{Z},& i=0.\end{array}$$

$$F{\mathbb{H}}^i(\Sigma \mathbb{R}{P}^3;{\mathcal{P}}_{\vec{p}})\otimes F{\mathbb{H}}^{4-i}(\Sigma \mathbb{R}{P}^3;{\mathcal{P}}_{D\vec{p}})$$

$$T{\mathbb{H}}^i(\Sigma \mathbb{R}{P}^3;{\mathcal{P}}_{\vec{p}})\otimes T{\mathbb{H}}^{4-i+1}(\Sigma \mathbb{R}{P}^3;{\mathcal{P}}_{D\vec{p}})$$

$${\mathbb{H}}^i(\Sigma \mathbb{R}{P}^3,{\mathcal{P}}_{\vec{p}}^{\ast })$$

$${\mathbb{H}}^i(\Sigma \mathbb{R}{P}^3,{\mathcal{P}}_{D\vec{p}}^{\ast }))$$

$${\mathbb{H}}^i(S^k\ast X;{\mathcal{P}}^{\ast })\cong \{\begin{array}{ll}{\mathbb{H}}^{i-k-1}(X;{\mathcal{P}}^{\ast }),& i>{\vec{p}}_1(S^k)+k+2,\\ {\mathbb{H}}^{{\vec{p}}_1(S^k)+1}(X;{\mathcal{P}}^{\ast })/T^{{\vec{p}}_2(S^k)}{\mathbb{H}}^{{\vec{p}}_1(S^k)+1}(X;{\mathcal{P}}^{\ast }),& i={\vec{p}}_1(S^k)+k+2,\\ 0,& {\vec{p}}_1(S^k)+1<i<{\vec{p}}_1(S^k)+k+2,\\ T^{{\vec{p}}_2(S^k)}{\mathbb{H}}^{{\vec{p}}_1(S^k)+1}(X;{\mathcal{P}}^{\ast }),& i={\vec{p}}_1(S^k)+1,\\ {\mathbb{H}}^i(X;{\mathcal{P}}^{\ast }),& i\le {\vec{p}}_1(S^k).\end{array}$$

$$H^i({\mathcal{P}}_x^{\ast })\cong \{\begin{array}{ll}0,& i>{\vec{p}}_1(S)+1,\\ T^{{\vec{p}}_2(S)}{\mathbb{H}}^i(L^{k-1};\mathcal{E}),& i={\vec{p}}_1(S)+1,\\ {\mathbb{H}}^i(L^{k-1};\mathcal{E}),& i\le {\vec{p}}_1(S).\end{array}$$

$${\mathbb{H}}^i(L^{k-1};\mathcal{E})\cong \{\begin{array}{ll}{H}^1({\mathcal{E}}_x),& i=k,\\ H^0({\mathcal{E}}_x),& i=k-1,\\ H^1({\mathcal{E}}_x),& i=1,\\ H^0({\mathcal{E}}_x),& i=0,\\ 0,& \text{otherwise.}\end{array}$$

$$0\to H^0({\mathcal{E}}_x)\to {\mathbb{H}}^1(L^1;\mathcal{E})\stackrel{q}{\to }{H}^1({\mathcal{E}}_x)\to 0.$$

Full text

1. 1 Introduction

2. 2 Definitions and background

3. 2.1 Spaces

4. 2.2 Algebra

5. 2.3 ts-Deligne sheaves

6. 2.4 Maximal ts-coefficient systems

7. 3 Topological invariance

8. 3.1 Pullback and pushforward perversities

9. 3.1.1 Pullback perversities

10. 3.1.2 Pushforward perversities

11. 3.2 Constrained perversities

12. 4 Necessity of the conditions

13. 4.1 Computational tools

14. 4.2 Singular strata in regular strata

15. 4.3 Singular strata in singular strata

2000 Mathematics Subject Classification: Primary: 55N33, 55N30, 57N80, 55M05

Keywords: intersection homology, intersection cohomology, Deligne sheaf, CS set, perversity, stratification

1 Introduction

In [GBF32] we introduced categories of torsion-sensitive perverse sheaves (more briefly ts-perverse sheaves) and studied their duality properties. In the classical category of perverse sheaves on a stratified pseudomanifold [BBD], the intermediate extensions of the coefficient systems are the ``Deligne sheaves'' whose hypercohomology groups are the intersection homology groups of Goresky and MacPherson. The primary motivation in [GBF32] was to create a generalization of these Deligne sheaves for which the various intersection homology duality theorems of Goresky-MacPherson [GM1, GM2], Goresky-Siegel [GS83], and Cappell-Shaneson [CS91] all arise as special cases of a single more general duality theorem that incorporates certain torsion phenomena into the sheaf complexes but does not require the special local cohomological conditions on spaces that are needed for some of the original theorems. Indeed, the ts-Deligne sheaves of [GBF32], which are the intermediate extensions of ts-coefficient systems, fulfill that goal, and furthermore they can be characterized by a simple set of axioms generalizing the Deligne sheaf axioms of Goresky and MacPherson.

After providing a generalization of Poincaré duality for singular spaces, the next most important property of intersection homology is its topological invariance: while the intersection homology groups are defined in terms of a stratification of the space, the resulting intersection homology groups are independent of the choice of stratification, at least assuming certain restrictions on the perversity parameters. In this paper we consider the topological invariance of the ts-Deligne sheaves up to quasi-isomorphism, including confirming a conjecture made in [GBF32]. In addition to extending versions of past topological invariance results to the torsion sensitive category, our techniques specialize to improve the previous known results about ordinary intersection homology.

In particular, our main theorem will be the following:

Theorem** (Theorem 3.6).**

Suppose that $X$ and $\mathbb{X}$ denote two CS set stratifications of the same underlying space with $\mathbb{X}$ coarsening $X$. Let $\mathcal{E}$ be a ts-coefficient system such that $\mathbb{X}$ is adapted to $\mathcal{E}$ (and hence so is $X$). Let $\vec{p}$ and $\vec{\mathbbm{p}}$ be respective ts-perversities on $X$ and $\mathbb{X}$ that are $\mathcal{E}$-compatible, and let $\mathcal{P}^{*}$ and $\mathbb{P}^{*}$ be the respective ts-Deligne sheaves with coefficients $\mathcal{E}$. Then $\mathcal{P}^{*}$ is quasi-isomorphic to $\mathbb{P}^{*}$.

Here CS sets are a class of stratified space generalizing pseudomanifolds, and so it includes irreducible algebraic and analytic varieties, and the $\mathcal{E}$-compatibility condition imposes both growth rate conditions on the perversities in relation to each other as well as conditions on how they interact with the coefficient system $\mathcal{E}$. While it will take some time below to explain all of the definitions in detail, we note as a corollary the special case in which our ts-perversity $\vec{p}$ is simply one of the original perversity parameters $\bar{p}$ of Goresky and MacPherson. In this case, our result implies, by a different route, the original topological invariance result for intersection homology; see [GBF44] for a detailed treatment of just this special case.

Corollary** (Goresky-MacPherson [GM2]).**

Let $X$ be an $n$-dimensional topological stratified pseudomanifold, e.g. a Whitney stratified irreducible complex variety, and let $Y$ be the same space with a different stratification as a stratified pseudomanifold. Let $\bar{p}$ be a perversity as defined by Goresky and MacPherson, i.e. a function $\bar{p}:\mathbb{Z}_{\geq 2}\to\mathbb{Z}$ such that $\bar{p}(2)=0$ and $\bar{p}(k)\leq\bar{p}(k+1)\leq\bar{p}(k)+1$. Then letting $I^{\bar{p}}H_{*}$ denote the Goresky-MacPherson intersection homology groups,

[TABLE]

To explain further, we briefly outline some of the history.

History.

The original intersection homology groups of Goresky and MacPherson [GM2] are defined on stratified pseudomanifolds and depend on perversity parameters $\bar{p}:\mathbb{Z}_{\geq 2}\to\mathbb{Z}$ satisfying the original Goresky-MacPherson conditions: $\bar{p}(2)=0$ and $\bar{p}(k)\leq\bar{p}(k+1)\leq\bar{p}(k)+1$. If $X$ is an $n$-dimensional stratified pseudomanifold, so in particular a filtered space $X=X^{n}\supset X^{n-2}\supset\cdots\supset X^{0}$ with each $X^{m}-X^{m-1}$ an $m$-manifold (possibly empty), then the Deligne sheaf is constructed beginning with a local system $E$ on $X-X^{n-2}$ and then performing a sequence of pushforwards and truncations over strata of increasing codimension. The perversity value $\bar{p}(k)$ determines the truncation degree following the pushforward to the codimension $k$ strata. In [GM2], Goresky and MacPherson showed that for a fixed perversity and local system the resulting sheaves are independent (up to quasi-isomorphism) of the precise choice of pseudomanifold stratification; a more detailed exposition was provided by Borel in [Bo, Section V.4].

King [Ki] later gave a proof of the topological invariance of intersection homology without using sheaves and requiring only that $\bar{p}$ be nonnegative as well as the growth condition. Furthermore, King worked in the broader category of CS sets and allowed strata of codimension one. However, it should be noted that when $\bar{p}$ has values such that $\bar{p}(k)>k-2$ the singular chain intersection homology of [Ki] is not quite the same thing as the sheaf-theoretic intersection homology of [GM2]; see [GBF26] for a discussion. In our book on singular chain intersection homology [GBF35], we call King's singular chain intersection homology GM intersection homology,'' while that arising from the Deligne sheaf hypercohomology is called non-GM intersection homology.'' If $\bar{p}(k)\leq k-2$ for all $k$, as is the case in the original work of Goresky and MacPherson [GM2], then these theories all agree. A sheaf theoretic approach to GM intersection homology and its topological invariance can be found in Habegger and Saper [HS91], while a singular chain approach to non-GM intersection homology has been developed in [GBF10, Sa05, GBF35]. Topological invariance of non-GM intersection homology is considered in [GBF11], where it is shown that topological invariance holds with $\bar{p}(1)>0$ and $\bar{p}(k)\leq\bar{p}(k+1)\leq\bar{p}(k)+1$ so long as all changes to the stratification occur within a fixed choice of $n-1$ skeleton $X^{n-1}$, but not in general otherwise.

It has since become apparent that it is useful to utilize perversities that depend not just on codimension but on the strata themselves so that we define $\bar{p}:\{\text{singular strata}\}\to\mathbb{Z}$ (if $X$ is an $n$-dimensional CS set, the $n$-dimensional strata are called regular and the lower dimensional strata are singular). A version of Deligne sheaves suited to such general perversities is defined in [GBF23], and the corresponding singular chain non-GM intersection homology is studied in this generality in [GBF35]. Clearly in this generality topological invariance becomes a more subtle issue. Nonetheless, there are such results, typically comparing just two stratifications of the same space, $X$ and $\mathbb{X}$, with $X$ refining $\mathbb{X}$ (or, equivalently, $\mathbb{X}$ coarsening $X$). In [Va14], Valette works with piecewise linear intersection homology on piecewise linear pseudomanifolds and arbitrary perversities $\bar{p}:\{\text{singular strata}\}\to\mathbb{N}$ satisfying $\bar{p}(S)\leq\text{codim}(S)-2$ for each singular stratum $S$. He shows, in our notation, that if $X$ refines $\mathbb{X}$ and if their respective perversities $\bar{p}$ and $\bar{\mathbbm{p}}$ satisfy $\bar{\mathbbm{p}}(\mathbb{S})\leq\bar{p}(S)\leq\bar{\mathbbm{p}}(\mathbb{S})+\text{codim}(S)-\text{codim}(\mathbb{S})$ whenever $S$ is a singular stratum of $X$ contained in the singular stratum $\mathbb{S}$ of $\mathbb{X}$ then the intersection homology groups agree, i.e. $I^{\bar{p}}H_{*}(X)\cong I^{\bar{\mathbbm{p}}}H_{*}(\mathbb{X})$. Note that with Valette's assumptions the GM and non-GM intersection homologies automatically agree.

More recently in [CST-inv], Chataur, Saralegi-Aranguren, and Tanré consider what they call $K^{*}$-perversities and show that a $K^{*}$-perversity on a CS set $X$ can be pushed forward to a perversity on the intrinsic coarsest stratification $X^{*}$ and that the two resulting intersection homology groups are isomorphic. This theorem holds for non-GM intersection homology (which is called ``tame intersection homology'' in [CST-inv]), and there is also a version for GM-intersection homology with fewer conditions on the perversities. They also show that it is similarly possibly to pull a $K^{*}$-perversity back to any refinement of $X$ and obtain isomorphic intersection homology groups. Our results below include the non-GM (tame) intersection homology 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 work throughout from the sheaf-theoretic point of view, which has the benefit of easily allowing for twisted coefficient systems and also in that quasi-isomorphism of sheaves implies isomorphism of the hypercohomology groups with any system of supports. Thus, in particular, our sheaf quasi-isomorphisms imply isomorphisms of intersection homology groups both with compact supports and with closed supports, the latter corresponding to intersection homology of locally-finite singular chains [GBF10, GBF23].

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. In particular, a ts-perversity is a function $\vec{p}:\{\text{singular strata}\}\to\mathbb{Z}\times 2^{P(R)}$, where $P(R)$ is the set of primes (up to unit) of our ground PID $R$ and $2^{P(R)}$ is its power set. We write $\vec{p}(S)=(\vec{p}_{1}(S),\vec{p}_{2}(S))$. Using this additional information about primes, torsion data is incorporated into the definition of the ts-Deligne sheaf utilizing the ``torsion-tipped truncation'' functor constructed in [GBF32] in place of the standard truncation. If $\vec{p}_{2}(S)=\emptyset$ for all $S$ and the ts-coefficient system is just a local system in degree [math] then the ts-Deligne sheaf reduces to the classical Deligne sheaf [GM2, GBF23]. Subsection 2.4 discusses some further natural assumptions about coefficient systems that will be utilized in our broadest topological invariance results.

In Section 3 we define what we call $\mathcal{E}$-compatibility between ts-perversities $\vec{\mathbbm{p}}$ and $\vec{p}$ on a CS set $\mathbb{X}$ and its refinement $X$. Here $\mathcal{E}$ is a ts-coefficient system common to $\mathbb{X}$ and $X$. This compatibility depends on $\mathcal{E}$ only over the regular strata of $\mathbb{X}$ and is necessary to get the compatibility started. From there compatibility is essentially a Goresky-MacPherson-type growth condition, but involving also the torsion information from $\vec{p}_{2}$ and $\vec{\mathbbm{p}}_{2}$. This compatibility condition generalizes that of Valette [Va14], which itself stems from the Goresky-MacPherson growth condition, by incorporating the torsion information and also allowing $\vec{p}_{1}(S)>\text{codim}(S)-2$. The central result of the paper is Theorem 3.6, which shows that the ts-Deligne sheaves from $\mathcal{E}$-compatible ts-perversities are quasi-isomorphic. In Sections 3.1.1 and 3.1.2 we apply Theorem 3.6 to pullback and pushforward perversities, recovering generalizations of the theorems of Chataur-Saralegi-Tanré [CST-inv]. In particular, we discuss pushforwards to arbitrary coarsenings, not just the intrinsic coarsening.

In Section 3.2 we consider quasi-isomorphisms of ts-Deligne sheaves arising from two CS set stratifications of a space without the assumption that one refines the other. This requires restricting ourselves to ts-perversities that depend only on codimension, i.e. functions $\vec{p}:\mathbb{Z}_{\geq 1}\to\mathbb{Z}\times 2^{P(R)}$, such that $\vec{p}_{1}$ satisfies the Goresky-MacPherson growth condition and $\vec{p}_{2}$ 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 $\vec{p}_{1}(2)$. For constrained ts-perversities that are also appropriately compatible with the ts-coefficient systems $\mathcal{E}$ (by a condition relating torsion information about $\mathcal{E}$ with $\vec{p}_{2}(2)$), we show in Theorem 3.23 that any two stratifications yield quasi-isomorphic ts-Deligne sheaves so long as the closures of their codimension one strata agree. In particular, this theorem holds if one makes the classical assumption that codimension one strata are forbidden. We also show in the same theorem that we can weaken the hypotheses to weakly constrained perversities and no compatibility requirement between $\vec{p}$ and $\mathcal{E}$ so long as the the two stratifications have the same regular strata (or, equivalently, the same codimension one skeleta). These two results generalize the classical Goresky-MacPherson topological invariance in [GM2] and that for ``superperversities'' in [GBF11]. The key idea is to apply Theorem 3.6 using appropriate common coarsenings of the two stratifications. Such intrinsic stratifications, relative to coefficient systems and fixed subspaces, are constructed in Section LABEL:S:_intrinsic, generalizing those in [Ki] and [HS91].

Section 4 concerns the extent to which the conditions for $\mathcal{E}$-compatibility between ts-perversities are necessary in order to obtain quasi-isomorphic ts-Deligne sheaves. We show that the conditions on singular strata of $X$ contained in regular strata of the coarsening $\mathbb{X}$ are strictly necessary: if they fail for any stratum the sheaves cannot be quasi-isomorphic. By contrast, the conditions on singular strata of $X$ contained in singular strata of the coarsening $\mathbb{X}$ are only ``necessary in general,'' meaning that we can construct examples in which failure of the conditions implies failure of quasi-isomorphism. However, these conditions may not be necessary in special cases, for example if certain stalk cohomology groups vanish due to the specific topology of some space; see Section 4.3 for further details. One of our main tools in this section will be a formula for computing the ts-Deligne sheaf hypercohomology for a join $S^{k}*X$ in terms of the ts-Deligne sheaf hypercohomology of $X$; see Corollary 4.6. This formula is obtained by first computing the hypercohomology for the suspension $\Sigma X$ in Proposition 4.4, which is illuminating in its own right, and then making a nice application of Theorem 3.6 to the iterated suspension.

The original Goresky-MacPherson proof of topological invariance involved support and cosupport'' axioms concerning the dimensions on which $H^{i}(f^{*}_{x}\mathcal{P}^{*})$ and $H^{i}(f^{!}_{x}\mathcal{P}^{*})$ may fail to vanish, $\mathcal{P}^{*}$ being the Deligne sheaf and $f_{x}$ the inclusion of the point $x$ into $X$. Our arguments to this point do not involve these axioms and so are fundamentally different from those in [GM2]. In Section LABEL:S:_dimension we develop versions of these support and cosupport axioms for ts-Deligne sheaves with strongly or weakly constrained ts-perversities. Strongly constrained ts-perversities require $\vec{p}_{1}(2)=0$ while simply constrained'' is a bit weaker; the stronger constraint in this section is not strictly necessary but simplifies the discussion. In the strongly constrained case we provide criteria to recognize ts-Deligne sheaves without reference to any specific stratification, leading to Theorem LABEL:T:_constrained2, a statement of topological invariance more analogous to the original Goresky-MacPherson invariance theorem of [GM2, Uniqueness Theorem] or [Bo, Theorem 4.15]. The weakly constrained version, Theorem LABEL:T:_constrainedS, again requires a fixed choice of the regular strata and is more analogous to the main theorem of [GBF11].

Lastly, Section LABEL:S:_intrinsic concerns the details about relative intrinsic stratifications.

Remarks.

When $\vec{p}_{2}(S)=\emptyset$ for all $S$ and $\mathcal{E}$ is a local system concentrated in degree [math], our ts-Deligne sheaves reduce to the Deligne sheaves of [GM2, GBF23]. With this assumption, many, though not all, of our results reduce to some previously-known theorems, as outlined above. However, we believe that even in these cases our proofs are quite different, as our main invariance results in Section 3 do not require analogues of the Goresky-MacPherson support and cosupport axioms. For the reader interested only in the classical Deligne sheaves and Goresky-MacPherson perversities, we have extracted a simplified version of this new argument and presented it in [GBF44] together with a very short second proof of the topological invariance of classical intersection homology that does use support and cosupport axioms.

More generally, some of the assumptions below simplify whenever $\mathcal{E}$ is just a globally defined local system of free modules, and we attempt to provide some flags in the exposition to help the reader primarily interested in that case.

We thank Jörg Schürmann for pointing out some very helpful references and Scott Nollet for many useful conversations. We also thank David Chataur, Martin Saralegi-Aranguren, and Daniel Tanré for both ongoing stimulating mathematical discussion and their generous hospitality. Finally, we thank the anonymous referee for suggesting several improvements to the exposition.

2 Definitions and background

2.1 Spaces

Our spaces will be paracompact dimensionally homogeneous CS sets, whose precise definition we recall below. CS sets include topological and piecewise linear pseudomanifolds. In fact, we have the hierarchy

[TABLE]

The primary difference between CS sets and topological stratified pseudomanifolds, given our additional dimensional homogeneity condition, is that the links of points of pseudomanifolds must themselves be stratified pseudomanifolds, while the links of points in CS sets need only be compact filtered spaces. PL pseudomanifolds are defined just as topological pseudomanifolds are with the added condition that all spaces and maps describing local conditions must be piecewise linear. Classical PL pseudomanifolds, which are those simplicial complexes consisting exclusively of $n$-simplices such that each $n-1$ face of each $n$-simplex is glued to exactly one $n-1$ face of another $n$-simplex, are a special case. All irreducible complex algebraic and analytic varieties can be given stratifications that realize them as PL pseudomanifolds; this is also true of real varieties that possess a dense manifold subset. Connected orbit spaces of manifolds under smooth actions of compact Lie groups are also PL pseudomanifolds. For more details and an overview of all these spaces, see [GBF35, Chapter 2].

We choose to work with CS sets both for their added generality but also because one of our key tools will be intrinsic stratification and the intrinsic stratification of a CS set is also a CS set. In [GM2], Goresky and MacPherson construct ``canonical $\bar{p}$-filtrations'' for topological pseudomanifolds, but, in addition to depending on the choice of perversity, the resulting filtrations do not necessarily give the space the structure of a stratified pseudomanifold. So, while Goresky-MacPherson [GM2] and Borel [Bo] treat topological pseudomanifolds as the primary objects, we follow King [Ki] by working in the even more general but self-contained class of CS sets. The class of PL pseudomanifolds is also preserved on passing to intrinsic stratifications (see [GBF35, Corollary 2.10.19.]), but we note that topological pseudomanifolds include such important non-PL examples as suspensions of topological manifolds and topological manifolds stratified by the inclusion of locally-flat, but not PL, embedded submanifolds. This includes, for example, locally flat topological knots in high dimensions. Moreover, as our fundamental language is sheaf theory, the added generality of CS sets does not much increase the difficulty of our results, and in some cases this choice provides simplifications. For example, we don't at any point need to take care about PL structures.

We now recall the definition of CS sets due to Siebenmann [Si72]. An $n$-dimensional CS set $X$ is a Hausdorff space equipped with a filtration

[TABLE]

such that $X_{k}:=X^{k}-X^{k-1}$ is a $k$-manifold (possibly empty) and for $x\in X_{k}$ there is an open neighborhood $U$ 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 $h:U\times cL\to N$ such that $h(U\times c(L^{j}))=X^{k+j+1}\cap N$ for all $j$. The space $L$ is called a link of $x$, and $N$ is called a distinguished neighborhood of $x$. Note that if $L=\emptyset$ then $cL=(cL)^{0}$ is a point. Dimensional homogeneity means that we assume $X-X^{n-1}$ is dense. Such spaces are locally compact [GBF35, Lemma 2.3.15], metrizable [CST-inv, Proposition 1.11], and of finite cohomological dimension ([GBF35, Lemma 6.3.46] and [Br, Theorem II.16.8]). See [GBF35, Section 2.3] for more details about CS sets in general. All CS sets in this paper will be assumed paracompact and dimensionally homogeneous without further mention. We also assume $X$ is $n$-dimensional unless specified otherwise.

Following Borel [Bo, Section V.2], we let $U_{k}=X-X^{n-k}$, and noting that $U_{k+1}$ is the disjoint union of $U_{k}$ and $X_{n-k}$, we also take $i_{k}:U_{k}\hookrightarrow U_{k+1}$ and $j_{k}:X_{n-k}\hookrightarrow U_{k+1}$. For any $x\in X$, we write $f_{x}:\{x\}\hookrightarrow X$. The connected components of $X_{k}$ are the $k$-dimensional strata. Strata in $X_{n}=X^{n}-X^{n-1}$ are regular strata and strata in $X_{k}$ for $k\leq n-1$ are singular strata. Note that strata may have codimension one, which is sometimes forbidden in other contexts.

We often abuse notation and use $X$ to refer both to the underlying space and to the space equipped with the stratification; when we wish to emphasize the underlying space or do not yet want to specify the stratification we also write $|X|$. If $X$ and $\mathbb{X}$ are two stratifications of the same space $|X|$, we say that $\mathbb{X}$ coarsens $X$, or that $X$ is a refinement of $\mathbb{X}$, if each stratum of $\mathbb{X}$ is a union of strata of $X$. Our standard notation will be $X$ for a CS set and $\mathbb{X}$ for a coarsening of $X$. We will use $\mathfrak{X}$ for the intrinsic stratifications constructed in Section LABEL:S:_intrinsic. If we wish to speak of the same space with two a priori unrelated stratifications, we write the stratifications $\mathcal{X}$ and $\mathcal{Y}$; if we construct a common coarsening of $\mathcal{X}$ and $\mathcal{Y}$, we will sometimes call that $\mathcal{Z}$.

2.2 Algebra

Algebraically, we fix a PID $R$ as our ground ring throughout, and we let $P(R)$ be the set of primes of $R$ up to unit. This means that the elements of $P(R)$ are technically equivalence classes such that $p\sim q$ if $p=uq$ for some unit $u$, though we will abuse notation by letting a prime stand for its equivalence class; cf. [GBF32, Section 2]. The following is Definition 2.1 of [GBF32].

Definition 2.1**.**

If $A$ is a finitely-generated $R$-module and $\wp\subset P(R)$, we define the $\wp$-torsion submodule of $A$ to be

[TABLE]

i.e. $T^{\wp}A$ is the submodule annihilated by products of powers of primes in $\wp$. If $T^{\wp}A=A$, we say that $A$ is $\wp$-torsion. If $T^{\wp}A=0$, we say that $A$ is $\wp$-torsion free. We take the empty product to be $1$, so in particular if $\wp=\emptyset$ then $T^{\wp}A=0$ and every $A$ is $\emptyset$-torsion free. If $\mathfrak{p}\in P(R)$ is a single element, we abuse notation and write $T^{\mathfrak{p}}A$ instead of $T^{\{\mathfrak{p}\}}A$.

2.3 ts-Deligne sheaves

We now recall some material from [GBF32], leading to the definition of ts-Deligne sheaves. All sheaves are sheaves of $R$-modules, and we think of ourselves as working in the derived category so that $\cong$ denotes quasi-isomorphism. If $\mathcal{S}^{*}$ is a sheaf complex, then $\mathcal{H}^{i}(\mathcal{S}^{*})$ denotes the derived cohomology sheaf and $\mathbb{H}^{i}(X;\mathcal{S}^{*})$ denotes hypercohomology.

We begin with ts-perversities [GBF32, Definitions 4.1 and 4.18]:

Definition 2.2**.**

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 $X$ is a function \vec{p}:\{\text{singular strata of X}\}\to\mathbb{Z}\times 2^{P(R)}. We denote the components of $\vec{p}(S)$ by $\vec{p}(S)=(\vec{p}_{1}(S),\vec{p}_{2}(S))$.

The complementary ts-perversity, or dual ts-perversity, $D\vec{p}$ is defined by $D\vec{p}(S)=(\text{codim}(S)-2-\vec{p}_{1}(S),P(R)-\vec{p}_{2}(S))$, i.e. the first component is the complementary perversity to $\bar{p}$ in the Goresky-MacPherson sense and the second component is the set of primes in $R$ complementary to $\vec{p}_{2}(S)$.

We also recall the notion of a $\wp$-coefficient system, slightly generalizing [GBF32, Definition 4.2]. On a pseudomanifold, these are the objects in the heart of the natural t-structures ${}^{\wp}D^{\heartsuit}$ constructed in [GBF32, Definition 5.1]. We give an explicit description here.

Definition 2.3**.**

Let $\wp\subset P(R)$ be a set of primes of the PID $R$. We will call a complex of sheaves111Even though $\mathcal{E}$ is a complex of sheaves, we do not write $\mathcal{E}^{*}$ in order to emphasize the role of $\mathcal{E}$ as coefficients. $\mathcal{E}$ on a space $M$ a $\wp$-coefficient system if

$\mathcal{H}^{1}(\mathcal{E})$ is a locally constant sheaf of finitely generated $\wp$-torsion modules, 2. 2.

$\mathcal{H}^{0}(\mathcal{E})$ is a locally constant sheaf of finitely generated $\wp$-torsion free modules, and 3. 3.

$\mathcal{H}^{i}(\mathcal{E})=0$ for $i\neq 0,1$.

More generally, if $M$ is a disjoint union of spaces, we call $\mathcal{E}$ a ts-coefficient system if it restricts on each component of $M$ to a $\wp$-coefficient system for some $\wp$ (which may vary by component).

Suppose $X$ is a CS set and $\mathcal{E}$ is a ts-coefficient system defined over a subset $U\subset X$. We call $U$ the domain of $\mathcal{E}$, denoted by $\text{Dom}(\mathcal{E})$. We say that the stratification of $X$ is adapted to $\mathcal{E}$ if $X-X^{n-1}\subset\text{Dom}(\mathcal{E})$, i.e. if $\mathcal{E}$ is defined on (at least) the regular strata of $X$; cf. [Bo, Section V.4.12]. Of course this is automatically satisfied if $\mathcal{E}$ is defined on all of $X$, for example if $\mathcal{E}$ is a local system on $X$.

Remark 2.4*.*

As noted in [Bo, Remark V.4.14.c], even if we restrict our coefficients to local systems concentrated in a single degree and defined on dense open submanifolds of pseudomanifolds, there can exist $\mathcal{E}$ for which there does not exist a CS set stratification adapted to $\mathcal{E}$. For example, define $\mathcal{E}$ to be the local system on $\mathbb{R}^{2}-(0,0)\cup\{(1/n,0)|n\in\mathbb{Z}_{\geq 1}\}$ with stalk $\mathbb{Z}$ and nontrivial monodromy on a small loop around each $(1/n,0)$.

Given a ts-perversity $\vec{p}$ and a ts-coefficient system $\mathcal{E}$ to which $X$ is adapted, the associated ts-Deligne sheaf is defined for pseudomanifolds in [GBF32, Definition 4.4] and shown there for pseudomanifolds to be an intermediate extension of $\mathcal{E}$ with respect to a certain $t$-structure [GBF32, Proposition 5.12]. The construction holds as well for CS sets, and the ts-Deligne sheaf $\mathcal{P}^{*}_{X,\vec{p},\mathcal{E}}$, often written simply as $\mathcal{P}^{*}$, is defined as

[TABLE]

Here each $\mathfrak{t}_{\leq\vec{p}}^{X_{k}}$ is a locally torsion-tipped truncation functor as defined in [GBF32, Section 3]. We refer the reader there for more details but note that for $\mathcal{S}^{*}$ defined on $U_{k+1}$ we have

$\left(\mathfrak{t}_{\leq\vec{p}}^{X_{n-k}}\mathcal{S}^{*}\right)_{x}=\mathcal{S}^{*}_{x}$ if $x\in U_{k}$, 2. 2.

if $x\in S\subset S_{n-k}$ for a singular stratum $S$ then

[TABLE]

If $\mathcal{E}$ is a local system (i.e. a locally constant sheaf of finitely generated $R$-modules) concentrated in degree [math], if $\vec{p}_{1}$ satisfies the Goresky-MacPherson conditions, and if $\vec{p}_{2}(S)=\emptyset$ for all $S$, then this is just the classical Deligne sheaf of Goresky and MacPherson from [GM2].

As for the traditional Deligne sheaves, the critical feature of the ts-Deligne sheaves is the ``cone formula.'' As we will prove below in Lemma 4.2, if $x$ is a point of a CS set with a neighborhood of the form $\mathbb{R}^{k}\times cL$ and if $S$ is the stratum containing the cone points, then

[TABLE]

We use this below to compute the hypercohomology of a suspension in Proposition 4.4.

Analogously to the Goresky-MacPherson Deligne sheaves, the ts-Deligne sheaves can be characterized by axioms. Here is the first set of axioms from [GBF32, Definition 4.7], generalized for CS sets. We write $\mathscr{S}^{*}_{k}$ for $\mathscr{S}^{*}|_{U_{k}}$.

Definition 2.5**.**

Let $X$ be an $n$-dimensional CS set, and let $\mathcal{E}$ be a ts-coefficient system on $U_{1}$. We say that the sheaf complex $\mathscr{S}^{*}$ on $X$ satisfies the Axioms TAx1$(X,\vec{p},\mathcal{E})$ if

1. a.

$\mathscr{S}^{*}$ is quasi-isomorphic to a complex that is bounded and that is [math] in negative degrees; 2. b.

$\mathscr{S}^{*}|_{U_{1}}\cong\mathcal{E}|_{U_{1}}$; 3. c.

if $x\in S\subset X_{n-k}$, where $S$ is a singular stratum, then $H^{i}(\mathscr{S}_{x})=0$ for $i>\vec{p}_{1}(S)+1$ and $H^{\vec{p}_{1}(S)+1}(\mathscr{S}_{x})$ is $\vec{p}_{2}(S)$-torsion; 4. d.

if $x\in S\subset X_{n-k}$, where $S$ is a singular stratum, then the attachment map $\alpha_{k}:\mathscr{S}_{k+1}\to Ri_{k*}\mathscr{S}_{k}$ induces stalkwise cohomology isomorphisms at $x$ in degrees $\leq\vec{p}_{1}(S)$ and it induces stalkwise cohomology isomorphisms $H^{\vec{p}_{1}(S)+1}(\mathscr{S}_{k+1,x})\to T^{\vec{p}_{2}(S)}H^{\vec{p}_{1}(S)+1}((Ri_{k*}\mathscr{S}_{k})_{x})$.

Theorem 4.8 of [GBF32], which also works for CS sets, shows that the ts-Deligne sheaf complex $\mathcal{P}^{*}_{X,\vec{p},\mathcal{E}}$ satisfies the axioms TAx1$(X,\vec{p},\mathcal{E})$, and conversely any sheaf complex satisfying TAx1$(X,\vec{p},\mathcal{E})$ is quasi-isomorphic to $\mathcal{P}^{*}_{X,\vec{p},\mathcal{E}}$. It is also observed in [GBF32, Theorem 4.10] that these sheaf complexes are $X$-clc, meaning that each sheaf $\mathcal{H}^{i}(\mathcal{P}^{*})$ is locally constant on each stratum. This also continues to hold for CS sets, which have the property that if $j$ is any inclusion of a locally closed subset that is a union of strata then $j^{*}$, $j^{!}$, $j_{!}$, and $Rj_{*}$ all preserve constructibility by [Sch03, Proposition 4.0.2.3] (see also [Sch03, Proposition 4.2.1.2.b]).

As in [GM2, Bo, GBF32], we can reformulate some of these axioms.

Definition 2.6**.**

We say $\mathscr{S}^{*}$ satisfies the Axioms TAx1'$(X,\vec{p},\mathcal{E})$ if

1. a.

$\mathscr{S}^{*}$ is $X$-clc and it is quasi-isomorphic to a complex that is bounded and that is [math] in negative degrees; 2. b.

$\mathscr{S}^{*}|_{U_{1}}\cong\mathcal{E}|_{U_{1}}$; 3. c.

if $x\in S\subset X_{n-k}$, where $S$ is a singular stratum, then $H^{i}(\mathscr{S}^{*}_{x})=0$ for $i>\vec{p}_{1}(S)+1$ and $H^{\vec{p}_{1}(S)+1}(\mathscr{S}^{*}_{x})$ is $\vec{p}_{2}(S)$-torsion; 4. d.

if $x\in S\subset X_{n-k}$, where $S$ is a singular stratum, and $f_{x}:x\hookrightarrow X$ is the inclusion, then

1. (a)

$H^{i}(f_{x}^{!}\mathscr{S})=0$ for $i\leq\vec{p}_{1}(S)+n-k+1$ 2. (b)

$H^{\vec{p}_{1}(S)+n-k+2}(f_{x}^{!}\mathscr{S})$ is $\vec{p}_{2}(S)$-torsion free.

The following theorem is a slight generalization of [GBF32, Theorem 4.13]:

Theorem 2.7**.**

On a CS set, the axioms TAx1'$(X,\vec{p},\mathcal{E})$ are equivalent to the axioms TAx1$(X,\vec{p},\mathcal{E})$ and so any sheaf complex satisfying TAx1'$(X,\vec{p},\mathcal{E})$ is quasi-isomorphic to $\mathcal{P}^{*}_{X,\vec{p},\mathcal{E}}$.

The proof is the same as that of [GBF32, Theorem 4.13], replacing the theorems about constructibility invoked from Borel (e.g. [Bo, Lemma V.3.10.d]) with Schürmann's [Sch03, Proposition 4.0.2.3].

2.4 Maximal ts-coefficient systems

Many of our theorems below compare ts-Deligne sheaves on two stratifications of a single CS set, one coarsening another. For this it suffices to have a ts-coefficient system $\mathcal{E}$ defined on the regular strata of the coarser of the two stratifications for then it restricts also to a ts-coefficient system on the regular strata of the finer stratification. However, we will also be interested in theorems concerning arbitrary stratifications, and in these cases we will need to construct common coarsenings that remain adapted to $\mathcal{E}$. The full details will be provided in Section LABEL:S:_intrinsic, though we discuss here some notions about coefficient systems that will be necessary at that point as these will also be needed in some of our earlier theorem statements. In particular, to construct these common coarsenings we will need to make some minor assumptions about the domains of our ts-coefficient systems.

To motivate our restrictions, we recall that for classical intersection homology theory on a stratified pseudomanifold $X$ it is observed by Borel in [Bo, Section V.4] that if $E$ is a local system (i.e. a locally constant sheaf of finitely-generated $R$-modules) defined on a dense open submanifold of $X$ whose complement has codimension $\geq 2$ then there is a unique largest submanifold of $X$ to which $E$ can be extended uniquely222This is no longer true if the local system is only defined on a dense open set whose complement has codimension 1. For example let $X=S^{1}$ with stratification $S^{1}\supset\{pt\}$. Suppose $E$ is the constant sheaf with stalk $\mathbb{Z}$ on $S^{1}-\{pt\}$. Then there are two non-isomorphic extensions of $E$ to $S^{1}$, namely the constant sheaf with stalk $\mathbb{Z}$ and the twisted sheaf with stalks $\mathbb{Z}$ such that a generator of $\pi_{1}(S^{1})$ acts by multiplication by $-1$. up to isomorphism [Bo, Lemma V.4.11] (though this submanifold may not necessarily be the largest $n$-dimensional manifold contained in $X$ due to monodromy). Since it is not clear that such a statement holds for the more general ts-coefficient systems, we instead build a maximality assumption into our coefficients when necessary. Since local systems have unique such maximal extensions, we can convince ourselves that we therefore do not lose much generality. Alternatively, if we limit ourselves to $\mathcal{E}$ composed of local systems, then Proposition 2.9 below shows that maximality can be guaranteed.

Definition 2.8**.**

Let $X$ be an $n$-dimensional CS set. We will call a sheaf complex $\mathcal{E}$ on $X$ a maximal ts-coefficient system if

$\text{Dom}(\mathcal{E})$ includes an open $n$-dimensional submanifold $U_{\mathcal{E}}$ of $X$ whose complement has codimension $\geq 2$, 2. 2.

$\mathcal{E}$ is a ts-coefficient system over $U_{\mathcal{E}}$ (see Definition 2.3), and 3. 3.

there is no larger submanifold of $X$ to which $\mathcal{E}$ extends as a ts-coefficient system.

Clearly ts-coefficient systems composed of constant sheaves defined on all of $X$ are maximal. The following lemma shows that ts-coefficient systems composed of locally constant sheaves (on open submanifolds of codimension at least 2) can be made maximal.

Proposition 2.9**.**

Suppose $\mathcal{E}$ is a ts-coefficient system defined on an open dense submanifold whose complement has codimension at least 2. If $\mathcal{E}$ is bounded (i.e. $\mathcal{E}^{i}=0$ for sufficiently large $|i|$) and each $\mathcal{E}^{i}$ is a local system (a locally constant sheaf of finitely-generated $R$-modules), then $\mathcal{E}$ has a maximal extension that is unique up to isomorphism. Furthermore, if $X$ is adapted to $\mathcal{E}$ then $X$ remains adapted to the extension.

Proof.

By assumption, each $\mathcal{E}^{i}$ is defined on an open dense submanifold $U\subset X$ whose complement has codimension at least 2, and so by [Bo, Lemma V.4.11] each $\mathcal{E}^{i}$ has a unique (up to isomorphism) extension $\tilde{\mathcal{E}}^{i}$ to a maximal open subset $U_{i}$. Let $W=\cap_{i}U_{i}$, which remains open and dense since all but finitely many of the $U_{i}$ will be the largest open submanifold of $X$. Since $U\subset U_{i}$, we also have $U\subset W$, and we let $\bar{\mathcal{E}}^{i}=\tilde{\mathcal{E}}^{i}|_{W}$. Also by [Bo, Lemma V.4.11], each boundary map $\mathcal{E}^{i}\to\mathcal{E}^{i+1}$ extends uniquely to a map $\bar{\mathcal{E}^{i}}\to\bar{\mathcal{E}}^{i+1}$. This gives us a unique (up to isomorphisms) complex $\bar{\mathcal{E}}^{*}$ 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 adapted to $E$ with no codimension one strata, and $U_{E}$ is the maximal submanifold over which $E$ is defined, then $U_{E}$ is a union of strata of $X$. This is shown at the bottom of [Bo, page 92]. We will also need a property like this to define our common coarsenings, which motivates the following definition. Once again we will then show that this condition is automatic when $\mathcal{E}$ consists of local systems and there are no codimension one strata.

Definition 2.10**.**

Suppose $\mathcal{E}$ is a maximal ts-coefficient system on $X$ and that $U_{\mathcal{E}}$ is the largest open submanifold on which $\mathcal{E}$ is defined. We say that the stratification of $X$ is fully adapted to $\mathcal{E}$ if

$X-X^{n-1}\subset\mathcal{U}_{\mathcal{E}}$, and 2. 2.

$\mathcal{U}_{\mathcal{E}}$ is a union of strata of $X$.

Proposition 2.11**.**

Suppose $\mathcal{E}$ is a maximal ts-coefficient system on $X$ such that each $\mathcal{E}^{i}$ is a local system. If $X$ has no codimension one strata and is adapted to $\mathcal{E}$ then it is fully adapted.

Proof.

The proof is essentially the same as the argument on [Bo, page 92]: We will proceed by contradiction. Let $S$ be a stratum of $X$ of minimal codimension so that $S$ intersects $U_{\mathcal{E}}$ but is not contained in it. Since $X$ is adapted to $\mathcal{E}$ we must have $\text{codim}(S)\geq 1$. Suppose $x\in S$ has a distinguished neighborhood $N\cong B\times cL$ such that there is some point $y\in B\times\{v\}$ (with $v$ the cone vertex) such that $y\in U_{\mathcal{E}}$. We claim that then $x\in U_{\mathcal{E}}$.

First, since $x$ and $y$ are in the same stratum of $X$ and as $y$ must have a Euclidean neighborhood in $X$ (since $y\in U_{\mathcal{E}}$), [GBF35, Lemma 2.10.4] implies that $x$ also has a Euclidean neighborhood; thus both $x$ and $y$ (and similarly all points of $B\times\{v\}$) are contained in the maximal submanifold of $X$. Furthermore, by assumption $U_{\mathcal{E}}$ must contain $N-(N\cap S)=B\times(cL-\{v\})$, and if $\pi:B\times cL\to cL$ is the projection then $\pi^{*}(\mathcal{E}|_{\{y\}\times cL})$ is a local system on $N$ whose restriction to $N-S$ is isomorphic to $\mathcal{E}|_{N-S}$. Since extension of local systems is unique when there are no codimension one strata by [Bo, Lemma V.4.11], $\pi^{*}(\mathcal{E}|_{\{y\}\times cL})$ must agree with $\mathcal{E}$ where they overlap, and so we must have $N\subset U_{\mathcal{E}}$ or else the maximality of $U_{\mathcal{E}}$ would be contradicted.

Now, since $U_{\mathcal{E}}$ is open in $X$, we have $U_{\mathcal{E}}\cap S$ open in $S$. The above argument shows that if $x$ is in the closure of $U_{\mathcal{E}}\cap S$ in $S$ then $x\in U_{\mathcal{E}}\cap S$. So $U_{\mathcal{E}}\cap S$ is open and closed in the connected set $S$ and is thus all of $S$. ∎

The preceding proposition can fail if there are codimension one strata:

Example 2.12*.*

Let $E$ be the local system on $\mathbb{R}^{2}-\{0\}$ with $\mathbb{Z}$ stalks and nontrivial monodromy around the origin. Let $X=\mathbb{R}^{2}$ filtered as \mathbb{R}^{2}\supset\text{x-axis}. Then $E$ is maximal and $X$ is adapted to $E$, but it is not fully adapted, though it can be refined to be so.

As a more dramatic example, consider the example from Remark 2.4 of a maximal local system $E$ that is defined on the complement in $\mathbb{R}^{2}$ of $(0,0)\cup\{(0,1/n)|n\in\mathbb{Z}_{\geq 1}\}$. If we again filter $X=\mathbb{R}^{2}$ as \mathbb{R}^{2}\supset\text{x-axis} then again $X$ is adapted to $E$, but there is no fully adapted refinement.

3 Topological invariance

In this section we prove our main topological invariance theorems. These are mostly sufficiency statements, demonstrating that if certain conditions hold between different perversities on different stratifications of the same space, as well as certain relations between the perversity on the more refined stratification and the ts-coefficient system, then the two corresponding ts-Deligne sheaves are quasi-isomorphic. We consider necessity in Section 4.

The following definition establishes our main criteria for comparison of ts-Deligne sheaves. The first set of conditions is in a sense more important, as the second set is satisfied automatically for sufficiently simple coefficient systems – see Remark 3.2.

Definition 3.1**.**

Suppose that $X$ and $\mathbb{X}$ denote two CS set stratifications of the same underlying space with $\mathbb{X}$ coarsening $X$. Let $\vec{p}$ and $\vec{\mathbbm{p}}$ be respective ts-perversities on $X$ and $\mathbb{X}$, and let $\mathcal{E}$ be a ts-coefficient system to which $\mathbb{X}$ (and hence also $X$) is adapted. We will say that $\vec{p}$ and $\vec{\mathbbm{p}}$ are $\mathcal{E}$-compatible ts-perversities if the following conditions hold whenever a singular stratum $S$ of $X$ is contained in a (singular or regular) stratum $\mathbb{S}$ of $\mathbb{X}$:

If $\mathbb{S}$ is singular then $\vec{\mathbbm{p}}_{1}(\mathbb{S})\leq\vec{p}_{1}(S)\leq\vec{\mathbbm{p}}_{1}(\mathbb{S})+\text{codim}(S)-\text{codim}(\mathbb{S})$, and furthermore

1. (a)

if $\vec{p}_{1}(S)=\vec{\mathbbm{p}}_{1}(\mathbb{S})$ then $\vec{p}_{2}(S)\supset\vec{\mathbbm{p}}_{2}(\mathbb{S})$, 2. (b)

if $\vec{p}_{1}(S)=\vec{\mathbbm{p}}_{1}(\mathbb{S})+\text{codim}(S)-\text{codim}(\mathbb{S})$, then $\vec{p}_{2}(S)\subset\vec{\mathbbm{p}}_{2}(\mathbb{S})$. 2. 2.

If $\mathbb{S}$ is regular then $-1\leq\vec{p}_{1}(S)\leq\text{codim}(S)-1$, and furthermore

1. (a)

if $\vec{p}_{1}(S)=-1$ then $H^{1}(\mathcal{E}_{x})=0$ and $H^{0}(\mathcal{E}_{x})$ is $\vec{p}_{2}(S)$-torsion for all $x\in S$, 2. (b)

if $\vec{p}_{1}(S)=0$ then $H^{1}(\mathcal{E}_{x})$ is $\vec{p}_{2}(S)$-torsion for all $x\in S$, 3. (c)

if $\vec{p}_{1}(S)=\text{codim}(S)-2$ then $H^{0}(\mathcal{E}_{x})$ is $\vec{p}_{2}(S)$-torsion free for all $x\in S$, 4. (d)

if $\vec{p}_{1}(S)=\text{codim}(S)-1$ then $H^{0}(\mathcal{E}_{x})=0$ and $H^{1}(\mathcal{E}_{x})$ is $\vec{p}_{2}(S)$-torsion free for all $x\in S$.

Remark 3.2*.*

If $0\leq\vec{p}_{1}(S)\leq\text{codim}(S)-2$ for all $S$ and $\mathcal{E}$ is a globally defined local system of free modules concentrated in degree [math], then all of condition 2 holds automatically.

Remark 3.3*.*

Since $\mathcal{E}$ is clc on the regular strata by assumption, the torsion conditions of property 2 hold for all $x\in S$ if and only if they hold for some $x\in S$.

Remark 3.4*.*

We notice that if $\mathbb{S}$ is singular then compatibility as just defined places no absolute constraints on the values of $\vec{p}(S)$ and $\vec{\mathbbm{p}}(\mathbb{S})$ but only relative constraints on how these must relate to each other. By contrast, if $\mathbb{S}$ is regular then there are absolute constraints on the values of $\vec{p}_{1}(S)$ and also, for the extreme values of $\vec{p}_{1}(S)$, constraints on how $\vec{p}_{2}(S)$ relates to $\mathcal{E}$. We will see such a dichotomy throughout. One consequence is that these conditions forbid any codimension one stratum of $X$ being contained in a regular stratum of $\mathbb{X}$ (unless $\mathcal{E}$ is trivial on that regular stratum) as this would require either $\vec{p}_{1}(S)=-1=\text{codim}(S)-2$ or $\vec{p}_{1}(S)=0=\text{codim}(S)-1$, and in either case the combined torsion assumptions imply each $H^{*}(\mathcal{E}_{x})=0$ so that $\mathcal{E}$ is trivial.

Remark 3.5*.*

One situation in which the conditions of Definition 3.1 are not necessary for topological invariance is when perversity values are so extreme that their specific values become irrelevant. For example, if $\vec{p}_{1}(S)<-1$ then $\mathcal{P}^{*}_{x}=0$ for any $x\in S$ regardless of the actual value of $\vec{p}(S)$. At the other extreme, [GBF32, Theorem 4.15] implies that if $X$ is a stratified pseudomanifold, $\mathcal{P}^{*}$ a ts-Deligne sheaf on $X$, and $x\in X_{n-k}$ then $H^{i}(\mathcal{P}^{*}_{x})=0$ for $i>k$. The main technical tool in the proof of that theorem is [GBF32, Lemma 4.14], but the argument for this lemma applies to any manifold stratified space (cf. [Bo, Lemma V.9.5]). The proof of [GBF32, Theorem 4.15] therefore generalizes to CS sets, using Lemma 4.1 below in the argument instead of the citation to [Bo, Lemma V.3.8.b]. Consequently, if $\vec{p}_{1}(S)\geq k$ then again the specific values don't matter. Therefore, for the purposes of Theorem 3.6 we could add to the conditions of Definition 3.1 the possibility that if $S\subset\mathbb{S}$ then either both $\vec{p}_{1}(S)$ and $\vec{\mathbbm{p}}_{1}(\mathbb{S})$ are $<-1$ or that they are both sufficiently large. However, rather than complicate Definition 3.1 even further, when considering the necessity of the conditions it is reasonable to assume that the ts-perversities are efficient, i.e. that $-1\leq\vec{p}_{1}(S)\leq\text{codim}(S)$ for all singular strata $S$. If $\vec{p}$ is not efficient, it can always be replaced by a ts-perversity that is efficient without altering $\mathcal{P}^{*}$. In any case, Theorem 3.6 does not require such assumptions.

We now come to the main theorem of the paper, which will be the basis for the results in the rest of Section 3.

Theorem 3.6**.**

Suppose that $X$ and $\mathbb{X}$ denote two CS set stratifications of the same underlying space with $\mathbb{X}$ coarsening $X$. Let $\mathcal{E}$ be a ts-coefficient system such that $\mathbb{X}$ is adapted to $\mathcal{E}$ (and hence so is $X$). Let $\vec{p}$ and $\vec{\mathbbm{p}}$ be respective ts-perversities on $X$ and $\mathbb{X}$ that are $\mathcal{E}$-compatible, and let $\mathcal{P}^{*}$ and $\mathbb{P}^{*}$ be the respective ts-Deligne sheaves with coefficients $\mathcal{E}$. Then $\mathcal{P}^{*}$ is quasi-isomorphic to $\mathbb{P}^{*}$.

Proof.

By Theorem 2.7, $\mathcal{P}^{*}$ is characterized uniquely up to quasi-isomorphism by the axioms TAx1'$(X,\vec{p},\mathcal{E})$. Therefore, to prove the proposition, it is sufficient to show that $\mathbb{P}^{*}$ also satisfies these axioms. We will use that $\mathbb{P}^{*}$ already satisfies the axioms TAx1'$(\mathbb{X},\vec{\mathbbm{p}},\mathcal{E})$.

Axiom a. Since $\mathbb{P}^{*}$ is $\mathbb{X}$-clc, it is also $X$-clc since $X$ refines $\mathbb{X}$. Furthermore, $\mathbb{P}^{*}$ is [math] for $*<0$ by construction, and the cohomology is nontrivial in a finite range of degrees by Axioms b and c, which we verify below, and by the definition of ts-coefficient systems.

Axiom b. We know $\mathbb{P}^{*}|_{\mathbb{X}-\mathbb{X}^{n-1}}\cong\mathcal{E}|_{\mathbb{X}-\mathbb{X}^{n-1}}$, but $X-X^{n-1}\subset\mathbb{X}-\mathbb{X}^{n-1}$ by assumption, so also $\mathbb{P}^{*}|_{X-X^{n-1}}\cong\mathcal{E}|_{X-X^{n-1}}$.

Axiom c. Suppose $x\in S\subset X_{n-k}$ for $k\geq 1$. We must show that $H^{i}(\mathbb{P}^{*}_{x})=0$ for $i>\vec{p}_{1}(S)+1$ and that $H^{\vec{p}_{1}(S)+1}(\mathbb{P}^{*}_{x})$ is $\vec{p}_{2}(S)$-torsion.

First suppose that $x$ is contained in a regular stratum of $\mathbb{X}$. Then $\mathbb{P}^{*}_{x}\cong\mathcal{E}_{x}$, so $H^{i}(\mathbb{P}^{*}_{x})\cong H^{i}(\mathcal{E}_{x})$. We recall that $H^{i}(\mathcal{E}_{x})$ is automatically [math] if $i\neq 0,1$ and that $-1\leq\vec{p}_{1}(S)\leq\text{codim}(S)-1$ by the compatibility assumption. If $\vec{p}_{1}(S)\geq 1$, then $\vec{p}_{1}(S)+1\geq 2$ and $H^{i}(\mathcal{E}_{x})=0$ for $i\geq\vec{p}_{1}(S)+1$. If $\vec{p}_{1}(S)=0$ then $H^{i}(\mathcal{E}_{x})=0$ for $i>\vec{p}_{1}(S)+1=1$ while $H^{\vec{p}_{1}(S)+1}(\mathcal{E}_{x})=H^{1}(\mathcal{E}_{x})$ is $\vec{p}_{2}(S)$-torsion in this case by the $\mathcal{E}$-compatibility assumptions. Finally, if $\vec{p}_{1}(S)=-1$, the compatibility assumptions imply that $H^{1}(\mathcal{E}_{x})=0$, and so $H^{i}(\mathcal{E}_{x})=0$ for $i>0$, while $H^{0}(\mathcal{E}_{x})$ is $\vec{p}_{2}(S)$-torsion333We see in this argument why $\vec{p}_{1}(S)\geq-1$ is required, as well as our torsion assumptions; see also Section 4.2, below..

Now suppose that $x$ is contained in the singular stratum $\mathbb{S}$ of $\mathbb{X}$. Then we know $H^{i}(\mathbb{P}^{*}_{x})=0$ for $i>\vec{\mathbbm{p}}_{1}(\mathbb{S})+1$ and $H^{\vec{\mathbbm{p}}(\mathbb{S})+1}(\mathbb{P}^{*}_{x})$ is $\vec{\mathbbm{p}}_{2}(\mathbb{S})$-torsion. But by assumption $\vec{\mathbbm{p}}_{1}(\mathbb{S})\leq\vec{p}_{1}(S)$, and if $\vec{\mathbbm{p}}_{1}(\mathbb{S})=\vec{p}_{1}(S)$ then $\vec{\mathbbm{p}}_{2}(\mathbb{S})\subset\vec{p}_{2}(S)$. It follows that $\mathbb{P}^{*}$ satisfies Axiom c of TAx1'$(X,\vec{p},\mathcal{E})$.

Axiom d. Again suppose $x\in S\subset X_{n-k}$ for $k\geq 1$, and let $f_{x}:\{x\}\hookrightarrow X$ be the inclusion. We must show that $H^{i}(f_{x}^{!}\mathbb{P}^{*})=0$ for $i\leq\vec{p}_{1}(S)+n-k+1$ and that it is $\vec{p}_{2}(S)$-torsion free when $i=\vec{p}_{1}(S)+n-k+2$.

Once again we first suppose that $x$ is contained in a regular stratum of $\mathbb{X}$. Then we have $f_{x}^{!}\mathbb{P}^{*}\cong f_{x}^{*}\mathbb{P}^{*}[-n]\cong\mathcal{E}_{x}[-n]$ by [Bo, Proposition V.3.7.b] and by assumption, and so $H^{i}(f_{x}^{!}\mathbb{P}^{*})\cong H^{i}(\mathcal{E}_{x}[-n])\cong H^{i-n}(\mathcal{E}_{x})$. So we must show $H^{j}(\mathcal{E}_{x})$ is [math] for $j\leq\vec{p}_{1}(S)-k+1$ and that it is $\vec{p}_{2}(S)$-torsion free when $j=\vec{p}_{1}(S)-k+2$. Recall that $H^{j}(\mathcal{E}_{x})=0$ automatically for $j\neq 0,1$. Now, as $\vec{p}$ and $\vec{\mathbbm{p}}$ are $\mathcal{E}$-compatible, we have that $\vec{p}_{1}(S)\leq\text{codim}(S)-1=k-1$. If $\vec{p}_{1}(S)\leq k-3$ then $\vec{p}_{1}(S)-k+2\leq-1$ and so $H^{j}(\mathcal{E}_{x})=0$ for $j\leq\vec{p}_{1}(S)-k+2$. If $\vec{p}_{1}(S)=k-2$, then $\vec{p}_{1}(S)-k+2=0$ so similarly $H^{j}(\mathcal{E}_{x})=0$ for $j<\vec{p}_{1}(S)-k+2$ while $H^{\vec{p}_{1}(S)-k+2}(\mathcal{E}_{x})=H^{0}(\mathcal{E}_{x})$ is $\vec{p}_{2}(S)$-torsion free in this case by the compatibility assumptions. Finally, if $\vec{p}_{1}(S)=k-1$ so that $\vec{p}_{1}(S)-k+2=1$, we have that $H^{0}(\mathcal{E}_{x})=0$ and $H^{1}(\mathcal{E}_{x})$ is $\vec{p}_{2}(S)$-torsion free again by the compatibility conditions444We see in this argument why $\vec{p}_{1}(S)\leq\text{codim}(S)-1$ is required, as well as our torsion-free assumptions; see also Section 4.2, below..

Next we suppose that $x\in S\subset X_{n-k}$ and that $S\subset\mathbb{S}$ for $\mathbb{S}\subset\mathbb{X}_{n-\ell}$ a singular stratum of $\mathbb{X}$. Let $U\cong\mathbb{R}^{n-\ell}\times cL$ be a distinguished neighborhood of $x$ in the $\mathbb{X}$ stratification. By abuse of notation, we can identify $U$ with $\mathbb{R}^{n-\ell}\times cL$, letting $\mathbb{P}^{*}$ also denote its pullback to this product neighborhood, which remains $\mathbb{X}$-clc. Let $x=(y,v)$ with $f_{y}:\{y\}\hookrightarrow\mathbb{R}^{n-\ell}$ and $f_{v}:\{v\}\hookrightarrow cL$ the vertex inclusion. Let $\pi_{1}:\mathbb{R}^{n-\ell}\times cL\to\mathbb{R}^{n-\ell}$ and $\pi_{2}:\mathbb{R}^{n-\ell}\times cL\to cL$ be the projections, and let $s:cL\hookrightarrow\{y\}\times cL$ be the inclusion. By [KS, Proposition 2.7.8] (letting the $Y_{n}$ there be close balls in $\mathbb{R}^{n-\ell}$), we have $\mathbb{P}^{*}\cong\pi_{2}^{*}R\pi_{2*}\mathbb{P}^{*}$. Let $\mathcal{R}_{A}$ denote the constant sheaf on the space $A$ with stalks in our ground ring $R$. Then

[TABLE]

By [Bo, Remark V.10.20.c], whose hypotheses are satisfied due to the constructibility (see [Sch03, Proposition 4.0.2.2]),

[TABLE]

For the first factor, by [Bo, Proposition V.3.7.b] we have $f^{!}_{y}\mathcal{R}_{\mathbb{R}^{n-\ell}}\cong f^{*}_{y}\mathcal{R}_{\mathbb{R}^{n-\ell}}[-(n-\ell)]=R[-(n-\ell)]$, and so

[TABLE]

To compute $H^{i}\left(f_{v}^{!}R\pi_{2*}\mathbb{P}^{*}\right)$, consider the long exact sequence [Bo, Section V.1.8]

[TABLE]

where $\bar{\mathfrak{i}}:cL-\{v\}\to cL$ fits into the Cartesian square

{diagram}

We can see that $\bar{\mathfrak{i}}^{*}R\pi_{2*}\mathbb{P}^{*}\cong R\bar{\pi}_{2*}\mathfrak{i}^{*}\mathbb{P}^{*}$ by replacing $\mathbb{P}^{*}$ with an injective resolution and then considering sections over open sets. It follows that $\alpha$ is isomorphic to the attaching map $\mathbb{H}^{i}(\mathbb{R}^{n-\ell}\times cL;\mathbb{P}^{*})\to\mathbb{H}^{i}(\mathbb{R}^{n-\ell}\times cL;R\mathfrak{i}_{*}\mathfrak{i}^{*}\mathbb{P}^{*})=\mathbb{H}^{i}(\mathbb{R}^{n-\ell}\times(cL-\{v\});\mathfrak{i}^{*}\mathbb{P}^{*})$. As $\mathbb{P}^{*}$ and $R\mathfrak{i}_{*}\mathfrak{i}^{*}\mathbb{P}^{*}$ are $\mathbb{X}$-clc, [Sch03, Proposition 4.0.2] implies that restriction to smaller distinguished neighborhoods of $x$ in $\mathbb{X}$ yields a constant map of constant direct systems, and so this hypercohomology attaching map is isomorphic to the map it induces stalkwise in the direct limit. And by the axioms TAx1$(\mathbb{X},\vec{\mathbbm{p}},\mathcal{E})$, which $\mathbb{P}^{*}$ satisfies, this attaching map induces stalk-wise cohomology isomorphisms for $i\leq\vec{\mathbbm{p}}_{1}(\mathbb{S})$ and an isomorphism onto the $\vec{\mathbbm{p}}_{2}(\mathbb{S})$-torsion module for $i=\vec{\mathbbm{p}}_{1}(\mathbb{S})+1$. So $H^{i}\left(f_{v}^{!}R\pi_{2*}\mathbb{P}^{*}\right)=0$ for $i\leq\vec{\mathbbm{p}}_{1}(\mathbb{S})+1$. We also know that $\mathbb{H}^{\vec{\mathbbm{p}}_{1}(\mathbb{S})+2}\left(cL;R\pi_{2*}\mathbb{P}^{*}\right)\cong\mathbb{H}^{\vec{\mathbbm{p}}_{1}(\mathbb{S})+2}\left(\mathbb{R}^{n-\ell}\times cL;\mathbb{P}^{*}\right)\cong\mathbb{H}^{\vec{\mathbbm{p}}_{1}(\mathbb{S})+2}\left(\mathbb{P}^{*}_{x}\right)$, the latter by [Sch03, Proposition 4.0.2.2] again, and this is [math] by axiom TAx1'c for $\mathbb{P}^{*}$. Since we have already noted that in degree $\vec{\mathbbm{p}}_{1}(\mathbb{S})+1$ the map $\alpha$ is an isomorphism onto the $\vec{\mathbbm{p}}_{2}(\mathbb{S})$-torsion module of $H^{\vec{\mathbbm{p}}_{1}(\mathbb{S})+1}\left(cL-\{v\};\bar{\mathfrak{i}}^{*}R\pi_{2*}\mathbb{P}^{*}\right)$, it follows that $H^{\vec{\mathbbm{p}}_{1}(\mathbb{S})+2}\left(f_{v}^{!}R\pi_{2*}\mathbb{P}^{*}\right)$ is $\vec{\mathbbm{p}}_{2}(\mathbb{S})$-torsion free.

Returning now to $H^{i}(f_{x}^{!}\mathbb{P}^{*})\cong H^{i}(f_{v}^{!}R\pi_{2*}\mathbb{P}^{*}[-(n-\ell)])=H^{i-n+\ell}(f_{v}^{!}R\pi_{2*}\mathbb{P}^{*})$, we conclude that $H^{i}(f_{x}^{!}\mathbb{P}^{*})=0$ when $i-n+\ell\leq\vec{\mathbbm{p}}_{1}(\mathbb{S})+1$, i.e. when $i\leq\vec{\mathbbm{p}}_{1}(\mathbb{S})+n-\ell+1$, and that $H^{\vec{\mathbbm{p}}_{1}(\mathbb{S})+n-\ell+2}(f_{x}^{!}\mathbb{P}^{*})$ is $\vec{\mathbbm{p}}_{2}(\mathbb{S})$-torsion free. By assumption, $\vec{p}_{1}(S)\leq\vec{\mathbbm{p}}_{1}(\mathbb{S})+\text{codim}(S)-\text{codim}(\mathbb{S})=\vec{\mathbbm{p}}_{1}(\mathbb{S})+k-\ell$, so $\vec{\mathbbm{p}}_{1}(\mathbb{S})+n-\ell+1\geq\vec{p}_{1}(S)-k+\ell+n-\ell+1=\vec{p}_{1}(S)-k+n+1$. So $H^{i}(f_{x}^{!}\mathbb{P}^{*})=0$ for $i\leq\vec{p}_{1}(S)-k+n+1$ as desired. Furthermore, $H^{\vec{p}_{1}(S)-k+n+2}(f_{x}^{!}\mathbb{P}^{*})$ will also be [math] unless the above inequalities are equalities, in which case $H^{\vec{p}_{1}(S)-k+n+2}(f_{x}^{!}\mathbb{P}^{*})=H^{\vec{\mathbbm{p}}_{1}(\mathbb{S})-\ell+n+2}(f_{x}^{!}\mathbb{P}^{*})$ is $\vec{\mathbbm{p}}_{2}(\mathbb{S})$-torsion free. But we have assumed for this scenario that $\vec{\mathbbm{p}}_{2}(\mathbb{S})\supset\vec{p}_{2}(S)$ so that this module is also $\vec{p}_{2}(S)$-torsion free.

We have now demonstrated all the axioms, completing the proof. ∎

3.1 Pullback and pushforward perversities

In [CST-inv], Chataur, Saralegi, and Tanré consider the invariance of intersection homology under refinement/coarsening when the perversity on the finer stratification is pulled back from a perversity on the coarser stratification or when the perversity on the coarser stratification is pushed forward from the finer stratification. In the following two subsections we consider such constructions for ts-perversities. We will see that pullback perversities can always be constructed and always result in quasi-isomorphic ts-Deligne sheaves, generalizing [CST-inv, Corollary 6.13]. By contrast, pushforward perversities require certain conditions to be defined and then further conditions to provide quasi-isomorphic ts-Deligne sheaves. Our results about pushforwards generalize [CST-inv, Theorem C].

In this section we assume the following situation: We suppose that $X$ and $\mathbb{X}$ denote two CS set stratifications of the same underlying space with $\mathbb{X}$ coarsening $X$. Let $\nu:X\to\mathbb{X}$ denote the identity map, which is a stratified map; we sometimes refer to $\nu$ as a coarsening map. If $S$ is a stratum of $X$, let $S^{\nu}$ denote the stratum of $\mathbb{X}$ containing it.

3.1.1 Pullback perversities

We first define pullback perversities.

Definition 3.7**.**

Let $\nu:X\to\mathbb{X}$ be a coarsening map. Suppose $\mathbb{X}$ is adapted to a ts-coefficient system $\mathcal{E}$, and let $\vec{\mathbbm{p}}$ be a ts-perversity on $\mathbb{X}$. We define the $\mathcal{E}$-compatible pullback perversity $\nu_{\mathcal{E}}^{*}\vec{\mathbbm{p}}$ on the refinement $X$ of $\mathbb{X}$ by:

if $S^{\nu}$ is singular then $\nu^{*}_{\mathcal{E}}\vec{\mathbbm{p}}(S)=\vec{\mathbbm{p}}(S^{\nu})$, 2. 2.

if $S^{\nu}$ is regular then $\nu^{*}_{\mathcal{E}}\vec{\mathbbm{p}}(S)=(0,\wp)$, where $\wp$ is the smallest subset of $P(R)$ such that $\mathcal{E}$ is a $\wp$-coefficient system on $S$.

Remark 3.8*.*

Since $X$ refines $\mathbb{X}$ and $\mathcal{E}$ is adapted to $\mathbb{X}$, for each singular stratum $S$ of $X$ contained in a regular stratum of $\mathbb{X}$ the $\wp$ in the second condition always exists. In fact, we could use in this condition any choice of $\wp\subset P(R)$ such that $\mathcal{E}$ is a $\wp$-coefficient system on $S$ for the purposes of the following theorem; we choose the smallest such $\wp$ just for definiteness.

Theorem 3.9**.**

If $X$ has no stratum of codimension one contained in a regular stratum of $\mathbb{X}$ then $\vec{\mathbbm{p}}$ and $\nu_{\mathcal{E}}^{*}\vec{\mathbbm{p}}$ are $\mathcal{E}$-compatible. Consequently $\mathcal{P}^{*}_{\nu^{*}_{\mathcal{E}}\vec{\mathbbm{p}}}\cong\mathbb{P}^{*}_{\vec{\mathbbm{p}}}$.

Proof.

The first statement is immediate from the definitions. The second follows by Theorem 3.6. ∎

Remark 3.10*.*

If we assume in Theorem 3.9 that $E$ is a local system concentrated in degree [math] (and so, in particular, a $\emptyset$-coefficient system) and if $\vec{p}_{2}(S)=\emptyset$ for all $S$, then the theorem becomes a statement about ordinary intersection homology that generalizes [CST-inv, Corollary 6.12].

3.1.2 Pushforward perversities

Chataur, Saralegi, and Tanré also introduce pushforward perversities in [CST-inv, Section 6]. More specifically, they establish conditions under which a perversity can be pushed forward from a stratification of a CS set to its intrinsic stratification and for which the corresponding intersection homology groups are isomorphic. We first generalize pushforwards and place them in our context:

Definition 3.11**.**

We will say that the ts-perversity $\vec{p}$ on $X$ can be pushed to $\mathbb{X}$ if $\vec{p}(Y)=\vec{p}(Z)$ for the two singular strata $Y$ and $Z$ of $X$ whenever all the following conditions hold:

$Y^{\nu}=Z^{\nu}$, 2. 2.

$Y^{\nu}$ is singular in $\mathbb{X}$, and 3. 3.

$\dim(Y)=\dim(Z)=\dim(Y^{\nu})$.

If this property holds then we define the pushforward $\nu_{*}\vec{p}$ by $\nu_{*}\vec{p}(\mathbb{S})=\vec{p}(S)$ if $S$ is a stratum of $X$ such that $\mathbb{S}=S^{\nu}$ and $\dim(S)=\dim(\mathbb{S})$. Every singular stratum of $\mathbb{X}$ must contain such a stratum $S$, so $\nu_{*}\vec{p}$ is well defined and without any ambiguity due to our assumptions.

Remark 3.12*.*

If $\vec{p}$ is a ts-perversity that depends only on codimension, i.e. $\vec{p}(S)=\vec{p}(T)$ whenever $\text{codim}(S)=\text{codim}(T)$, then $\vec{p}$ can be pushed forward to any coarsening. In this case we may abuse notation and also write $\nu_{*}\vec{p}$ simply as $\vec{p}$. As a further abuse, we can also treat $\mathbb{Z}_{\geq 1}$ as the domain of $\vec{p}$ writing $\vec{p}(\text{codim}(S))=\vec{p}(S)$.

Unfortunately, in contrast to Theorem 3.9, a perversity and its pushforward are not necessarily $\mathcal{E}$-compatible, even when the pushforward is defined. For example, we need only let $\mathbb{X}$ be a trivially filtered manifold and let $X$ be a refinement with a stratum $S$ on which $\vec{p}_{1}(S)<-1$.

Of course Theorem 3.6 and the necessity condition in Section 4 show that $\mathcal{E}$-compatibility is our most general criterion for topological invariance, but for comparison with earlier results, especially [CST-inv, Theorem C], it is useful to delineate in terms of $\vec{p}$ exactly when $\vec{p}$ and $\nu_{*}\vec{p}$ will be $\mathcal{E}$-compatible. Inspection of the definitions yields the following (cf. the definition of $K^{*}$-perversities in [CST-inv, Definition 6.8]):

Proposition 3.13**.**

Let $\nu:X\to\mathbb{X}$ be a coarsening map, and suppose $\mathbb{X}$ is adapted to the ts-coefficient system $\mathcal{E}$. Suppose $\vec{p}$ is a perversity on $X$ that can be pushed to $\mathbb{X}$. For any singular stratum $S\subset X$, let $\tilde{S}$ denote555These $\tilde{S}$ are called source strata in [CST-inv]. Note that the precise choice of $\tilde{S}$ does not matter in the conditions that follow as the assumption that $\vec{p}$ can be pushed forward assures us that any two choices would give the same perversity conditions. any stratum of $X$ such that $(\tilde{S})^{\nu}=S^{\nu}$ and $\dim(\tilde{S})=\dim(S^{\nu})$. Then $\vec{p}$ and $\nu_{*}\vec{p}$ are $\mathcal{E}$-compatible if the following conditions hold on $\vec{p}$:

If $\tilde{S}$ is singular then $\vec{p}_{1}(\tilde{S})\leq\vec{p}_{1}(S)\leq\vec{p}_{1}(\tilde{S})+\text{codim}(S)-\text{codim}(\tilde{S})$, and furthermore

1. (a)

if $\vec{p}_{1}(\tilde{S})=\vec{p}_{1}(S)$ then $\vec{p}_{2}(\tilde{S})\subset\vec{p}_{2}(S)$, 2. (b)

if $\vec{p}_{1}(S)=\vec{p}_{1}(\tilde{S})+\text{codim}(S)-\text{codim}(\tilde{S})$, then $\vec{p}_{2}(\tilde{S})\supset\vec{p}_{2}(S)$. 2. 2.

If $\tilde{S}$ is regular then $-1\leq\vec{p}_{1}(S)\leq\text{codim}(S)-1$, and furthermore

1. (a)

If $\vec{p}_{1}(S)=-1$ then $H^{1}(\mathcal{E}_{x})=0$ and $H^{0}(\mathcal{E}_{x})$ is $\vec{p}_{2}(S)$-torsion for all $x\in S$, 2. (b)

if $\vec{p}_{1}(S)=0$ then $H^{1}(\mathcal{E}_{x})$ is $p_{2}(S)$-torsion for all $x\in S$ 3. (c)

if $\vec{p}_{1}(S)=\text{codim}(S)-2$, then $H^{0}(\mathcal{E}_{x})$ is $p_{2}(S)$-torsion free for all $x\in S$, 4. (d)

if $\vec{p}_{1}(S)=\text{codim}(S)-1$ then $H^{0}(\mathcal{E}_{x})=0$ and $H^{1}(\mathcal{E}_{x})$ is $\vec{p}_{2}(S)$-torsion free for all $x\in S$.

Remark 3.14*.*

Condition 1 on $\vec{p}_{1}$ is essentially Condition B of [CST-inv, Definition 6.8], while Condition 2 on $\vec{p}_{1}$ corresponds there to Conditions A and D. Condition C of [CST-inv] is built into our assumption that $\vec{p}$ can be pushed forward. Note, however, that in [CST-inv] only pushforwards to the intrinsic stratification are considered.

Remark 3.15*.*

If $0\leq\vec{p}_{1}(S)\leq\text{codim}(S)-2$ for all $S$ and $\mathcal{E}$ is a globally defined local system of free modules in degree [math], then all of condition 2 holds automatically.

3.2 Constrained perversities

So far we have considered only the situation in which we have two stratifications, $X$ and $\mathbb{X}$, one refining the other. In this case we have a point of comparison between any two ts-perversities $\vec{p}$ and $\vec{\mathbbm{p}}$ on these stratifications since any stratum $S$ of $X$ is contained in a stratum $\mathbb{S}$ of $\mathbb{X}$, allowing us to compare $\vec{p}(S)$ with $\vec{\mathbbm{p}}(\mathbb{S})$. If we are given instead two arbitrary stratifications, then our best hope for relating them seems to be if we can find a common refinement or coarsening, in which case we can perhaps apply Theorem 3.6 twice, connecting each given stratification with our common intermediary. More generally, we can attempt to compare all stratifications of a given space by finding a universal common refinement or coarsening. We will see that such common refinements do not always exist, even for just two stratifications (Remark 3.27), but intrinsic common coarsenings do exist, as we'll show in Section LABEL:S:_intrinsic.

Of course we also need to be able to assign sufficiently compatible perversities to all these stratifications. The simplest way to proceed when faced with all possible stratifications seems to be to revert more closely to the original definition of Goresky and MacPherson [GM1] in which perversities were assumed to be functions only of codimension. This allows one to define perversities without any reference to a specific stratification: the perversity simply assigns the same predetermined value to all strata of the same codimension. In such a setting, Goresky and MacPherson proved a topological invariance statement of the form, if a perversity satisfies certain conditions then all stratifications (without codimension one strata) of the same space with that perversity and a fixed coefficient system yield quasi-isomorphic Deligne sheaves'' [GM2, Theorem 4.1]. This theorem was slightly refined by Borel [Bo, Section V.4]. In both cases, the proofs utilize common coarsenings, with the assumptions about the perversities being strong enough to imply (in our language) $\mathcal{E}$-compatibility among the manifestations of the same'' perversity on all stratifications.

Analogously, our goal in this section is to construct ts-perversities that depend only on codimension and such that all stratifications yield quasi-isomorphic ts-Deligne sheaves from these ts-perversities. Of course we must also take into account adaptability to coefficient systems and place certain limitations on the behavior of codimension one strata. The result will be a sequence of theorems each with a pair of results. One of each pair places stricter conditions on the perversities but allows for any stratifications up to some limitations on compatibility of codimension one strata; the other removes some of the restrictions on the perversities but forces us to fix the regular strata of the stratifications.

We start with the following definitions:

Definition 3.16**.**

We call a ts-perversity $\vec{p}$ constrained if it satisfies all of the following:

$\vec{p}$ depends only on the codimension of strata (so we can write $\vec{p}=(\vec{p}_{1},\vec{p}_{2}):\mathbb{Z}_{\geq 1}\to\mathbb{Z}\times 2^{P(R)}$), 2. 2.

$\vec{p}_{1}$ satisfies the Goresky-MacPherson growth condition $\vec{p}_{1}(k)\leq\vec{p}_{1}(k+1)\leq\vec{p}_{1}(k)+1$ for $k\geq 1$, 3. 3.

$\vec{p}_{1}(2)\in\{-1,0,1\}$, 4. 4.

if $\vec{p}_{1}(k+1)=\vec{p}_{1}(k)$ then $\vec{p}_{2}(k+1)\supset\vec{p}_{2}(k)$, 5. 5.

if $\vec{p}_{1}(k+1)=\vec{p}_{1}(k)+1$ then $\vec{p}_{2}(k+1)\subset\vec{p}_{2}(k)$.

If $\vec{p}$ satisfies all these conditions except (3) then we say that $\vec{p}$ is weakly constrained. If $\vec{p}_{1}(2)=0$, we say that $\vec{p}$ is strongly constrained.

When $\vec{p}$ is constrained, it will also often be useful for us to consider ts-coefficient systems $\mathcal{E}$ that are ts-perverse with respect to $\vec{p}(2)$, in other words $\mathcal{E}\in{}^{\vec{p}(2)}D^{\heartsuit}(\text{Dom}(\mathcal{E}))$ using the t-structure of [GBF32, Remark 5.4] with the ts-perversity there taking the constant value $\vec{p}(2)$ on the domain of $\mathcal{E}$. Explicitly this means the following:

if $\vec{p}_{1}(2)=-1$ then $H^{1}(\mathcal{E}_{x})=0$ and $H^{0}(\mathcal{E}_{x})$ is $\vec{p}_{2}(2)$-torsion for all $x\in\text{Dom}(\mathcal{E})$, 2. 2.

if $\vec{p}_{1}(2)=0$ then $H^{1}(\mathcal{E}_{x})$ is $\vec{p}_{2}(2)$-torsion and $H^{0}(\mathcal{E}_{x})$ is $\vec{p}_{2}(2)$-torsion free for all $x\in\text{Dom}(\mathcal{E})$, 3. 3.

if $\vec{p}_{1}(2)=1$ then $H^{0}(\mathcal{E}_{x})=0$ and $H^{1}(\mathcal{E}_{x})$ is $\vec{p}_{2}(2)$-torsion free for all $x\in\text{Dom}(\mathcal{E})$.

Remark 3.17*.*

If $\vec{p}_{1}(2)=0$ and $\mathcal{E}$ is a globally defined local system of free modules in degree [math], then these conditions hold automatically.

Remark 3.18*.*

Note that there are a few conditions specific to codimension $2$, even though $\vec{p}$ is also defined for codimension one. This is because these conditions will govern what happens when a singular stratum of $X$ is contained in a regular stratum of $\mathbb{X}$, but we know that for topological invariance we must preclude this for codimension one strata of $X$ anyway and so these conditions only come into play starting at codimension $2$.

Remark 3.19*.*

We will not utilize strongly constrained ts-perversities until Section LABEL:S:_dimension, where they will be convenient.

We begin by considering when a constrained $\vec{p}$ is $\mathcal{E}$-compatible with itself across two stratifications.

Proposition 3.20**.**

Suppose $\mathbb{X}$ is a coarsening of $X$ and that $\mathbb{X}$ is adapted to a ts-coefficient system $\mathcal{E}$. Suppose either

no codimension one stratum of $X$ is contained in a regular stratum of $\mathbb{X}$, $\vec{p}$ is constrained, and $\mathcal{E}\in{}^{\vec{p}(2)}D^{\heartsuit}(\text{Dom}(\mathcal{E}))$, or 2. 2.

$\mathbb{X}^{n-1}=X^{n-1}$* and $\vec{p}$ is weakly constrained.*

Then $\vec{p}$ on $X$ and $\vec{p}$ on $\mathbb{X}$ are $\mathcal{E}$-compatible and so the ts-perversity $\vec{p}$ ts-Deligne sheaves $\mathcal{P}^{*}$ on $X$ and $\mathbb{P}^{*}$ on $\mathbb{X}$ are quasi-isomorphic.

Remark 3.21*.*

The dichotomy here is a reflection of that of Remark 3.4. If we want to allow singular strata of $X$ in regular strata of $\mathbb{X}$ then we need an assumption that will force the perversities on such strata into the absolute bounds $-1\leq\vec{p}_{1}(S)\leq\text{codim}(S)-1$ of Definition 3.1. The requirement $\vec{p}_{1}(2)\in\{-1,0,1\}$, together with the Goresky-MacPherson growth 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 requirement once we have eliminated codimension one strata of $X$ in regular strata of $\mathbb{X}$ (see Remark 3.4). If we wish to dispense with this additional constraint on $\vec{p}_{1}(2)$ then we can instead ask that no singular stratum of $X$ be contained in a regular one of $\mathbb{X}$, leading to the second alternative of the proposition.

Of course we could also dabble in many more specific cases — for example we might allow $\vec{p}(2)=-2$ and $\vec{p}(3)=-1$ but then simply forbid codimension $2$ strata of $X$ from appearing in regular strata of $\mathbb{X}$ (plus conditions involving $\mathcal{E}$). Such scenarios are in any case still captured by Theorem 3.6, so we leave the reader to formulate his or her own variants and instead consider just these two broad situations that are closest in keeping to the previous theorems of [GM2, GBF11, CST-inv].

Proof of Proposition 3.20.

In each case we check the conditions of Definition 3.1. As noted in Remark 3.12, in this case we simply write $\vec{p}$ for both perversities, as well as writing $\vec{p}(\text{codim}(S))=\vec{p}(S)$ when convenient.

Assuming the first hypotheses, suppose $S$ is a singular stratum of $X$ in the singular stratum $\mathbb{S}$ of $\mathbb{X}$. Then $\text{codim}(\mathbb{S})\leq\text{codim}(S)$ so the first condition of Definition 3.1 follows by iterating Condition 2 of Definition 3.16. Furthermore, since $\vec{p}_{1}$ is non-decreasing, $\vec{p}_{1}(S)=\vec{p}_{1}(\mathbb{S})$ only if $\vec{p}_{1}(k)$ is constant from $\text{codim}(\mathbb{S})$ to $\text{codim}(S)$, in which case Condition 4 of Definition 3.16 inductively implies that $\vec{p}_{2}(S)\supset\vec{p}_{2}(\mathbb{S})$ as needed. Similarly, if $\vec{p}_{1}(S)=\vec{p}_{1}(\mathbb{S})+\text{codim}(S)-\text{codim}(\mathbb{S})$ then $\vec{p}_{1}(k+1)=\vec{p}_{1}(k)+1$ for $\text{codim}(\mathbb{S})\leq k<\text{codim}(S)$ and so Condition 5 of Definition 3.16 implies $\vec{p}_{2}(S)\subset\vec{p}_{2}(\mathbb{S})$.

Now suppose $\mathbb{S}$ is regular. By assumption $\text{codim}(S)\geq 2$ and so the Goresky-MacPherson growth condition on $\vec{p}_{1}$, together with $\vec{p}_{1}(2)\in\{-1,0,1\}$, ensures that $-1\leq\vec{p}_{1}(S)\leq\text{codim}(S)-1$.

• •

If $\vec{p}_{1}(S)=-1$, the growth condition implies that $\vec{p}_{1}(k)=-1$ for $2\leq k\leq\text{codim}(S)$. Since $\mathcal{E}\in{}^{\vec{p}(2)}D^{\heartsuit}(\text{Dom}(\mathcal{E}))$, we have that $H^{1}(\mathcal{E}_{x})=0$ and $H^{0}(\mathcal{E}_{x})$ is $\vec{p}_{2}(2)$-torsion for all $x$ in the domain of $\mathcal{E}$, so in particular for $x\in S$. Now Condition 4 of Definition 3.16 implies that $\vec{p}_{2}(S)\supset\vec{p}_{2}(2)$, so $H^{0}(\mathcal{E}_{x})$ is also $\vec{p}_{2}(S)$-torsion for all $x\in S$.

• •

If $\vec{p}_{1}(S)=\text{codim}(S)-1$, the growth condition implies that $\vec{p}_{1}(k)=k-1$ for $2\leq k\leq\text{codim}(S)$ and in particular that $\vec{p}_{1}(2)=1$. Since $\mathcal{E}\in{}^{\vec{p}(2)}D^{\heartsuit}(\text{Dom}(\mathcal{E}))$, we have $H^{0}(\mathcal{E}_{x})=0$ and $H^{1}(\mathcal{E}_{x})$ is $\vec{p}_{2}(2)$-torsion free for all $x$ in the domain of $\mathcal{E}$, so in particular for $x\in S$. Condition 5 of Definition 3.16 implies that $\vec{p}_{2}(S)\subset\vec{p}_{2}(2)$, so $H^{1}(\mathcal{E}_{x})$ is also $\vec{p}_{2}(S)$-torsion free for all $x\in S$.

• •

If $\vec{p}_{1}(S)=0$ the growth condition implies that $\vec{p}_{1}(2)\in\{-1,0\}$. If $\vec{p}_{1}(2)=-1$ then since $\mathcal{E}\in{}^{\vec{p}(2)}D^{\heartsuit}(\text{Dom}(\mathcal{E}))$, we have $H^{1}(\mathcal{E}_{x})=0$ for all $x$ in the domain of $\mathcal{E}$, in which case certainly $H^{1}(\mathcal{E}_{x})$ is $\vec{p}_{2}(S)$-torsion for all $x\in S$. If $\vec{p}_{1}(2)=0$ the growth condition implies that $\vec{p}_{1}(k)=0$ for $2\leq k\leq\text{codim}(S)$. Also since $\mathcal{E}\in{}^{\vec{p}(2)}D^{\heartsuit}(\text{Dom}(\mathcal{E}))$, $H^{1}(\mathcal{E}_{x})$ is $\vec{p}_{2}(2)$-torsion. Condition 4 of Definition 3.16 implies that $\vec{p}_{2}(S)\supset\vec{p}_{2}(2)$, so $H^{1}(\mathcal{E}_{x})$ is also $\vec{p}_{2}(S)$-torsion for all $x\in S$.

• •

If $\vec{p}_{1}(S)=\text{codim}(S)-2$ the growth condition implies that $\vec{p}_{1}(2)\in\{0,1\}$. If $\vec{p}_{1}(2)=1$ then since $\mathcal{E}\in{}^{\vec{p}(2)}D^{\heartsuit}(\text{Dom}(\mathcal{E}))$, we have $H^{0}(\mathcal{E}_{x})=0$ for all $x$ in the domain of $\mathcal{E}$, in which case certainly $H^{0}(\mathcal{E}_{x})$ is $\vec{p}_{2}(S)$-torsion free for all $x\in S$. If $\vec{p}_{1}(2)=0$ the growth condition implies that $\vec{p}_{1}(k)=k-2$ for $2\leq k\leq\text{codim}(S)$. Since $\mathcal{E}\in{}^{\vec{p}(2)}D^{\heartsuit}(\text{Dom}(\mathcal{E}))$, $H^{0}(\mathcal{E}_{x})$ is $\vec{p}_{2}(2)$-torsion free. Condition 5 of Definition 3.16 implies that $\vec{p}_{2}(S)\subset\vec{p}_{2}(2)$, so $H^{0}(\mathcal{E}_{x})$ is also $\vec{p}_{2}(S)$-torsion free for all $x\in S$.

In the second situation of the lemma, no singular stratum of $X$ is contained in a regular stratum of $\mathbb{X}$. The argument is therefore exactly as above except that we don't need to verify any of Condition 2 of Definition 3.1. Therefore, we don't need Condition 3 of Definition 3.16 nor the assumption that $\mathcal{E}\in{}^{\vec{p}(2)}D^{\heartsuit}(\text{Dom}(\mathcal{E}))$.

The final statement of the proposition follows from Theorem 3.6. ∎

Corollary 3.22**.**

Suppose that $\mathcal{X}$ and $\mathcal{Y}$ are any two CS set stratifications of the same space, both of which are adapted to the ts-coefficient system $\mathcal{E}$. Suppose either

$\vec{p}$* is a constrained ts-perversity, $\mathcal{E}\in{}^{\vec{p}(2)}D^{\heartsuit}(\text{Dom}(\mathcal{E}))$, and $\mathcal{X}$ and $\mathcal{Y}$ possess a common coarsening $\mathcal{Z}$ that is adapted to $\mathcal{E}$ such that no codimension one stratum of $\mathcal{X}$ or $\mathcal{Y}$ is contained in a regular stratum of $\mathcal{Z}$, or* 2. 2.

$\vec{p}$* is a weakly constrained ts-perversity and $\mathcal{X}$ and $\mathcal{Y}$ possess a common coarsening $\mathcal{Z}$ that is adapted to $\mathcal{E}$ and such that $\mathcal{X}^{n-1}=\mathcal{Z}^{n-1}=\mathcal{Y}^{n-1}$.*

Then $\mathcal{X}$ and $\mathcal{Y}$ have quasi-isomorphic ts-perversity $\vec{p}$ ts-Deligne sheaves, i.e. $\mathcal{P}^{*}_{\mathcal{X},\vec{p},\mathcal{E}}\cong\mathcal{P}^{*}_{\mathcal{Y},\vec{p},\mathcal{E}}$.

Proof.

For each case apply Proposition 3.20 twice. ∎

The preceding corollary shows that we can compare Deligne sheaves from different stratifications provided appropriate common coarsenings exist. With minor hypotheses, such common coarsenings are constructed in Section LABEL:S:_intrinsic with the following result. Recall the definitions of maximal ts-coefficient systems from Definition 2.8 and of ``fully adapted'' from Definition 2.10.

Theorem 3.23**.**

Suppose that $\mathcal{X}$ and $\mathcal{Y}$ are any two CS set stratifications of the same space, both fully adapted to the maximal ts-coefficient system $\mathcal{E}$. Suppose either

$\vec{p}$* is a constrained ts-perversity, $\mathcal{E}\in{}^{\vec{p}(2)}D^{\heartsuit}(\text{Dom}(\mathcal{E}))$, and the closure of the union of the codimension one strata of $\mathcal{X}$ (which may be empty) is equal to the closure of the union of the codimension one strata of $\mathcal{Y}$, or* 2. 2.

$\vec{p}$* is a weakly constrained ts-perversity and $\mathcal{X}^{n-1}=\mathcal{Y}^{n-1}$.*

Then $\mathcal{X}$ and $\mathcal{Y}$ have quasi-isomorphic ts-perversity $\vec{p}$ ts-Deligne sheaves, i.e. $\mathcal{P}^{*}_{\mathcal{X},\vec{p},\mathcal{E}}\cong\mathcal{P}^{*}_{\mathcal{Y},\vec{p},\mathcal{E}}$.

Proof.

Let $C_{1}$ denote the closure of the union of the codimension one strata of $\mathcal{X}$. In the first case, Proposition LABEL:P:_intrinsic yields an intrinsic stratification $\mathfrak{X}_{\mathcal{E},C_{1}}$ that is a common coarsening of $\mathcal{X}$ and $\mathcal{Y}$ that is adapted to $\mathcal{E}$ and such that no codimension one stratum of $\mathcal{X}$ or $\mathcal{Y}$ is contained in a regular stratum of $\mathfrak{X}$ (in fact the closures of the unions of the codimension one strata of $\mathcal{X}$, $\mathcal{Y}$, and $\mathfrak{X}$ are all the same set). In the second case, Proposition LABEL:P:_intrinsic yields an intrinsic stratification $\mathfrak{X}_{\mathcal{E},\mathcal{X}^{n-1}}=\mathfrak{X}_{\mathcal{E},\mathcal{X}^{n-1}}$ that is a common coarsening of $\mathcal{X}$ and $\mathcal{Y}$ that is adapted to $\mathcal{E}$ and such that $\mathfrak{X}_{\mathcal{E},\mathcal{X}^{n-1}}^{n-1}=\mathcal{X}^{n-1}=\mathcal{Y}^{n-1}$. This last statement holds because Property LABEL:I:_int3 of Proposition LABEL:P:_intrinsic shows that if $U_{\mathcal{E}}$ is the domain of $\mathcal{E}$ then in this case $\mathfrak{X}_{\mathcal{E},\mathcal{X}^{n-1}}-\mathfrak{X}^{n-1}_{\mathcal{E},\mathcal{X}^{n-1}}=U_{\mathcal{E}}-\mathcal{X}^{n-1}$, which must be $\mathcal{X}-\mathcal{X}^{n-1}$ as $\mathcal{X}$ is adapted to $\mathcal{E}$. Thus $\mathcal{X}$ and $\mathfrak{X}_{\mathcal{E},\mathcal{X}^{n-1}}$ have the same regular strata and hence the same $n-1$ skeleta, and similarly for $\mathcal{Y}$.

The theorem now follows from the preceding corollary. ∎

Remark 3.24*.*

If we take $\mathcal{E}$ to be a constant sheaf concentrated in degree [math], set $\vec{p}_{2}(k)=\emptyset$ 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 of Goresky-MacPherson [GM2, Theorem 4.1], though with a fairly different proof (see also [GBF44]). Similarly, with the same assumptions on $\mathcal{E}$ and $\vec{p}_{2}$ and with $\vec{p}_{1}(2)>0$, the second case of the theorem recovers the main result of [GBF11].

Remark 3.25*.*

In general, if we are given two CS set stratifications $\mathcal{X}$ and $\mathcal{Y}$ of the same space without any conditions on their codimension one strata there is not necessarily a common coarsening $\mathfrak{X}$ such that no codimension one stratum of $\mathcal{X}$ or $\mathcal{Y}$ is contained in a regular stratum of $\mathfrak{X}$. For example, just let $\mathcal{X}$ be the plane $\mathbb{R}^{2}$ with its trivial stratification and let $\mathcal{Y}$ be any other stratification with a codimension one stratum. The only common coarsening is $\mathcal{X}$ itself, and a regular stratum of $\mathcal{X}$ already contains a codimension one stratum of $\mathcal{Y}$. Even if we rule out starting with such a bad situation, consider letting $\mathcal{X}$ and $\mathcal{Y}$ be stratifications of $\mathbb{R}^{2}$, one with the $x$-axis as singular stratum and one with the $y$-axis as singular stratum. Again, there is no appropriate common coarsening.

We have a similar result to Corollary 3.22 concerning common refinements instead of common coarsenings, though this is generally less useful as common refinements do not exist in general, even with restrictions on codimension one strata; see Remark 3.27. Nonetheless, we include this corollary as it may occasionally be useful when working with spaces for which the hypotheses of the preceding corollaries do not apply.

Corollary 3.26**.**

Suppose that $\vec{p}$ is any ts-perversity depending only on codimension and that $\mathcal{X}$ and $\mathcal{Y}$ are any two CS set stratifications of the same space, both of which are adapted to the ts-coefficient system $\mathcal{E}$. If $\mathcal{X}$ and $\mathcal{Y}$ posses a common refinement $\mathcal{Z}$ such that no codimension one stratum of $\mathcal{Z}$ is contained in a regular stratum of $\mathcal{X}$ or $\mathcal{Y}$, then the ts-perversity $\vec{p}$ ts-Deligne sheaves on $\mathcal{X}$ and $\mathcal{Y}$ are quasi-isomorphic.

Proof.

This follows from Theorem 3.9 using $\mathcal{Z}$ as an intermediary. ∎

Remark 3.27*.*

Unfortunately, unlike the case of coarsenings for which any two stratifications have a common coarsening (given appropriate assumptions on coefficients and codimension one strata), common refinements do not exist in general. For example, let $\mathcal{X}$ be $\mathbb{R}^{n}$, $n\geq 2$, stratified as $\mathbb{R}^{n}\supset\{x_{1}\text{-axis}\}$, and let $\mathcal{Y}$ be $\mathbb{R}^{n}$ stratified by $\mathbb{R}^{n}\supset A$, where $A$ is the image of $f:\mathbb{R}\to\mathbb{R}^{n}$ given by $f(t)=(t,t^{3}\sin(1/t),0,\ldots,0)$ for $t\neq 0$ and $f(0)=\vec{0}$. Then $A$ is a $C^{1}$ submanifold and so possesses a tubular neighborhood, making $\mathcal{Y}$ a CS set. But the intersection of $A$ and the $x_{1}$-axis is the union of the origin with the points $\left\{\frac{1}{n\pi},0,\ldots,0\right\}$ and so any common refinement $\mathcal{Z}$ of the stratifications $\mathcal{X}$ and $\mathcal{Y}$ would require $\mathcal{Z}^{0}$ to have an accumulation point, which is not possible for CS sets.

4 Necessity of the conditions

In this section we consider the necessity of the $\mathcal{E}$-compatibility conditions on ts-perversities (Definition 3.1) in order for our main topological invariance result (Theorem 3.6) to hold. We will show that the singular-in-regular'' conditions are strictly necessary, using that the links 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 in general, meaning that we will construct examples that show that failure of the conditions of Definition 3.1 can result in non-quasi-isomorphic ts-Deligne sheaves unless certain stalk cohomology modules (or some of their torsion submodules or torsion-free quotient modules) vanish. As noted in Remark 3.5, such vanishing is assured when the first component of a ts-perversity has values less than $-1$ or greater than $\text{codim}(S)-1$, but for efficient ts-perversities (those for which $-1\leq\vec{p}_{1}(S)\leq\text{codim}(S)-1$ for all singular strata $S$) such vanishing depends on the particulars of $X$, $\mathcal{E}$, and $\vec{p}$. There are certainly stronger conditions that can be imposed on $(X,\mathbb{X},\mathcal{E})$ that would allow some weakening of Definition 3.1. As an extreme example we could take $\mathcal{E}=0$ in which case $\mathcal{P}^{*}=\mathbb{P}^{*}=0$ regardless of the stratifications or perversities. Another less trivial, though still somewhat artificial, example is noted in Remark 3.21. In fact all such extra conditions are likely to be somewhat artificial in our current general context.

For historical context, we recall that if $\mathcal{E}$ is concentrated in degree [math], if $\vec{p}$ and $\vec{\mathbbm{p}}$ depend only on codimension, and if $\vec{p}_{2}(k)=\vec{\mathbbm{p}}_{2}(k)=\emptyset$ for all $k\geq 2$ (with no codimension one strata allowed) then $\mathcal{P}^{*}$ and $\mathbb{P}^{*}$ will be the original Deligne sheaves of Goresky and MacPherson [GM2]. In this case the hypercohomology groups with compact supports will be the classical intersection homology groups. If $\bar{p}(k)\leq k-2$ for all $k$, as in [GM2], then these will agree with the singular intersection homology groups of King [Ki], called ``GM intersection homology'' in [GBF35]. For GM intersection homology, the Goresky-MacPherson growth condition $\bar{p}(k)\leq\bar{p}(k+1)\leq\bar{p}(k)+1$ is known to be necessary in general for topological invariance; see King [Ki, Section 2]. The main idea is to compare $c(\Sigma X)$ with $\mathbb{R}\times cX$, where $cX$ is the open cone on $X$ and $\Sigma X$ is the suspension. For compact stratified $X$ these spaces are topologically homeomorphic but with different natural stratifications coming from that on $X$. Goresky and MacPherson also assume $\bar{p}(2)=0$ (and no codimension one strata), though King shows that this assumption is not necessary for the topological invariance of singular chain GM intersection homology, only that $\bar{p}(1)\geq 0$.

If $\bar{p}(k)>k-2$ for any $k$ the hypercohomology of the Deligne sheaf is called ``non-GM intersection homology'' in [GBF35]. Non-GM intersection homology can only be obtained using singular chains after making some additional modifications; see [GBF35, Chapter 6]. We showed in [GBF11, Section 3] that if $\bar{p}(2)>0$ then the non-GM intersection homology cannot be a topological invariant in general, but that if the Goresky-MacPherson growth condition holds then we will have invariance under refinements such that $X^{n-1}=\mathbb{X}^{n-1}$. The general necessity of the growth condition for non-GM intersection homology can be argued identically as in the GM case, and we will use essentially the same basic argument below for the case of singular strata of $X$ in singular strata of $\mathbb{X}$.

In the remainder of this section, we first develop some computational machinery in Section 4.1. We then consider the singular-in-regular'' situation in Section [4.2](#S4.SS2) and the singular-in-singular'' situation in Section 4.3. Throughout we continue our notation from the preceding sections, namely $\mathbb{X}$ coarsens $X$ with corresponding ts-perversities $\vec{\mathbbm{p}}$ and $\vec{p}$ and ts-Deligne sheaves $\mathbb{P}^{*}$ and $\mathcal{P}^{*}$. We also assume $\mathbb{X}$ adapted to a ts-coefficient system $\mathcal{E}$.

4.1 Computational tools

We need some tools for computation. Our first lemma is standard:

Lemma 4.1**.**

Suppose $Y$ is a filtered space, and let $X=\mathbb{R}^{k}\times Y$ have the product filtration $X^{j}=\mathbb{R}^{k}\times X^{j-k}$. If $\mathcal{S}^{*}$ is $X$-clc then $\mathbb{H}^{i}(X;\mathcal{S}^{*})\cong\mathbb{H}^{i}(Y;\mathcal{S}^{*}|_{Y})$, identifying $Y$ with any $\{z\}\times Y\subset\mathbb{R}^{k}\times Y$.

Proof.

Let $\pi:\mathbb{R}^{k}\times Y\to Y$ be the projection and let $s:\{z\}\times Y\hookrightarrow\mathbb{R}^{k}\times Y$ be the inclusion. Since $\mathcal{S}^{*}$ is $X$-clc, in particular $H^{j}(\mathcal{S}^{*})$ is constant along each $\mathbb{R}^{k}\times\{y\}$, and so $\mathcal{S}^{*}=\pi^{*}R\pi_{*}\mathcal{S}^{*}$ by [KS, Proposition 2.7.8] (letting the $Y_{n}$ there be closed balls in $\mathbb{R}^{k}$). Then $s^{*}\mathcal{S}^{*}\cong s^{*}\pi^{*}R\pi_{*}\mathcal{S}^{*}=R\pi_{*}\mathcal{S}^{*}$, so $\mathbb{H}^{i}(\mathbb{R}^{k}\times Y;\mathcal{S}^{*})\cong\mathbb{H}^{i}(Y;R\pi_{*}\mathcal{S}^{*})\cong\mathbb{H}^{i}(Y;s^{*}\mathcal{S}^{*})=\mathbb{H}^{i}(Y;\mathcal{S}^{*}|_{Y})$. ∎

Lemma 4.2**.**

Let $L$ be a compact filtered set such that $X=\mathbb{R}^{k}\times cL$ is a CS set. Let $x=(s,v)\in\mathbb{R}^{k}\times cL$ with $s\in\mathbb{R}^{k}$ arbitrary and $v\in cL$ the cone vertex, and let $S$ be the stratum $\mathbb{R}^{k}\times\{v\}$. If $\mathcal{P}^{*}$ is a ts-Deligne sheaf on $\mathbb{R}^{k}\times cL$, then identifying $L$ with some copy $(z,t,L)\subset\mathbb{R}^{k}\times(0,1)\times L\subset\mathbb{R}^{k}\times cL$, we have

[TABLE]

Proof.

Let $W=X-S=\mathbb{R}^{k}\times(cL-\{v\})\cong\mathbb{R}^{k+1}\times L$, let $i:W\hookrightarrow X$, and let $\mathcal{P}^{*}_{W}=\mathcal{P}^{*}|_{W}$. From the definition of the ts-Deligne sheaf and the torsion tipped truncation, we have

[TABLE]

Since $\mathcal{P}^{*}_{W}$ is $W$-clc, $Ri_{*}\mathcal{P}^{*}_{W}$ is $X$-clc by [Sch03, Proposition 4.0.2.3], and so $H^{i}((Ri_{*}\mathcal{P}^{*}_{W})_{x})\cong\mathbb{H}^{i}(X;Ri_{*}\mathcal{P}^{*}_{W})$ by [Sch03, Proposition 4.0.2.2]. But then

[TABLE]

using Lemma 4.1. ∎

If $L$ is itself a CS set, then $\mathcal{P}^{*}|_{L}$ will itself be a ts-Deligne sheaf, as we show in the next lemma. To establish notation, suppose $X$ is a CS set and $\vec{p},\mathcal{E},\mathcal{P}^{*}$ are a ts-perversity, ts-coefficient system, and ts-Deligne sheaf on $X$. If $Y\subset X$ is a CS set with the induced stratification (in the sense of the statement of the lemma below) we will denote by $\vec{p}_{Y}$ the ts-perversity on $Y$ such that $\vec{p}_{Y}(S)=\vec{p}(Z)$ if the singular stratum $S$ of $Y$ is contained in the singular stratum $Z$ of $X$. We also write $\mathcal{E}_{Y}$ for the restriction of $\mathcal{E}$ to the intersection of its domain with $Y$, and we will write $\mathcal{P}^{*}_{Y}$ for the Deligne sheaf on $Y$ with respect to $\vec{p}_{Y},\mathcal{E}_{Y}$.

Lemma 4.3**.**

Suppose that either

$Y$* is an open subset of the CS set $X$ stratified by $Y^{j}=Y\cap X^{j}$ or* 2. 2.

$Y$* is a CS set and $X=\mathbb{R}^{m}\times Y$ with $X^{j}=\mathbb{R}^{m}\times X^{j-m}$ and we identify $Y$ with $\{z\}\times Y$ for some $z\in\mathbb{R}^{m}$.*

Let $\vec{p},\mathcal{E}$ be a ts-perversity and ts-coefficient system on $X$. Then $\mathcal{P}^{*}|_{Y}\cong\mathcal{P}^{*}_{Y}$, i.e. the restriction of the ts-Deligne sheaf on $X$ to $Y$ is quasi-isomorphic to the $\vec{p}_{Y},\mathcal{E}_{Y}$ ts-Deligne sheaf on $Y$.

Proof.

Note that in both cases both $X$ and $Y$ are CS sets by [GBF35, Lemmas 2.3.13 and 2.11.4]. We know that ts-Deligne sheaves are characterized up to quasi-isomorphic by the axioms TAx1'. That $\mathcal{P}^{*}$ satisfies these axioms on $X$ implies that $\mathcal{P}^{*}|_{Y}$ satisfies the axioms on $Y$. The only axiom that is not obvious is the last axiom when $X=\mathbb{R}^{m}\times Y$. In this case suppose $X$ is $n$-dimensional so that $Y$ is $n-m$ dimensional. Let $x\in Y_{(n-m)-k}\subset X_{n-k}$. Let $f_{x}:\{x\}\hookrightarrow X$ and $g_{x}:\{x\}\hookrightarrow Y$ for $x$ in a singular stratum. Now, exactly as in the proof of Theorem 3.6, if $\pi:\mathbb{R}^{m}\times Y\to Y$ is the projection we have $f^{!}_{x}\mathcal{P}^{*}\cong g_{x}^{!}R\pi_{*}\mathcal{P}^{*}[-m],$ while $R\pi_{*}\mathcal{P}^{*}\cong\mathcal{P}^{*}|_{Y}$ by the proof of Lemma 4.1. Thus

[TABLE]

The axiom for $\mathcal{P}^{*}|_{Y}$ now follows from the axiom for $\mathcal{P}^{*}$. ∎

In what follows we shall abuse notation and write the restrictions $\vec{p}_{Y}$, $\mathcal{E}_{Y}$, and $\mathcal{P}^{*}_{Y}$ as simply $\vec{p}$, $\mathcal{E}$, and $\mathcal{P}^{*}$ if it is clear what is meant from context. We let $\Sigma X$ denote the (unreduced) suspension, stratified by $(\Sigma X)^{i}=\Sigma(X^{i-1})$ with $(\Sigma X)^{0}=\{\mathfrak{n},\mathfrak{s}\}$, the union of the two suspension points.

Proposition 4.4**.**

Let $X^{n-1}$ be a compact CS set with suspension $\Sigma X$. Let $\vec{p}$ be a $ts$-perversity on $\Sigma X$ such that $\vec{p}(\mathfrak{n})=\vec{p}(\mathfrak{s})$, and denote the common value $(p,\wp)$. Let $\mathcal{E}$ be a ts-coefficient system to which $\Sigma X$ is adapted, and let $\mathcal{P}^{*}$ be the associated ts-Deligne sheaf. Then

[TABLE]

Proof.

Let $U_{1}=\Sigma X-\{\mathfrak{s}\}\cong cX$ and $U_{2}=\Sigma X-\{\mathfrak{n}\}\cong cX$. We consider the Mayer-Vietoris sequence [KS, Remark 2.6.10]

[TABLE]

Note that $U_{1}\cap U_{2}\cong(0,1)\times X$ so we have $\mathbb{H}^{i}(U_{1}\cap U_{2};\mathcal{P}^{*})\cong H^{i}(X;\mathcal{P}^{*})$ by Lemma 4.1.

Now consider $\mathbb{H}^{i}(U_{1};\mathcal{P}^{*})$. Since $\mathcal{P}^{*}$ is $\Sigma X$-clc and since $U_{1}$ is a distinguished neighborhood of $\mathfrak{n}$, we have $\mathbb{H}^{i}(U_{1};\mathcal{P}^{*})\cong H^{i}(\mathcal{P}^{*}_{\mathfrak{n}})$ by [Sch03, Proposition 4.0.2.2]. Since this is [math] for $i>p+1$, and similarly for $U_{2}$, we have $\mathbb{H}^{i}(\Sigma X;\mathcal{P}^{*})\cong\mathbb{H}^{i-1}(U_{1}\cap U_{2};\mathcal{P}^{*})\cong\mathbb{H}^{i-1}(X;\mathcal{P}^{*})$ for $i>p+2$.

For $i\leq p$, using [Sch03, Proposition 4.0.2.2] again implies that the hypercohomology map from $U_{1}$ to $U_{1}-\{\mathfrak{n}\}=U_{1}\cap U_{2}$ is isomorphic to the attaching map, which is an isomorphism by axiom TAx1d, and similarly for $U_{2}$. It follows that in this range $\mathbb{H}^{i}(\Sigma X;\mathcal{P}^{*})\cong\mathbb{H}^{i}(U_{j};\mathcal{P}^{*})\cong\mathbb{H}^{i}(U;\mathcal{P}^{*})\cong\mathbb{H}^{i}(X;\mathcal{P}^{*})$ for $j=1,2$.

In the middle range we have {diagram} Once again we have that $\mathbb{H}^{p+1}(U_{1};\mathcal{P}^{*})\cong H^{p+1}(\mathcal{P}^{*}_{\mathfrak{n}})$, which in this case gives the group $T^{\wp}\mathbb{H}^{p+1}(U_{1}\cap U_{2};\mathcal{P}^{*})\cong T^{\wp}\mathbb{H}^{p+1}(X;\mathcal{P}^{*})$ by Lemma 4.2. Each of the maps $\mathbb{H}^{p+1}(U_{j};\mathcal{P}^{*})\to\mathbb{H}^{p+1}(U_{1}\cap U_{2};\mathcal{P}^{*})$ thus corresponds to the inclusion (up to sign) of the $\wp$-torsion subgroup. So we have $\mathbb{H}^{p+1}(\Sigma X;\mathcal{P}^{*})\cong T^{\wp}\mathbb{H}^{p+1}(X;\mathcal{P}^{*})$ and $\mathbb{H}^{p+2}(\Sigma X;\mathcal{P}^{*})\cong\mathbb{H}^{p+1}(X;\mathcal{P}^{*})/T^{\wp}\mathbb{H}^{p+1}(X;\mathcal{P}^{*})$. ∎

Example 4.5*.*

We provide an example of a computation using Proposition 4.4 that also demonstrates how Poincaré duality can fail for CS sets, both for classical homology and cohomology and for ordinary intersection cohomology, while duality is recovered using ts-Deligne sheaves.

Let $X=\mathbb{R}P^{3}$, which is orientable. Then, by standard computations, we have

[TABLE]

This computation demonstrates the failure, in general, of classical Poincaré duality for singular spaces. Even with classical intersection homology, Poincaré duality typically only holds rationally; see [GM1, GM2, Bo] and our remarks below the following computation.

However, let now $\mathcal{E}$ be the constant coefficient system with stalk $\mathbb{Z}$, and let $\vec{p}$ take the value $(1,\emptyset)$ on the suspension points. The dual perversity $D\vec{p}$ will then take the value $(1,P(\mathbb{Z}))$. By [GBF32, Theorem 4.19], we know $\mathcal{D}\mathcal{P}_{\vec{p}}^{*}[-n]\cong\mathcal{P}^{*}_{D\vec{p}}$, which implies by [GBF32, Corollary 4.21] the nonsingular pairings

[TABLE]

where $T\mathbb{H}^{i}$ denotes the torsion subgroup and $F\mathbb{H}^{i}$ denotes the torsion-free quotient. Indeed, we can compute by Proposition 4.4:

[TABLE]

This is consistent with the existence of the claimed nonsingular pairings. By contrast, the first of these hypercohomology computations with ts-perversity $\vec{p}$ agrees with the classical Goresky-MacPherson intersection cohomology for the self dual perversity $\bar{p}$ with $\bar{p}(4)=1$, but we see that these intersection cohomology groups cannot be self-dual. ∎

As a corollary, and as a nice example of an application of Theorem 3.6, we compute $\mathbb{H}^{i}(S^{k}*X;\mathcal{P}^{*})$ for $k>0$, where $X$ is a compact CS set, $S^{k}$ is the $k$-sphere with trivial stratification, and $S^{k}*X$ is the join. Rather than use the join stratification of [GBF35, Section 2.11], however, it will be more natural for us below to use the following stratification. Recall that we can decompose $S^{k}*X$ into $cS^{k}\times X$ and $S^{k}\times cX$ (see [GBF35, Section 2.11]). We give $S^{k}\times cX\subset S^{k}*X$ the stratification it obtains from the cone and product stratifications, while we stratify $cS^{k}\times X$ as $D^{k+1}\times X$, where $D^{k+1}$ is the interior of the unit disk with the trivial stratification. These two stratifications agree on the overlap $S^{k}\times(0,1)\times X$ and so patch together to give a CS set stratification of $S^{k}*X$. Letting $v$ denote the cone vertex, we identify $S^{k}$ with the stratum $S^{k}\times\{v\}\subset S^{k}\times cX\subset S^{k}*X$.

Corollary 4.6**.**

Let $X$ be a compact CS set, let $S^{k}$, $k>0$, be the $k$-sphere with trivial stratification, and let $S^{k}*X$ be the join with the stratification as above. Let $\vec{p}$ be a ts-perversity on $S^{k}*X$, and let $\mathcal{E}$ be a ts-coefficient system to which $S^{k}*X$ is adapted. Then

[TABLE]

Proof.

Let $\Sigma^{k+1}X$ be the $k+1$ times iterated suspension of $X$. Topologically (ignoring stratifications) $\Sigma^{k+1}X\cong S^{k}*X$. Furthermore, the stratification of $\Sigma^{k+1}(X)$ as an iterated suspension refines the stratification of $S^{k}*X$. In fact, the stratifications are identical on $S^{k}*X-S^{k}$, which is $D^{k+1}\times X$ with the product stratification. In particular, no singular stratum of $\Sigma^{k+1}X$ is contained in a regular stratum of $S^{k}*X$. Let $\vec{q}$ be the ts-perversity on $\Sigma^{k}X$ such that $\vec{q}(Z)=\vec{p}(Z)$ for strata $S$ shared by $\Sigma^{k+1}X$ and $S^{k}*X$ and such that $\vec{q}(Z)=\vec{p}(S^{k})$ if $Z$ is a stratum of $\Sigma^{k+1}X$ contained in $S^{k}$. Then $\vec{p}$ and $\vec{q}$ are $\mathcal{E}$-compatible for any $\mathcal{E}$, and so by Theorem 3.6 the $\vec{p}$ ts-Deligne sheaf on $S^{k}*X$ and the $\vec{q}$ ts-Deligne sheaf on $\Sigma^{k+1}X$ are quasi-isomorphic. The computation now follows by applying Proposition 4.4 iteratively. ∎

4.2 Singular strata in regular strata

In this section we will show that if $S$ is a singular stratum of $X$ contained in a regular stratum $\mathbb{S}$ of $\mathbb{X}$ then the conditions of Definition 3.1 for such strata are strictly necessary in order to have $\mathcal{P}^{*}\cong\mathbb{P}^{*}$ over $S$. On $\mathbb{S}$ we have $\mathbb{P}^{*}\cong\mathcal{E}$ so we will assume also that $\mathcal{P}^{*}\cong\mathcal{E}$ over $\mathbb{S}$ and see that contradictions occur if any of the conditions of Definition 3.1 for this scenario fail.

Suppose $x\in S$. Then $x$ has a distinguished neighborhood $\mathbb{R}^{n-k}\times cL$, $k>0$. Topologically this is homeomorphic to $cS^{n-k-1}\times cL\cong c(S^{n-k-1}*L)$. Furthermore, since $x$ is in a regular stratum of $\mathbb{X}$, we have $(c(S^{n-k-1}*L),x)\cong(\mathbb{R}^{n},x)$ by [GBF35, Corollary 2.10.2] and its proof. Thus $H^{*}(S^{n-k-1}*L)\cong H^{*}(S^{n-1})$, and so $L$ must be a $k-1$ homology sphere. We will write $L$ as $L^{k-1}$ to remind us of this.

Returning to our ts-Deligne sheaves, we have $H^{*}(\mathbb{P}^{*}_{x})=H^{*}(\mathcal{E}_{x})$. Since $x$ has a Euclidean neighborhood, $\mathcal{E}$ is clc on our neighborhood of $x$ and so each derived cohomology sheaf $\mathcal{H}^{i}(\mathcal{E})$ is constant on this neighborhood. By assumption there is some $\wp\subset P(R)$ such that $H^{1}(\mathcal{E}_{x})$ is $\wp$-torsion while $H^{0}(\mathcal{E}_{x})$ is $\wp$-torsion free. All other $H^{i}(\mathcal{E}_{x})$ are [math].

On the other hand, by Lemma 4.2,

[TABLE]

Since there is some $\wp$ such that $H^{1}(\mathcal{E}_{x})$ is $\wp$-torsion and $H^{0}(\mathcal{E}_{x})$ is $\wp$-torsion free, any map $H^{0}(L^{k-1},H^{1}(\mathcal{E}_{x}))\to H^{k-1}(L^{k-1},H^{0}(\mathcal{E}_{x}))$ must be trivial, so the hypercohomology spectral sequence for $\mathbb{H}^{*}(L^{k-1};\mathcal{E})$ degenerates (using also that $L^{k-1}$ has the cohomology of a $k-1$ sphere). It follows that if $k=1$ we have $\mathbb{H}^{i}(L^{0},\mathcal{E})\cong H^{i}(\mathcal{E}_{x})\oplus H^{i}(\mathcal{E}_{x})$, and if $k\geq 3$ we have

[TABLE]

Similarly if $k=2$, the only possibly nontrivial groups are for $i=0,1,2$ with $\mathbb{H}^{0}(L^{1};\mathcal{E})\cong H^{0}(L^{1};\mathcal{H}^{0}(\mathcal{E}))\cong H^{0}(\mathcal{E}_{x})$ and $\mathbb{H}^{2}(L^{1};\mathcal{E})\cong H^{1}(L^{1};\mathcal{H}^{1}(\mathcal{E}))\cong H^{1}(\mathcal{E}_{x})$. But the only easy information we have about $\mathbb{H}^{1}(L^{1};\mathcal{E})$ is that it must fit in the extension problem

[TABLE]

Given these preliminaries, we can now consider the necessity of the conditions of Definition 3.1, separately in the cases $k=1$, $k=2$, and $k\geq 3$, by comparing $H^{*}(\mathcal{E}_{x})$ with $H^{*}(\mathcal{P}^{*}_{x})$ as obtained from the preceding computations.

$k=1$.

In this case $H^{i}(\mathcal{P}_{x})$ comes by torsion-tipped truncating $H^{i}(L^{0};\mathcal{E})\cong H^{i}(\mathcal{E}_{x})\oplus H^{i}(\mathcal{E}_{x})$. Thus there is no way that $H^{i}(\mathcal{P}_{x})$ can equal $H^{i}(\mathbb{P}^{*}_{x})\cong H^{i}(\mathcal{E}_{x})$, regardless of perversity unless $\mathcal{E}$ is trivial. This shows why we must always rule out codimension one strata of $X$ in regular strata of $\mathbb{X}$.

$k\geq 3$.

Given the computations above, we can readily see that if $\mathcal{E}$ is not trivial then $H^{*}(\mathcal{P}_{x})\cong H^{*}(\mathcal{E}_{x})$ (and hence $H^{*}(\mathcal{P}_{x})\cong H^{*}(\mathbb{P}_{x})$) if and only one of the following scenarios holds:

$\vec{p}_{1}(S)=-1$, $H^{0}(\mathcal{E}_{x})$ is $\vec{p}_{2}(S)$-torsion, $H^{1}(\mathcal{E}_{x})=0$, 2. 2.

$\vec{p}_{1}(S)=0$, $H^{1}(\mathcal{E}_{x})$ is $\vec{p}_{2}(S)$-torsion, $H^{0}(\mathcal{E}_{x})$ arbitrary, 3. 3.

$1\leq\vec{p}_{1}(S)\leq k-3$, $H^{*}(\mathcal{E}_{x})$ arbitrary, 4. 4.

$\vec{p}_{1}(S)=k-2$, $H^{0}(\mathcal{E}_{x})$ is $\vec{p}_{2}(S)$-torsion free, $H^{1}(\mathcal{E}_{x})$ is arbitrary, 5. 5.

$\vec{p}_{1}(S)=k-1$, $H^{0}(\mathcal{E}_{x})=0$, $H^{1}(\mathcal{E}_{x})$ is $\vec{p}_{2}(S)$-torsion free.

Of course here ``arbitrary'' still means within the limitations of $\mathcal{E}$ being a $\wp$-coefficient system for some $\wp\subset P(R)$, and we have recovered precisely the conditions from Definition 3.1.

$k=2$.

This case is a bit more delicate as we can't pin down $\mathbb{H}^{1}(L^{1};\mathcal{E})$ in general. However, we see again that to have $H^{*}(\mathcal{P}_{x})\cong H^{*}(\mathbb{P}^{*}_{x})\cong H^{*}(\mathcal{E}_{x})$ with $\mathcal{E}$ nontrivial, it is first of all necessary to have $\vec{p}_{1}(S)\geq-1$, and also by the same arguments if $\vec{p}_{1}(S)=-1$, then we must have $H^{1}(\mathcal{E}_{x})=0$, and $H^{0}(\mathcal{E}_{x})$ must be $\vec{p}_{2}(S)$ torsion.

If $\vec{p}_{1}(S)=1=k-1$, then to have $H^{1}(\mathcal{P}_{x})\cong H^{1}(\mathcal{E}_{x})$ requires $\mathbb{H}^{1}(L^{1};\mathcal{E})\cong H^{1}(\mathcal{E}_{x})$, which from the short exact sequence above requires $H^{0}(\mathcal{E}_{x})=0$. And looking at degree $2$ we must again have that $H^{1}(\mathcal{E}_{x})$ is $\vec{p}_{2}(S)$-torsion free.

If $\vec{p}_{1}(S)\geq 2$, then we must have $H^{1}(\mathcal{E}_{x})=0$ for degree $2$ to work but again we also need $\mathbb{H}^{1}(L^{1};\mathcal{E})\cong H^{1}(\mathcal{E}_{x})$ and so $H^{0}(\mathcal{E}_{x})=0$, which forces $\mathcal{E}$ to be trivial.

Finally, suppose $\vec{p}_{1}(S)=0=k-2$. Then to have $H^{*}(\mathcal{P}_{x})\cong H^{*}(\mathbb{P}^{*}_{x})$ we must have $T^{\vec{p}_{2}(S)}\mathbb{H}^{1}(L^{1};\mathcal{E})\cong H^{1}(\mathcal{E}_{x})$. In particular, $H^{1}(\mathcal{E}_{x})$ must be $\vec{p}_{2}(S)$-torsion. Let $k:H^{1}(\mathcal{E}_{x})\hookrightarrow\mathbb{H}^{1}(L^{1};\mathcal{E})$ take $H^{1}(\mathcal{E}_{x})$ isomorphically onto $T^{\vec{p}_{2}(S)}\mathbb{H}^{1}(S;\mathcal{E})$, and consider the composition $qk$, where $q$ is the quotient map in (1). If $z\in H^{1}(\mathcal{E}_{x})$, $z\neq 0$, then $z$ is $\wp$-torsion for some $\wp$ such that $\mathcal{E}$ is a $\wp$-coefficient system, so $qk(z)\neq 0$ or else there would be a $\wp$-torsion element in $\ker(q)\cong H^{0}(\mathcal{E}_{z})$, violating that $\mathcal{E}$ is a $\wp$-coefficient system. Thus $qk$ is injective. Since $H^{1}(\mathcal{E}_{x})$ is a finitely-generated torsion module over a PID, it is Artinian666In fact, since $H^{1}(\mathcal{E}_{x})$ is torsion, we can treat it as a module over $R/\text{Ann}(H^{1}(\mathcal{E}_{x}))$. If $H^{1}(\mathcal{E}_{x})\neq 0$ then Ann$(H^{1}(\mathcal{E}_{x}))\neq 0$, and this is an Artinian ring since $R$ is a PID. If $H^{1}(\mathcal{E}_{x})=0$, it is clearly Artinian. and so every injective endomorphism is an isomorphism [RIB, Lemma II.4.$\alpha$]. Precomposing $k$ with the inverse of this isomorphism provides a splitting of (1). Consequently we also have $H^{0}(\mathcal{E}_{x})\cong\mathbb{H}^{1}(L^{1};\mathcal{E})/T^{\vec{p}_{2}(S)}\mathbb{H}^{1}(L^{1};\mathcal{E})$, and so $H^{0}(\mathcal{E}_{x})$ must be $\vec{p}_{2}(S)$-torsion free.

4.3 Singular strata in singular strata

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 $\mathbb{S}$ of $\mathbb{X}$. As noted above, these conditions won't always be necessary in the strictest sense since weaker conditions might suffice depending on the local cohomology computations resulting from specific choices of spaces, perversities, and coefficient systems. Instead, we show the conditions to be ``necessary in general,'' meaning that we will demonstrate the existence of examples where $\mathcal{P}^{*}$ is not quasi-isomorphic to $\mathbb{P}^{*}$ because the conditions fail. Accordingly, we can choose to work in relatively simple settings.

We first discuss necessity in the codimension [math] setting, followed by the codimension $>0$ case. For simplicity, we assume that our ts-perversities are efficient; see Remark 3.5. This allows us to focus on the degree ranges where the conditions of Definition 3.1 may be relevant rather than cases that are degenerate due to extreme degree values.