Local Cohomology and Stratification
Abstract
We outline an algorithm to recover the canonical (or, coarsest) stratification of a given finitedimensional regular CW complex into cohomology manifolds, each of which is a union of cells. The construction proceeds by iteratively localizing the poset of cells about a family of subposets; these subposets are in turn determined by a collection of cosheaves which capture variations in cohomology of cellular neighborhoods across the underlying complex. The result is a nested sequence of categories, each containing all the cells as its set of objects, with the property that two cells are isomorphic in the last category if and only if they lie in the same canonical stratum. The entire process is amenable to efficient distributed computation.
Keywords
Canonical stratification Local cohomologyMathematics Subject Classification
52S60 55N30 18E351 Introduction
The twodimensional singular space below—let us call it \(\mathrm {Y}\)—is built by pinching a torus along a meridian and attaching a disk across an equator:
Any regular CW structure, such as the illustrated decomposition into little squares, constitutes a stratification of \(\mathrm {Y}\) where ddimensional strata are precisely the dcells; passing to a subdivision further refines this stratification in the sense that every new cell is entirely contained in the interior of an old cell. On the other hand, one can discover a much coarser stratification by examining the topology of small neighborhoods around points of \(\mathrm {Y}\). Up to homeomorphism, these fall into three different classes depending on whether the central point is at the pinch, on the singular equator, or on one of the manifoldlike twodimensional regions:
The neighborhoods above deformation retract onto their central vertices and are therefore contractible; however, their onepoint compactifications (obtained by collapsing their boundaries to points) are new stratified spaces with potentially interesting topology. The compactified neighborhood around the pinch point is homeomorphic to two 2spheres joined at their north and south poles with a spanning disk across the middle. The compactified neighborhood of any point in the singular equator resembles a 2sphere whose interior has been partitioned into two by a disk. And finally, the compactified neighborhood around any point in either of the twodimensional regions is homeomorphic to an ordinary 2sphere:
Our main result here involves algorithmically recovering the coarsest stratification of a finite regular CW complex where all strata are unions of cells. This is called the canonical stratification of the complex; its existence and uniqueness for a special class of spaces (called pseudomanifolds) plays a central role in Goresky and MacPherson’s proof of the topological invariance of intersection homology [14, Sec 4]. Our argument, much like theirs, has an intuitive geometric core but invokes algebraic and categorical machinery. For the purposes of this introductory section, we focus on geometry and ask: Given a finite cellulation of \(\mathrm {Y}\), how might one identify the canonical strata and determine which cells lie in each canonical stratum, as shown below?
(Since the pinch point has a different compactified neighborhood than a generic point on the singular equator, it must constitute a separate stratum.)
In light of the discussion above, one hopes to recover canonical strata by clustering together cells whenever they exhibit similar compactified neighborhoods. But already in this example, we encounter two significant difficulties: First, the compactified neighborhoods do not distinguish cells in the two 2strata from each other. The second difficulty is somewhat subtler—although the compactified neighborhoods of cells in the 0stratum and 1stratum are not homeomorphic, both are homotopyequivalent to a wedge of two 2spheres. Therefore, they cannot be distinguished by weaker, more computable topological invariants such as cohomology. We tackle the first problem by constructing a complex of cosheaves which encodes how local topology varies across cells, and we bypass the second problem by working exclusively in the category of cohomologically stratified spaces.
Theorem 1.1
There is a category \(\varvec{S}\) obtained from \({{\mathbf{Fc}}}(\mathrm {X})\) by formally inverting a particular subset of those face relations \(x \ge y\) for which \(\varvec{L}^\bullet (x \ge y)\) induces isomorphisms on cohomology; two cells lie in the same canonical stratum of \(\mathrm {X}\) if and only if they are isomorphic in \(\varvec{S}\).
This paper is organized as follows: Sect. 2 tersely collects relevant background material involving stratifications and cosheaves, while Sect. 3 describes the local cohomology complex of cosheaves \(\varvec{L}^\bullet \) associated with a regular CW complex \(\mathrm {X}\). In Sect. 4 we use \(\varvec{L}^\bullet \) to define an initial functor \({{\mathbf{Fc}}}(\mathrm {X}) \rightarrow \varvec{S}_0\) via categorical localization and prove that the topdimensional canonical strata of \(\mathrm {X}\) correspond bijectively with isomorphism classes of (images of) topcells in \(\varvec{S}_0\). Section 5 contains the heart of our argument: It concludes the proof of Theorem 1.1 by describing the inductive construction of \({{\mathbf{Fc}}}(\mathrm {X}) \rightarrow \varvec{S}_{d}\) from the previous \({{\mathbf{Fc}}}(\mathrm {X}) \rightarrow \varvec{S}_{d1}\). The final Sect. 6 may be read independently of the preceding ones—it is devoted to computational matters and includes an efficient distributed algorithm for recovering canonical strata in practice.
1.1 Context
Our goal is to provide a principled topological preprocessor for various manifold learning algorithms. The fundamental aim of manifold learning is to automatically infer the intrinsic dimension of a compact submanifold M (often with boundary) of Euclidean space from a finite point sample \(P \subset \mathbb {R}^n\) lying on, or at least near, M. An enormous amount of work has been done in this field, and we will make no attempt to even summarize it here—the curious reader is referred to [19] and its myriad citations. In most cases, one constructs a graph around the points in P by inserting all edges of Euclidean length bounded above by a parameter threshold \(\epsilon > 0\). Shortest paths along this graph then serve as proxies for intrinsic geodesics of M and are used to find distancepreserving projections to lowerdimensional subspaces of \(\mathbb {R}^n\) via spectral methods.
When M is not a single manifold, but rather a union of different manifolds across several dimensions, our standard manifold learning methods necessarily result in overfitting along regions where M has low intrinsic dimension. Theorem 1.1 provides a remedy by conferring the ability to partition the Delaunay triangulation [23, Ch 5] around P at radius \(\epsilon \) into distinct clusters, each of which is guaranteed to lie on a cohomology manifold of known dimension.^{1} These clusters may then be independently subjected to standard manifold learning techniques. Perhaps more important from a practical perspective is the fact that the computations involved in partitioning P are easily distributed across several processors. And as a straightforward byproduct, one also gains the capacity to identify (those points of P which lie in) every anomalous, singular region of M.
There now exists a substantial body of work where filtered (co)homology groups of cell complexes built around data points play a central role [3, 8, 12]. With the knowledge of canonical strata of such cell complexes comes the prospect of efficiently computing far more refined topological invariants such as intersection cohomology [13] groups. Also accessible, thanks to suitable flavors of Morse theory [21, 22], is the ability to cluster canonical dstrata by \((d+1)\)dimensional cobordisms internal to the ambient dataset.
2 Preliminaries
Our primary references for stratification theory are [15, 18], and [30]; for categorical localization see [7] and [10]; and for cellular (co)sheaves see [4] and [25]. The interplay between cell structures and general stratifications on the same underlying space has been explored thoroughly in [26]. Here we will be concerned entirely with cohomologically stratified spaces, which the reader may have encountered before in [14, Sec 4.1], [11, Sec 3.3] or [24, Sec 5].
2.1 Cohomological Stratifications
Definition 2.1

Frontier if a stratum \(\sigma \) intersects the closure of another stratum \(\tau \), then in fact \(\sigma \) is completely contained in the closure of \(\tau \) and \(\dim \tau \ge \dim \sigma \) (with equality occurring if and only if \(\tau = \sigma \)). This relation, denoted by \(\tau \succeq \sigma \) henceforth, forms a partial order on the set of all strata.
 Link for each dstratum \(\sigma \), there exists an \((nd1)\)dimensional stratified space \(\mathrm {L}= \mathrm {L}(\sigma )\), called the link of \(\sigma \):so that every open neighborhood around a point p in \(\sigma \) admits a basic open subneighborhood \(\mathrm {U}_p \subset \mathrm {X}\) with the following structure. The intersections \(\mathrm {U}_p \cap \mathrm {X}_i\) are empty for \(i < d\), while for \(d \le i \le n\) there are stratified quasiisomorphisms of compactly supported singular cochain complexes (of Rmodules): where \(\mathscr {C}\mathrm {L}_\bullet \) denotes the open cone^{2} on \(\mathrm {L}_\bullet \).$$\begin{aligned} \varnothing = \mathrm {L}_{1} \subset \mathrm {L}_0 \subset \cdots \subset \mathrm {L}_{nd1} = \mathrm {L}, \end{aligned}$$
 (1)They should respect (contravariant maps induced by) the inclusions \(\mathrm {L}_\bullet \subset \mathrm {L}_{\bullet +1}\) in their domains and \(\mathrm {X}_\bullet \subset \mathrm {X}_{\bullet +1}\) their codomains so that the following squares commute:
 (2)They should preserve strata in the sense that there exist surjective setmapsso that for each istratum \(\tau \succeq \sigma \), the cochain maps from (2) restrict to quasiisomorphisms$$\begin{aligned} \Phi _i: \left\{ (id1)\text {strata of } \mathrm {L}\right\} \twoheadrightarrow \left\{ i\text {strata } \tau \succeq \sigma \text { of } \mathrm {X}\right\} \end{aligned}$$$$\begin{aligned} {C}_{\mathrm{c}}^\bullet (\mathscr {C}\Phi ^{1}_i(\tau ) \times \mathbb {R}^d) {\mathop {\longrightarrow }\limits ^{\simeq }} {C}_{\mathrm{c}}^\bullet (\mathrm {U}_p \cap \tau ). \end{aligned}$$
 (3)They should refine to smaller basic neighborhoods \(\mathrm {V}_p\) so that the following triangle of cochain maps commutes: Here the vertical map is induced by the inclusion \(\mathrm {V}_p \subset \mathrm {U}_p\), and since the triangle commutes, this map is forced to also be a quasiisomorphism.
Example 2.2
The decomposition of the space \(\mathrm {Y}\) from the Introduction into the pinch point, the equatorial circle, a disk 2stratum and a toral 2stratum is a cohomological stratification. The pinch point lies at the frontier of the equatorial circle, which in turn lies at the frontier of both 2strata. The link \(\mathrm {L}\) of the singular equator (which we will call \(\sigma \) here) is a threepoint space. An open cone on \(\mathrm {L}\) is therefore the union of three halfopen intervals along a common boundary point. The product of this cone with the real line is homeomorphic to small neighborhoods around points in \(\sigma \):

filtered, provided we stratify the open cone \(\mathscr {C}\mathrm {L}\) with one 0stratum and three 1strata (and the real line by a single 1stratum),

stratumpreserving, provided that the map \(\Phi _2\) sends two of the points in \(\mathrm {L}\) to the toral 2stratum and the third point to the disk 2stratum, and

refinable, because one can shrink the neighborhood around \(\sigma \) by a small amount while still preserving the homeomorphism.
Remark 2.3
To see a nonexample of a cohomological stratification, try combining the pinch point p of \(\mathrm {Y}\) with the singular equator \(\sigma \) into a single 1stratum. The refinability constraint will not be satisfied in this case. In particular, given any point \(q \in \sigma \) near p with two basic neighborhoods \(V_q \subset U_q\) satisfying \(p \in U_q  V_q\), the inclusion of \(V_q\) into \(U_q\) fails to induce an isomorphism on twodimensional compactly supported cohomology. Instead, one obtains a rank one map \(R^2 \rightarrow R^2\).
 (1)
By a theorem of Eilenberg and Zilber [9], links are well defined (only) up to filtered quasiisomorphism type: We may replace \(\mathrm {L}\) by a filtered cochain complex by substituting a tensor product for the left side of (2).
 (2)
The link axiom (when \(i = d\)) guarantees that each dstratum is indeed a ddimensional Rcohomology manifold.
 (3)
This definition, unlike similar notions which typically appear in the intersection homology literature, does not require \(\mathrm {X}\) to be a pseudomanifold (see [14, Sec 1.1] for instance). In other words, \(\mathrm {X}_{n1}\) need not equal \(\mathrm {X}_{n2}\).
Definition 2.4

an open star \({\mathbf{st}}(y)\) containing all cells x which satisfy \(x \ge y\), and

a link \({\mathbf{lk}}(y)\), containing all cells x that share a coface but no face with y.
We call one stratification a coarsening of another whenever each stratum of the former is a union of strata of the latter. All stratifications encountered henceforth will be coarsenings of the skeletal stratification for a fixed finitedimensional regular CW complex \(\mathrm {X}\).
Definition 2.5
The canonical stratification of a finitedimensional regular CW complex \(\mathrm {X}\) is the coarsest stratification of \(\mathrm {X}\) whose strata are all unions of cells.
It may not be immediately clear why canonical stratifications should exist at all. We will establish both their existence and uniqueness for regular CW complexes in the sequel via an explicit construction, which can be made efficiently algorithmic whenever the number of cells in \(\mathrm {X}\) is finite. The core of our construction is heavily inspired by Goresky and MacPherson’s proof that canonical\(\bar{p}\)filtrations of stratified pseudomanifolds exist and are unique [14, Sec 4.2].
Remark 2.6
 (1)
since our starting point is a regular CW complex rather than a stratified pseudomanifold, we have direct recourse to the combinatorics of cell incidences which determine the topology of the underlying space, and
 (2)
here we use the sheaf of compactly supported cellular cochains on the underlying space rather than the sheaf of intersection chains with respect to a perversity function \(\bar{p}:\mathbb {Z}_{\ge 2} \rightarrow \mathbb {Z}\).
The following result is a direct consequence of the frontier axiom from Definition 2.1 for cells lying in top strata.
Proposition 2.7
Let \(\mathrm {X}\) be a regular CW complex of dimension n equipped with any stratification coarser than its skeletal stratification. If a cell y of \(\mathrm {X}\) lies in an ndimensional stratum \(\sigma \), then so must every cell x which satisfies \(x \ge y\).
Proof
Let \(\tau \) be the unique coarse stratum containing x. Since y lies in the boundary of x by assumption, the closure of \(\tau \) intersects \(\sigma \) nontrivially at y. The desired conclusion now follows from the axiom of the frontier in Definition 2.1 and the fact that there are no strata of dimension exceeding n, since \(\dim \tau \ge \dim \sigma = n\) forces \(\tau = \sigma \). \(\square \)
Thus, membership of cells in topdimensional canonical strata is upwardclosed with respect to the face partial order.
2.2 Localizations of Posets
The localization of a poset \(\varvec{P}\) about a subcollection \(\Sigma \) of its order relations is the minimal category containing \(\varvec{P}\) in which the relations of \(\Sigma \) have been rendered invertible [10, Ch I.1].
Definition 2.8
 (1)
only relations in \(\Sigma \) and equalities can point backward (i.e., \(\le \)),
 (2)
composition is given by concatenation, and
 (3)
the trivial zigzag \(p = p\) represents the identity of each element p.
 horizontally if one is obtained from the other by removal of intermediate equalities:$$\begin{aligned} \left( \cdots \le x \ge y = y \ge x' \le \cdots \right)&\sim \left( \cdots \le x \ge x' \le \cdots \right) , \\ \left( \cdots \ge y \le x = x \le y' \ge \cdots \right)&\sim \left( \cdots \ge y \le y' \ge \cdots \right) , \end{aligned}$$
 or vertically, if they form the rows in a grid: where all vertical face relations (also) lie in \(\Sigma \).
The category \(\varvec{P}[\Sigma ^{1}]\) enjoys the following universal property [10, Lem I.1.3]—there is a canonical localization functor\(\varvec{P}\rightarrow \varvec{P}[\Sigma ^{1}]\) which acts by inclusion. In particular, it sends each element of \(\varvec{P}\) to itself and each relation \(x \ge y\) to its own equivalence class of \(\Sigma \)zigzags. Moreover, any functor \(\varvec{P}\rightarrow \varvec{C}\) which maps order relations in \(\Sigma \) to isomorphisms in the target category \(\varvec{C}\) factorizes uniquely across the localization functor.
2.3 Cellular Cosheaves
Sheaves (and their dual cosheaves) assign algebraic objects to open sets of topological spaces [2]. Our interest here is in a particularly tame and computable subclass of cosheaves, where the base space is always a finitedimensional regular CW complex \(\mathrm {X}\) and the assigned algebraic objects are cellwise constant. We write \({{\mathbf{Fc}}}(\mathrm {X})\) to indicate the poset of cells with order relation \(x > y\) denoting that y is a face of x.
Definition 2.9

\(\varvec{F}(x = x)\) is the identity on \(\varvec{F}(x)\) for each cell x, and

\(\varvec{F}(y \ge z) \circ \varvec{F}(x \ge y)\) equals \(\varvec{F}(x \ge z)\) across any triple of cells \(x \ge y \ge z\).
Definition 2.10
3 Local Cohomology of CW Complexes
Let \(\mathrm {X}\) be a finitedimensional regular CW complex with face poset \({{\mathbf{Fc}}}(\mathrm {X})\) and R a fixed nontrivial commutative ring with identity \(1_R\). We write \({\mathbf{Mod}}(R)\) to denote the category of Rmodules, and \({{\mathbf{Ch}}}(R)\)—rather than the cumbersome \({{\mathbf{Ch}}}({\mathbf{Mod}}(R))\)—to indicate the category of cochain complexes of Rmodules indexed by the nonnegative natural numbers.
Definition 3.1
 (1)For each dimension \(d \ge 0\) and cell x of \(\mathrm {X}\), the cosheaf \(\varvec{L}^d\) has as its stalk \(\varvec{L}^d(x)\) the free Rmodule with basisWhen \(x \ge y\), the extension \(\varvec{L}^d(x \ge y):\varvec{L}^d(x) \hookrightarrow \varvec{L}^d(y)\) is determined by the obvious inclusion of basis cells.$$\begin{aligned} \left\{ z \in {{\mathbf{Fc}}}(\mathrm {X}) \mid z \ge x \text { and } \dim z = d\right\} . \end{aligned}$$
 (2)The cosheaf morphism \(\beta ^d\) assigns to each cell x the map \(\beta ^d_x:\varvec{L}^d(x) \rightarrow \varvec{L}^{d+1}(x)\) defined by (linearly extending) the following action on basis cells. For each dcell \(z \ge x\), we havewhere the sum is taken over all \((d+1)\)cells \(w \ge x\), and \(\langle {w,z}\rangle _R\) is the Rvalued degree of the attaching map in \(\mathrm {X}\) from the boundary of w onto z. (Since we have assumed that \(\mathrm {X}\) is regular, this number \(\langle {w,z}\rangle _R\) takes values in \(\{0,\pm 1_R\}\) for all cells w and z.)$$\begin{aligned} \beta ^d_x(z) = \sum _{w} \langle {w,z}\rangle _R \cdot w, \end{aligned}$$
Definition 3.2
By Definition 2.1, if a cell x lies in some topdimensional stratum of \(\mathrm {X}\), then we must have an isomorphism \(H^\bullet \varvec{L}(x) \simeq H^\bullet R[\dim \mathrm {X}]\). For cells of high dimension, this requirement has strong consequences.
Proposition 3.3
If \(n = \dim \mathrm {X}\), then every ncell x has \(\varvec{L}^\bullet (x) = R[n]^\bullet \). And moreover, every \((n1)\)cell y with \(H^\bullet \varvec{L}(y) \simeq H^\bullet R[n]\) lies in the boundary of exactly two ncells.
Proof
Example 3.4
Consider the twodimensional simplicial complex depicted below: It is obtained by subdividing a parallelogram into four 2simplices along diagonals and attaching an extra 2simplex along one of the four resulting halfdiagonals.
4 Extracting Top Strata

dimensionality retain only those face relations in which both cells have the local cohomology (isomorphic to that) of a topdimensional cell, and

upwardclosure remove \((x \ge y)\) if there exists some \(x' \ge y\) for which \(\varvec{L}^\bullet (x' \ge y)\) is not a quasiisomorphism. (Recall that cells lying in top strata are upwardclosed by Proposition 2.7.)
Definition 4.1
The category \(\varvec{S}_0\) is the localization of the face poset \({{\mathbf{Fc}}}(\mathrm {X})\) about \(W_0\).
Recall that \(\varvec{S}_0\) has the cells of \(\mathrm {X}\) as objects, while its morphisms are equivalence classes of \(W_0\)zigzags (as described in Definition 2.8). And, there is a canonical functor \({{\mathbf{Fc}}}(\mathrm {X}) \rightarrow \varvec{S}_0\) which is universal with respect to rendering all the face relations from \(W_0\) invertible.
Proposition 4.2
Proof
Here is the main result of this section.
Proposition 4.3
Two ndimensional cells lie in the same canonical nstratum of \(\mathrm {X}\) if and only if they are isomorphic in \(\varvec{S}_0\).
Proof
Finally, note that if an arbitrary cell y lies in a (not necessarily canonical) nstratum \(\sigma \) of \(\mathrm {X}\), then there must exist an ncell \(w \ge y\) also lying in \(\sigma \) by Proposition 2.7; otherwise, we arrive at the contradiction \(H^n\varvec{L}(y) = 0\). Thus, we have the following consequence of Proposition 4.3.
Corollary 4.4
The canonical nstrata of \(\mathrm {X}\) correspond bijectively with isomorphism classes of its ncells in \(\varvec{S}_0\).
5 Uncovering Lower Strata
Definition 5.1
Note that \(W_\bullet \) is an increasing sequence of sets while both \(\mathrm {Y}_\bullet \) and \(E_\bullet \) are decreasing.
Example 5.2
To motivate the sets which appear in Definition 5.1, consider the twodimensional space \(\mathrm {Y}_0\) pictured below: It is the closed cone over a wedge of two circles.
Remark 5.3
 (1)
To see that \(\mathrm {Y}_d \subset \mathrm {Y}_{d1}\) is a regular CW subcomplex, note that if a cell y lies in the difference \(\mathrm {Y}_{d1}  \mathrm {Y}_{d}\), then we have \((y = y) \in W_{d1}\). And given any cell \(x \ge y\) in \(\mathrm {Y}_{d1}\), we have \((x \ge y) \in W_{d1}\) by upwardclosure, whence \((x = x) \in W_{d1}\). Thus, x also lies in the difference \(\mathrm {Y}_{d1}  \mathrm {Y}_{d}\). Since the collection of cells removed from \(\mathrm {Y}_{d1}\) to obtain \(\mathrm {Y}_d\) is upwardclosed with respect to the face partial order, \(\mathrm {Y}_d\) is a regular CW subcomplex of \(\mathrm {Y}_{d1}\).
 (2)
To see that \(\dim \mathrm {Y}_d \le (nd)\), assume \(\dim \mathrm {Y}_{d1} \le (nd+1)\). But now, for each cell w in \(\mathrm {Y}_{d1}\) of dimension \((nd+1)\), we have \((w = w) \in W_{d1}\) by Proposition 3.3, so w is not in \(\mathrm {Y}_d\). Since \(\mathrm {Y}_d\) contains no cells of dimension \((nd+1)\), its dimension cannot exceed \((nd)\).
Definition 5.4
For each \(d > 0\) in [n], the category \(\varvec{S}_d\) is the localization of \({{\mathbf{Fc}}}(\mathrm {X})\) about the set \(W_d\) from (7).
Lemma 5.5
Proof
The cells of \(\mathrm {X}\) are ordered by dimension in the sense that the poset \({{\mathbf{Fc}}}(\mathrm {X})\) is only allowed to have morphisms from higherdimensional to lowerdimensional cells. Since morphisms in \(\varvec{S}_\bullet \) are zigzags, one cannot expect such a dimensional monotonicity to hold verbatim. But the \(W_\bullet \)induced monotonicity from the previous lemma precludes the existence of certain morphisms in \(\varvec{S}_\bullet \).
Corollary 5.6
Given \(i \le j\) in [n] along with cells w in \(\mathrm {Y}_j\) and z in \(\mathrm {Y}_i  \mathrm {Y}_j\), there are no morphisms in \(\varvec{S}_d\) from w to z for any d in [n].
Proof
As a direct consequence of the preceding result, isomorphism classes in \(\varvec{S}_i\) of \((ni)\)dimensional cells from \(\mathrm {Y}_i\) remain unchanged across all the inclusions \(\varvec{S}_i \hookrightarrow \varvec{S}_j\) for \(i \le j\). The next result gives convenient alternate descriptions of such isomorphism classes across the entire sequence \(\varvec{S}_\bullet \).
Proposition 5.7
 (1)
lie in the same connected component of \(\mathrm {Y}_d  \mathrm {Y}_{d+1}\),
 (2)
are connected by a \(W_d\)zigzag, whose relations (\(\le \) and \(\ge \)) all lie in \(W_d  W_{d1}\),
 (3)
are isomorphic in \(\varvec{S}_d\) to a common \((nd)\)dimensional cell from \(\mathrm {Y}_d\).
Proof
Our next result describes the canonical strata of \(\mathrm {X}\) as isomorphism classes in \(\varvec{S}_\bullet \).
Proposition 5.8
For each \(d \in [n]\) and canonical \((nd)\)stratum \(\sigma \) of \(\mathrm {X}\), every cell lying in \(\sigma \) is isomorphic in \(\varvec{S}_d\) to some \((nd)\)cell from \(\mathrm {Y}_d\).
Proof
The following result establishes the converse of Proposition 5.8.
Proposition 5.9
For each d in [n], the isomorphism class in \(\varvec{S}_d\) of any \((nd)\)dimensional cell from \(\mathrm {Y}_d\) is a canonical \((nd)\)stratum of \(\mathrm {X}\).
Proof
Finally, to see that \(\sigma \) is canonical, we use Proposition 5.8 to pick an \((nd)\)cell w from \(\mathrm {Y}_d\) so that all cells in \(\sigma \) are isomorphic in \(\varvec{S}_d\) to w. Proposition 5.7 now guarantees that no additional cells may be added to \(\sigma \) while preserving its connectedness. \(\square \)
Theorem 5.10
Given a finite regular ndimensional CW complex \(\mathrm {X}\), let \(\mathrm {Y}_\bullet \) and \(W_\bullet \) be as in Definition 5.1, and let \(\varvec{S}_\bullet \) denote the localization of the face poset \({{\mathbf{Fc}}}(\mathrm {X})\) about \(W_\bullet \). For each \(d \in [n]\), the canonical \((nd)\)strata of \(\mathrm {X}\) correspond bijectively with isomorphism classes of \((nd)\)dimensional cells from \(\mathrm {Y}_d\) in \(\varvec{S}_k\) for any \(k \ge d\).
The morphisms in the last category \(\varvec{S}= \varvec{S}_n\) also recover the frontier partial order among canonical strata.
Proposition 5.11
The frontier relation \(\tau \succeq \sigma \) holds among two canonical strata of \(\mathrm {X}\) if and only if there is a morphism in \(\varvec{S}\) from a cell lying in \(\tau \) to a cell lying in \(\sigma \).
Proof
The argument above shows that the zigzag paths which form morphisms in \(\varvec{S}\) may remain indefinitely in a single stratum or descend to lowerdimensional adjacent strata, but they can never ascend to higher strata. In other words, \(\varvec{S}\) is a (cellular, 1categorical) version of the entrance path category associated with the canonical stratification of \(\mathrm {X}\)—see [21, Def 3.1], [27, Sec 7] or [32, Sec 2].
6 Algorithms
From a computational perspective, the local cohomology \(\varvec{L}^\bullet \) of a finite regular CW complex enjoys the obvious, but enormous, advantage of being local. One only ever needs to construct cochain complexes corresponding to open stars of cells (rather than holding the entire complex in system memory). Since each such starcomplex may be processed independently of the others, the construction of \(\varvec{L}^\bullet \) and the computation of \(H^\bullet \varvec{L}\) are inherently distributable operations.
6.1 Subroutines
The next required subroutine is Cohom, which accepts a poset of cells as input and returns the sequence of Rmodules corresponding to the cohomology of the associated cochain complex. When R is a field, the cohomology modules are all vector spaces and in this case the output can just be a sequence of integers storing their dimensions. If R is the ring of integers, then one has to encode the torsion subgroups (if any) in addition to the ranks of the free parts in every dimension. In either case, since the efficient computation of (co)homology has been extensively discussed elsewhere [6, 16, 17, 20], we will treat this subroutine as a black box and not explicitly write it down here.
6.2 Description
The main loop (in line 01) increments the current codimension d from 0 up to n; as it executes each iteration, the cells lying on \((nd)\)dimensional canonical strata are identified and removed from \(\mathrm {X}\), so that in the dth iteration we pass from \(\mathrm {Y}_d\) to \(\mathrm {Y}_{d+1}\) in the language of Definition 5.1. There are three secondary loops (at lines 02, 04 and 09); all three are trivially distributable across several processors, as indicated by the dist comments in those lines. The first loop in line 04 is easiest to explain: It uses our subroutines UpSet and Cohom to compute the compactly supported cohomology of each open star \({\mathbf{st}}(w) \cap \mathrm {Y}_d\) where w is a cell in \(\mathrm {Y}_d \subset \mathrm {X}\).
The second loop from line 04 computes whether or not inclusions of open stars \({\mathbf{st}}(x) \subset {\mathbf{st}}(y)\) for \(x >_1 y\) induce isomorphisms on compactly supported cohomology by examining their cokernels. In other words, we use the fact that the map induced by \(\varvec{L}_d^\bullet (x >_1 y)\) (namely, the inclusion \({\mathbf{st}}(x) \cap \mathrm {Y}_d \hookrightarrow {\mathbf{st}}(y) \cap \mathrm {Y}_d\)) on compactly supported cohomology is an isomorphism if and only if the difference \(({\mathbf{st}}(y)  {\mathbf{st}}(x)) \cap \mathrm {Y}_d\) has trivial compactly supported cohomology. We begin by assigning to each face relation \((x >_1 y)\) the number \(\ell (x >_1 y) = 1\). As the algorithm executes, \(\ell (x >_1 y)\) gets incremented to the largest integer k so that \((x >_1 y)\) lies in \(E_k\), where \(E_\bullet \) denotes the sets from (6). Since this loop is completely independent of the loop in line 02, it may be executed simultaneously with that loop.
The third loop, which spans lines 09 through 12, must be executed after the first two intermediate loops have terminated. It uses the cohomology groups \(h^\bullet \) computed by the loop in line 02 and the \(\ell \)values computed by the loop in line 04 to select cells lying on canonical strata of dimension \((nd)\) via (7). Note that line 10 ensures the correct dimensionality and line 11 enforces upwardclosure. Line 13 removes all cells lying on canonical \((nd)\)strata from \(\mathrm {Y}_d\), so we are left with \(\mathrm {Y}_{d+1}\) and the outer loop of line 01 enters its next iteration.
6.3 Complexity
The third intermediate loop (on line 09) may be reinterpreted as two nested loops: an outer iloop which decrements dimension from \((nd1)\) to 0, and an inner uloop which examines all remaining cells of dimension i. Since we require knowledge of codim values of all cells \(v >_1 u\) while processing each u, only the uloop is actually distributable. During each iteration of the uloop, one incurs an O(p) cost while checking all remaining cells \(v >_1 u\) in line 11, and an Rdependent cost of testing Rmodule isomorphisms in line 10. At least when R is a finite field, the rationals or the integers, checking whether a given module has rank one or zero incurs O(1) cost. Thus, in typical cases we expect line 11 to dominate the burden of executing the uloop. Noting that the iloop runs at most n times regardless of d, each iteration of the loop on line 09 incurs a computational cost of O(np).
Since all cells with codim\( = d\) are identified on lines 08 and 12 in a given iteration of the outer dloop of StratCast, we assume that removing them from \(\mathrm {X}\) incurs no additional cost. Putting all our estimates together, each iteration of the dloop incurs a worstcase cost of \(O(p^3 + np)\). Noting that this loop executes exactly \((n+1)\) times, the total complexity of running StratCast—assuming that the number of available processors exceeds the number of cells m plus the maximum number mp of codimensionone face relations in \(\mathrm {X}\)—is \(O\left( (n+1)(p^3+np)\right) \). And once StratCast has terminated, one may find all desired canonical strata of \(\mathrm {X}\) by computing connected components of cells with the same codim value. This is linear in the number of cells in \(\mathrm {X}\), so we obtain the following result.
Proposition 6.1
 (1)
ring operations in R incur O(1) cost,
 (2)
isomorphismtesting against 0 and R in \({\mathbf{Mod}}(R)\) is O(1), and
 (3)
the number of available processors exceeds \(m(p+1)\).
The first two conditions above are satisfied by typical choices of R (such as the integers or finite fields). In practice, one expects to have \(m \gg (n+1)(p^3+np)\), so the observed cost of computing canonical strata is essentially linear in the size of \(\mathrm {X}\).
Footnotes
 1.
The probabilistic version of such a framework is described in [1].
 2.
The open cone on \(\mathrm {Z}\) is the quotient of \(\mathrm {Z}\times [0,1)\) by the relation which identifies (z, 0) with \((z',0)\) for any z and \(z'\) in \(\mathrm {Z}\); by convention, the open cone on the empty set is the onepoint space.
 3.
Regularity in this context means that the characteristic map of every cell is an embedding.
 4.
That is, if \((p \ge q)\) and \((q \ge r)\) both lie in \(\Sigma \), then so does \((p \ge r)\).
Notes
Acknowledgements
I am indebted to Justin Curry, Rob Ghrist and Ulrike Tillmann for insightful discussions, and to Adam Brown and Yossi Bokor for carefully reading (and discovering oversights in) earlier drafts of this paper. I am also grateful to the two anonymous referees for their corrections, comments and suggestions for improvement. This work was partially supported by The Alan Turing Institute under the EPSRC Grant EP/N510129/1 and by the Friends of the Institute for Advanced Study. It is dedicated to Mark Goresky and Bob MacPherson, whose efforts over four decades have built some of my favorite playgrounds.
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