Stratified Cell Complexes

• Stratified cell complexes are topological spaces that integrate cellular decompositions with poset stratification, capturing both combinatorial and singular features.
• They enable the construction of face categories and classifying spaces, providing a combinatorial framework for analyzing intricate topological invariants.
• Applied in configuration spaces, Morse theory, and intersection cohomology, these complexes offer precise models for complex singular and stratified phenomena.

A stratified cell complex is a topological space equipped with both a cell complex structure and a compatible stratification by a poset, thereby unifying the combinatorics of cellular decompositions with the singularity theory of stratified spaces. This synthesis supports and extends fundamental constructions in topology, homotopy theory, and singularity theory; it allows cell-wise and strata-wise analysis and provides combinatorial models for intricate topological features of singular and moduli spaces, as exemplified in configuration spaces, stratified Morse theory, and intersection space cohomology (Tamaki, 2011, Tamaki, 2016, Waas, 2024, Knudson et al., 2018, Banagl, 2011).

1. Axiomatic Foundations of Stratified Cell Complexes

A stratified cell complex is formalized as a Hausdorff topological space $X$ together with:

• A surjective, continuous stratification map $s: X \to P$ to a poset $P$ (endowed with the Alexandroff topology), such that fibers $e_p = s^{-1}(p)$ are locally closed and called strata.
• Each stratum $e_p$ is further partitioned into open cells $e_\alpha$; each cell comes with a characteristic map $$\varphi_\alpha: \operatorname{Int} D^{n_\alpha} \to e_\alpha$$ (open $n_\alpha$-disks), and the closure $\overline{e_\alpha}$ is a finite union of cells.
• A partial order on cells is induced by closure relations; for $e_\beta \subset \overline{e_\alpha}$, there exists unique face morphism $e_\alpha \to e_\beta$.
• The stratification is normal if $e_q \cap \overline{e_p} \neq \varnothing$ implies $q \leq p$; it is totally normal if the combinatorics of incidence among strata extends to the (cellular) structure of faces and closures (Tamaki, 2016, Tamaki, 2011, Waas, 2024).

Cellular stratified spaces can include polyhedral decompositions, handlebodies, manifolds with corners, arrangements complements, and spaces arising in Morse or Floer theory. A key feature is the compatibility between the stratification and the attaching maps for each cell: the attaching sphere of a cell of dimension $n$ embeds into the $(n-1)$-skeleton formed by lower-dimensional strata (Tamaki, 2011).

2. Face Categories, Classifying Spaces, and Structural Properties

To encode both the face and strata data, a face category $\mathbf{C}(X)$ is constructed with:

• Objects: the cells $e_\alpha$ of $X$.
• Morphisms: a unique morphism $e_\alpha \to e_\beta$ if $e_\beta \subset \overline{e_\alpha}$; composition arises from transitivity of closure.
• In totally normal situations, $\mathbf{C}(X)$ is an acyclic category, and its nerve $N(\mathbf{C}(X))$ is a simplicial set.

The classifying space $B\mathbf{C}(X) = |N(\mathbf{C}(X))|$ admits a canonical embedding into $X$, which, for regular CW complexes, is a homeomorphism; more generally, $B\mathbf{C}(X)$ is a strong deformation retract of $X$ under suitable normality and polyhedrality conditions (Tamaki, 2016, Tamaki, 2011). For manifolds with corners, or arrangements, $B\mathbf{C}(X)$ generalizes barycentric subdivision, providing a combinatorial model for moduli and configuration spaces.

Stellar stratifications generalize globular cells to stellar cells (star-shaped domains in disks), leading to cylindrical face categories with topological morphism spaces, encoding richer structure and permitting the modeling of more general singular spaces (Tamaki, 2016).

3. Homotopy Theory: Homotopy Links and Model Structures

Stratified cell complexes serve as a robust setting for stratified homotopy theory. For a poset $P$, a $P$-stratified cell complex $(X, s)$ with cell structure as above supports the construction of generalized homotopy links:

• For any flag $\mathcal{I} = [p_0 < \cdots < p_n] \subset P$, one considers the stratified simplex $\Delta^{\mathcal{I}} \to P$ and the functor

$\mathrm{Hol}_P^{\mathcal{I}}(X) = \{\text{$P$-stratum-preserving maps } |\Delta^{\mathcal{I}}| \to X\}.$

• After barycentric subdivision, one builds subcomplexes $\mathcal{N}_{\mathcal{I}}(X)$ such that $$\mathrm{Hol}_P^{\mathcal{I}}(X) \simeq (\mathcal{N}_{\mathcal{I}}(X))_{p_n}$$ (weak homotopy equivalence).
• Pushout diagrams of stratified cell complexes induce homotopy pushouts on homotopy links.

This supports the construction of semi-model structures on the category of stratified spaces, where cofibrations are cellular, and weak equivalences are detected on all generalized homotopy links. Classical examples such as Whitney stratified manifolds and conical PL stratifications are bifibrant in this context (Waas, 2024). The so-called topological stratified homotopy hypothesis identifies the exit-path $\infty$-category of a Whitney stratified manifold with its stratified homotopy type.

4. Stratified Cell Complexes in Stratified Morse Theory

Stratified and cellular structures organize the critical loci and gradient data of Morse-type functions on singular or stratified spaces:

• In discrete stratified Morse theory, a finite simplicial complex $K$ together with a stratification $s: K \to \mathcal{S}$ by a poset of strata and a function $f: K \to \mathbb{R}$ lead to the definition of discrete stratified Morse functions (DSMFs), where the Morse conditions of Forman are required stratum-wise (Knudson et al., 2018).
• The union of gradient pairs in all strata defines the cell-attachment structure, and the topology of $K$ is modeled by a CW-complex with one $p$-cell for each critical $p$-simplex.
• Morse inequalities stratify as $\#\{\text{critical$p$-simplices}\} \geq b_p(K)$, and sublevel set inclusions are realized as stratum-preserving collapses.
• Algorithms construct maximal stratifications supporting DSMFs given arbitrary initial functions.
• The framework extends classical stratified Morse theory (Goresky–MacPherson) and recovers the behavior of critical points and attaching cells for stratified singular spaces (Knudson et al., 2018, Grinberg, 2010).

5. Constructions, Operations, and Duality in Stratified Cell Complexes

Stratified cell complexes admit rich families of constructions:

• Subdivisions: Barycentric subdivisions compatible with the face category and stratification, preserving all relevant structures.
• Products: Under polyhedral or compact-parameter hypotheses, the product $X \times Y$ of two stratified cell complexes is equipped with the product stratification and a compatible cell structure, extending the parameter spaces accordingly.
• Duality: The stellar dual of $X$ is $BC(X)$ with the stratum poset reversed; if $X$ is totally normal, the iterated dual is homeomorphic to $X$.
• Cellular Subcomplexes: Formed by downward-closed unions of cells under $\prec$; serves as the cellular basis for colimits and homological computations.
• Salvetti-Type Models: The face category construction reproduces the Salvetti complex for arrangement complements as $BC(X)$, and more generally models for configuration spaces, Floer theories, and other stratified moduli spaces (Tamaki, 2011, Tamaki, 2016).

6. Stratified Cell Complexes in Intersection and Perverse Cohomology

The construction of intersection spaces associated with stratified pseudomanifolds employs cell complexes stratified by perversity data:

• Given a stratified pseudomanifold $X$ of depth 1, one forms a CW-complex $I^pX$ by truncating the link of the singular stratum with respect to the perversity $p$, and attaching cones over this truncated space.
• The resulting intersection space cohomology $HI$ differs from intersection cohomology $IH$; $HI$ admits a perversity-internal cup product, and exhibits Poincaré duality with $HI^{n-r}_q(X)$ for complementary perversities $p+q = \text{top}$ (Banagl, 2011).
• These models use the precise cellular data of the stratified cell complex, and de Rham complexes of differential forms can be constructed on the regular part of $X$, with explicit conditions reflecting the stratified structure.
• Mirror symmetry interchanges $HI$ and $IH$, connecting these theories to dualities in string theory, with applications in the analysis of moduli spaces and Lagrangian brane counts.

7. Examples and Applications

Stratified cell complexes and their face category models encompass a wide variety of geometric and topological examples:

• Regular CW complexes (trivial stratification).
• Poset realization: the order complex of a poset becomes a regular CW model.
• Configuration spaces of finite graphs, where totally normal cellular stratifications provide finite regular CW models (Tamaki, 2016).
• Complements to hyperplane (and braid) arrangements, with stratification by intersection types and face categories corresponding to the braid category, yielding the Salvetti complex (Tamaki, 2011).
• Toric and projective manifolds, which admit minimal cell decompositions modeled by cylindrically normal stratified structures.
• Stratified Morse and Floer moduli spaces, where unstable/critical manifolds provide the backbone of the stratified cell structure.
• Singularity theory and intersection space constructions, via the explicit stratification and cellular attachment data of underlying pseudomanifolds (Banagl, 2011).

In all these settings, the stratified cell complex formalism supports both a rigorous combinatorial encoding (face/cylindrical category, nerve, subdivision structures) and a robust analytical framework for singular, moduli, or intersection-theoretic invariants. The generality of this approach supports a unified treatment of geometric, combinatorial, and homotopical features essential to modern topology, as demonstrated in recent developments in stratified homotopy theory and applications to configuration and intersection space models (Tamaki, 2016, Tamaki, 2011, Waas, 2024, Knudson et al., 2018, Banagl, 2011).