Abstract Regular Cell Complexes (ARCCs)

• Abstract Regular Cell Complexes (ARCCs) are algebraic and combinatorial representations of regular CW-complexes, defined by graded cells, boundary operators, and flag-transitivity.
• They enforce rigorous shellability and regularity conditions to ensure acyclicity and predictable homological invariants essential for topological analysis.
• ARCCs facilitate advanced applications in signal processing, network science, and higher-order data analysis through computable Hodge Laplacians and group-theoretic automorphisms.

An Abstract Regular Cell Complex (ARCC) is an algebraic and combinatorial generalization of regular CW-complexes, designed to capture higher-order, highly symmetric, and topologically regular structures in a framework well-suited to both algebraic topology and applied data analysis. ARCCs unify the homological and combinatorial features of regular cell complexes, abstract polytopes, incidence geometries, and modern signal processing domains including graphs, simplicial complexes, and cubical complexes. ARCCs are characterized by purely algebraic data—graded sets of cells with boundary operators encoded as signed incidence matrices or posets—governed by strict regularity, shellability, and group-theoretic flag-transitivity conditions (Grenzebach et al., 2011, Schulte, 2017, Roddenberry et al., 2021, Hoppe et al., 11 Jun 2025, Hammack et al., 2023). This makes them an indispensable tool for both structural combinatorics and algorithmic tasks involving higher-order network data.

1. Algebraic and Combinatorial Foundations

An ARCC of maximal dimension $n$ consists of a finite graded collection $\{C_k\}_{k=0}^n$ of $k$–cells and corresponding boundary matrices $B_k \in \{0, \pm1\}^{|C_{k-1}|\times|C_k|}$ for $k=1,\ldots,n$, encoding oriented incidence between $k$– and $(k-1)$–cells. Each column of $B_1$ must have exactly one $+1$ and one $-1$, enforcing that 1-cells (edges) are incident to exactly two 0-cells (vertices). For each $\{C_k\}_{k=0}^n$0-cell $\{C_k\}_{k=0}^n$1, the closure of $\{C_k\}_{k=0}^n$2—the subcomplex generated by $\{C_k\}_{k=0}^n$3 and all its faces—must satisfy the acyclicity and connectivity properties: the relevant boundary maps form a complex with $\{C_k\}_{k=0}^n$4 for all $\{C_k\}_{k=0}^n$5, and $\{C_k\}_{k=0}^n$6 is isomorphic to $\{C_k\}_{k=0}^n$7 (Hoppe et al., 11 Jun 2025, Roddenberry et al., 2021). Equivalently, ARCCs can be viewed as ranked posets satisfying strong flag-connectedness and diamond conditions, with the regularity property formalized by automorphism group flag-transitivity (Schulte, 2017).

Regularity conditions in ARCCs admit a purely algebraic formulation: a shellable chain complex is regular if, upon ordering maximal basis elements ("cells"), the boundary maps satisfy linear independence and leading coefficient $\{C_k\}_{k=0}^n$8 conditions that guarantee all attaching maps are homeomorphisms of spheres, mirroring CW-regularity (Grenzebach et al., 2011). In the group-theoretic model, the automorphism group $\{C_k\}_{k=0}^n$9 of an ARCC acts transitively on the set of flags (maximal chains), with distinguished stabilizer subgroups $k$0 preserving all faces except at rank $k$1, satisfying intersection and string-commutation properties—resulting in a "generalized string C-group" structure (Schulte, 2017).

2. Shellability, Regularity, and Homological Invariants

The concept of shellability provides a combinatorial stratification of the ARCC. A shelling is a total ordering on the set of maximal cells such that, at each step, the intersection of the newly added cell with the union of previous cells is "pure" of one lower dimension. Shellability generalizes classical shellings of regular CW-complexes, but in the pure algebraic setting it proves to be a strictly weaker invariant: shellable but non-regular complexes can display torsion or homological pathologies (Grenzebach et al., 2011).

Regularity imposes strong constraints, most crucially that whenever a $k$2–cell's boundary lies in the span of previous boundaries in the shelling order, it does so via a unit coefficient, corresponding to injective characteristic maps in the topological realization. Totally regular ARCCs further require every single-cell subcomplex to be acyclic, which allows explicit computation of homology ranks based on critical or precritical cell counts. Precisely, for a totally regular pure complex of order $k$3 with $k$4 precritical $k$5–cells, one obtains $k$6, intermediate homology vanishes, and $k$7 (Grenzebach et al., 2011).

Shellability is preserved under taking skeleta (lower-dimensional truncations), and every shellable ARCC admits a dimensionally non-increasing shelling order (Grenzebach et al., 2011).

Property	Shellable ARCC	Regular (Totally Regular) ARCC
Homology	Possibly torsion	Free modules, predictable rank formulas
Skeleta	Preserved	Preserved
Characterization	Shelling structure	Unit boundary conditions, acyclicity

3. Group-Theoretic Realization and Automorphism Structure

Abstract Regular Cell Complexes can be presented equivalently as flag-transitive incidence complexes associated to generalized string C-groups. Given such a group $k$8 with distinguished subgroups $k$9, one defines $B_k \in \{0, \pm1\}^{|C_{k-1}|\times|C_k|}$0–faces as cosets $B_k \in \{0, \pm1\}^{|C_{k-1}|\times|C_k|}$1, ordered by coset inclusion. The intersection and string-commutativity axioms on these subgroups guarantee that the resulting poset is an ARCC of rank $B_k \in \{0, \pm1\}^{|C_{k-1}|\times|C_k|}$2, and the full group acts transitively on flags (Schulte, 2017).

Classical regular polytopes (e.g., $B_k \in \{0, \pm1\}^{|C_{k-1}|\times|C_k|}$3-simplices, hypercubes), as well as incidence geometries such as finite projective planes and spherical buildings, realize as ARCCs under this framework. In the special case of abstract regular polytopes, the "diamond condition" stipulates that for each chain of faces of ranks $B_k \in \{0, \pm1\}^{|C_{k-1}|\times|C_k|}$4, there are exactly two intermediate $B_k \in \{0, \pm1\}^{|C_{k-1}|\times|C_k|}$5–faces, corresponding to involutive generators (Schulte, 2017).

Extensions and amalgamations—such as symmetric-group extensions and free product constructions—produce infinite families of ARCCs of higher rank or controlled facet incidence, preserving the regularity and combinatorial structure (Schulte, 2017).

4. Homology, Hodge Theory, and Discrete Laplacians

Each ARCC admits an associated chain complex $B_k \in \{0, \pm1\}^{|C_{k-1}|\times|C_k|}$6, where the boundary operators are given by the signed incidence matrices described above. The cochain complex is the dual, and standard definitions of homology groups $B_k \in \{0, \pm1\}^{|C_{k-1}|\times|C_k|}$7 apply (Hoppe et al., 11 Jun 2025, Roddenberry et al., 2021).

A central algebraic object on ARCCs is the combinatorial Hodge Laplacian: $B_k \in \{0, \pm1\}^{|C_{k-1}|\times|C_k|}$8 which is always positive semidefinite and whose kernel is isomorphic to $B_k \in \{0, \pm1\}^{|C_{k-1}|\times|C_k|}$9. This yields an orthogonal Hodge decomposition of each chain space into subspaces of gradients, curls, and harmonic $k=1,\ldots,n$0–cochains—directly paralleling the continuous de Rham theory. In signal processing terms, $k=1,\ldots,n$1 reduces to the standard graph Laplacian, while $k=1,\ldots,n$2 interpolates between simplicial, cubical, and mesh Laplacians in various geometric contexts (Roddenberry et al., 2021, Hoppe et al., 11 Jun 2025).

Operator	Formula	Interpretation
Up Laplacian	$k=1,\ldots,n$3	From $k=1,\ldots,n$4- to $k=1,\ldots,n$5-cells
Down Laplacian	$k=1,\ldots,n$6	From $k=1,\ldots,n$7- to $k=1,\ldots,n$8-cells
Combinatorial Hodge Lap	$k=1,\ldots,n$9	Full $k$0–cell Laplacian

The algebraic rigor of ARCCs ensures that all these operators are well-defined and efficiently computable from the sparse data of boundary matrices, facilitating spectral analysis, harmonic representative computation, and higher-order signal processing (Hoppe et al., 11 Jun 2025).

5. Topological and Incidence-Geometric Characterization

Topologically, an ARCC is a finite regular CW-complex where each $k$1–cell is attached via a homeomorphism of $k$2 onto its boundary, which must be a subcomplex of the $k$3–skeleton. Purity requires every face to be included in at least one maximal facet, and strong connectivity of the ARCC is defined via the connectivity of the dual graph, constructed from $k$4–cells and shared $k$5–cells.

The incidence numbers $k$6 provide the combinatorial counterpart of attaching maps, allowing for explicit computation of cellular boundaries and the f-vector $k$7, which enumerates numbers of $k$8–cells (Hammack et al., 2023). Regularity ensures the characteristic maps are injective and attaching spheres are connected, resulting in well-behaved homological invariants.

A significant consequence of this structure is the existence of a unified Euler–Poincaré relation

$k$9

for finite ARCCs, and a categorical equivalence (up to dimension 2) of ARCCs and regular CW-complexes, guaranteeing that algebraic data encodes all topologically relevant information (Hoppe et al., 11 Jun 2025). Additionally, evenness of facet degrees leads to generalized cycle decompositions and Euler covers, mirroring classical Eulerian tour theory in arbitrary dimension (Hammack et al., 2023).

6. Applications: Signal Processing, Network Science, and Higher-Order Data

Modern applications of ARCCs focus on the analysis of signals supported on higher-order structures. Signal processing on ARCCs generalizes graph signal processing by enabling the manipulation of data on $(k-1)$0–cells, for all $(k-1)$1, using shift-invariant polynomial filters defined via the Hodge Laplacians $(k-1)$2 (Roddenberry et al., 2021, Hoppe et al., 11 Jun 2025). The spectrum of $(k-1)$3 supports efficient denoising, flow decomposition, and sparse harmonic analysis.

Neural architectures have been generalized to ARCCs: convolutional neural network layers are constructed for features on $(k-1)$4–cells using learned polynomial filters in $(k-1)$5, pointwise nonlinearities, and potentially cross-dimensional messages via boundary and coboundary operators. These models naturally subsume graph neural networks (when $(k-1)$6), simplicial and cubical architectures, and extend to more elaborate mesh and polyhedral data (Roddenberry et al., 2021).

ARCCs also serve as foundational objects for constructing higher-order network models, encoding geometric and combinatorial constraints critical in manifold learning, topological data analysis, robotics, trajectory planning, and the study of hypergraphs or designs (Hammack et al., 2023, Hoppe et al., 11 Jun 2025).

7. Examples, Constructions, and Further Directions

Concrete examples of ARCCs include:

• Graphs: 1-dimensional ARCCs with $(k-1)$7 as the standard incidence matrix.
• Simplicial complexes: ARCCs where all cells are simplices, and boundary matrices reflect simplex face relations.
• Cubical complexes: ARCCs with cells products of intervals, boundary maps given via Kronecker sums.
• Polytope complexes: Incidence posets corresponding to flag-transitive automorphism groups with strictly controlled face figures (Schulte, 2017).

ARCCs can be generated from raw data (e.g., via clique complexes, cubical products, planar window liftings), and all attaching/topological data is encoded algebraically, enabling practical implementation via sparse matrix computation (Hoppe et al., 11 Jun 2025). Extensions include symmetric group-based constructions, amalgamations to produce infinite complexes with prescribed facet adjacency, and coverings analogous to universal covering spaces in topology (Schulte, 2017).

Potential research directions include Morse theory on ARCCs via algebraic gradient matchings, extension to mixed or stratified complexes over general rings, deeper connections to combinatorial representation theory, and the efficient implementation of ARCC-based learning systems in non-Euclidean domains (Grenzebach et al., 2011).