Topological Arrangements: Theory & Applications

• Topological Arrangements are generalized collections of subspaces in a manifold that relax linear constraints while enforcing local regularity.
• They unify topology, combinatorics, and algebra via intersection posets, face categories, and effective CW complex constructions.
• These arrangements enable modeling of spaces with complex intersections, advancing studies in cohomology, valuation theory, and homotopy types.

A topological arrangement is a finite collection of subsets or subspaces within a manifold or CW complex that generalizes classical (linear) hyperplane arrangements to settings where the "hyperplanes" are allowed to be arbitrary submanifolds, pseudohyperplanes, spheres, disks, topoplanes, or similar objects, subject to local flatness and intersection regularity. This generalized framework provides a unified language and robust toolkit to analyze the topological, combinatorial, and algebraic properties of spaces partitioned by such objects. Research on topological arrangements spans geometry, combinatorics, topology, oriented matroids, and valuation theory, and has led to significant generalizations of the foundational results from the theory of linear arrangements.

1. Foundations and Definitions

Topological arrangements extend the hyperplane arrangement paradigm by relaxing linearity and allowing the "hyperplanes" to be more general subspaces—such as locally flat codimension-1 submanifolds, topological spheres, topodisks, or even objects with nontrivial topology—while enforcing strong intersection locality and combinatorial regularity. Key frameworks include:

• Submanifold arrangements: A finite collection $\mathcal{A} = \{N_1,\dots,N_k\}$ of codimension-1, locally flat submanifolds in a smooth manifold $X$ intersecting locally as hyperplanes (i.e., around every point, there is a chart mapping intersections to a finite union of linear hyperplanes). The arrangement is required to stratify $X$ into a totally normal cellular decomposition, with combinatorial data fully encoded by the intersection poset and the face category (Deshpande, 2011).
• Topoplane/topological hyperplane arrangements: A finite collection of topoplanes (subsets homeomorphic to balls of codimension 1), typically embedded in some contractible manifold or ball, with all nonempty intersections ('flats') also homeomorphic to balls, and with intersections that are 'clean', i.e., without local knottings or wild embeddings (Randriamaro, 2020).
• Arrangements of pseudohyperplanes/spheres: Codimension-1 spheres or pseudospheres in a sphere or manifold, embedded tamely so that each intersection is again a (possibly lower-dimensional) subsphere, and the stratification yields a regular CW complex. These generalize to both realizable and nonrealizable oriented matroids (Deshpande, 2012, Deshpande, 2012, Dobbins, 2017).
• General topological arrangements: As formalized in (Khovanskii et al., 6 Jan 2026), a topological arrangement $(X,A)$ consists of a compact, nonsingular semiregular CW complex $X$ and a finite collection $A = \{H_1, \ldots, H_m\}$ of subcomplexes ("hyperplanes") such that the atoms (connected components of $X \setminus \cup H_i$) are open cells of $X$, and the intersection of any subfamily of hyperplanes yields a well-behaved subcomplex (see details in (Khovanskii et al., 6 Jan 2026)).

2. Combinatorial Structures: Intersection Posets and Face Categories

Irrespective of the embedding or smoothness category, every topological arrangement admits two principal combinatorial invariants:

• Intersection poset (or lattice): The set $L(\mathcal{A})$ of all nonempty connected components of intersections of collections of the basic subspaces ("hyperplanes"), with partial order given by reverse inclusion. The rank function typically records codimension. This object controls much of the algebraic topology, including Betti numbers and cohomological invariants (Deshpande, 2011, Khovanskii et al., 6 Jan 2026, Deshpande, 2012, Deshpande, 2012, Deshpande, 2018).
• Face poset/category: The poset or acyclic category $\mathcal{F}(\mathcal{A})$ of faces (connected components of the partitions induced by the arrangement). In regular cases, morphisms correspond to inclusion of faces into closure of higher-dimensional faces. These combinatorial models control the construction of CW complexes modeling arrangement complements, e.g., the Salvetti complex (Deshpande, 2011, Deshpande, 2012).

A summary table of essential combinatorial data is as follows:

Structure	Definition	Role
Intersection poset $L$	Connected comps. of intersections, ordered by reverse inclusion	Controls cohomology, Möbius invariants, stratifications
Face poset/category	Open faces formed by stratification	Indexes cells in CW complexes modeling complements
Atoms	Connected comps. of the complement of all hyperplanes	Building blocks for valuation theory, atom algebra

3. Topological Invariants and Homotopy Type

Key invariants of interest for topological arrangements include:

• Homotopy type and cell complex models: The complement space (e.g., the tangent bundle complement, microbundle complement, or open set $X \setminus \cup H_i$) is homotopy equivalent to a regular finite CW complex constructed via the Salvetti complex or similar. The cells correspond to chains in the face category or intersection poset (Deshpande, 2011, Deshpande, 2012, Deshpande, 2012). For arrangements in spheres, the Salvetti construction uses the dual face poset to build CW models for the complement in the tangent bundle (Deshpande, 2012).
• Cohomological invariants (Orlik–Solomon-type rings, Betti numbers, Möbius functions): The additive and, often, multiplicative structure of the (co)homology groups of the complement is determined by combinatorial data (intersection poset, Möbius function). For example, the Möbius function captures the Betti numbers and Euler characteristic via generalized Goresky–MacPherson formulas (Deshpande, 2018, Deshpande, 2012, Deshpande, 2012, Vassiliev, 2014, Khovanskii et al., 6 Jan 2026).
• Valuations and convex chain algebras: For a topological (hyperplane) arrangement, valuations on atoms (integer-valued, additive functions) are classified by a filtration and reciprocity/period conditions, and the space of simple valuations is universally described by combinatorial data—generalizing the Hadwiger theorem in convex geometry (Khovanskii et al., 6 Jan 2026).
• Face enumeration (generalized Zaslavsky formulas): In transsective (or regular) settings, the $f$-vector (numbers of faces of all dimensions) is computed via Möbius inversion on the lattice of flats, and the chamber-count is determined combinatorially (Randriamaro, 2020).

4. Arrangements Beyond Linearity: Pseudohyperplanes, Spheres, Topodisks

Topological arrangements encompass a range of settings, which often require different technical tools:

• Pseudohyperplane and pseudosphere arrangements: Arrangements arising as realizations of (possibly non-realizable) oriented matroids. The Folkman–Lawrence representation theorem states that every oriented matroid may be constructed from an arrangement of pseudospheres, and the "microbundle complement" generalizes the classical complexified complement for linear arrangements (Deshpande, 2012). The resulting topological models (Salvetti complexes) remain fully combinatorial, though their fundamental groups, cohomology rings, and realization spaces exhibit new algebraic and geometric phenomena (Dobbins, 2017).
• Arrangements of topodisks, spheres, or general submanifolds: Arrangements of topological disks, spheres, or higher-codimension submanifolds require careful control of local intersections and regularity. Results on arrangements of disks provide explicit bounds for dual graph diameter and chamber structure, with complexity controlled by overlap numbers and intersection multiplicities (Abiad et al., 20 Oct 2025). For arrangements of spheres, the complement in the ambient manifold's tangent bundle is explicitly computed, and its homotopy type is fully combinatorial except for possible subtleties in the "mirrored" or centrally symmetric case (Deshpande, 2012).

5. Applications: Topology, Combinatorics, and Valuations

The theory of topological arrangements provides concrete results and methodologies across several domains:

• Classification of complements: In the real affine case, the diffeomorphism and homotopy type of the complement is determined solely by the number of components and combinatorial data on multiple points (Ishikawa et al., 2019). For complex line arrangements, generic arrangements sharing combinatorics yield topologically equivalent complements after small perturbation, with explicit control over critical values, Euler characteristics, and fiber structure (Bodin, 2012).
• CW complexes and computational models: Algebraic-topological algorithms based on chain and cochain complexes, and matrix representations of incidence data, yield dimension-independent, efficient combinatorial descriptions of arrangement topologies for planar and volumetric settings (Paoluzzi et al., 2019).
• Valuation theory and scissors congruence: The algebra of "convex chains" or "polytope functions" on arrangements, along with filtered differential complexes, provides a topological generalization of the Varchenko–Gelfand and Hadwiger theories, unifying translation-invariant valuations, inclusion–exclusion, and reciprocity constraints across polynomial, disk, or general arrangements (Khovanskii et al., 6 Jan 2026).
• Generalization of classical theorems: All of the main results of classical arrangement theory—Salvetti complex construction, Orlik–Solomon algebra, Goresky–MacPherson formulas, stratified Morse theory—are shown to hold, in the appropriate form, in the much broader class of topological arrangements (Deshpande, 2012, Deshpande, 2011, Khovanskii et al., 6 Jan 2026, Randriamaro, 2020).

6. Realization Spaces, Contractibility, and Further Directions

Topological arrangements provide insights into realization problems and moduli spaces:

• Contractibility and universality: In the setting of weighted pseudosphere arrangements, realization spaces for oriented matroids of rank 3 are always contractible, in sharp contrast with the possibly arbitrary homotopy types of realization spaces for vector configurations (Mnëv universality). This makes weighted pseudosphere arrangements powerful for classifying spaces and bundle theories, rapidly generalizing the role of Stiefel and Grassmannian manifolds (Dobbins, 2017).
• Complexification and symplectic realization: Any real pseudoline arrangement can be complexified to yield a smooth (even symplectic) arrangement of spheres in $\mathbb{C}P^2$ which faithfully represents its incidence data. However, not all combinatorial arrangements are even topologically realizable in $\mathbb{C}P^2$; existence of non-realizable examples is detected via branched-cover and signature defects (Ruberman et al., 2016).
• Open problems and future research: Crucial unresolved areas include classifying arrangements up to homeomorphism when only combinatorics is fixed (as in the existence of Zariski pairs), understanding the limitations of the topological representation theorem in higher dimensions, the role of arrangement combinatorics in the formality and group-theoretic properties of complements, and the interplay between topological arrangements and valuation theory (Deshpande, 2012, Khovanskii et al., 6 Jan 2026).

7. Summary Table: Major Types of Topological Arrangements

Category	Objects	Key Topological Model	Invariant-combinatorics link
Submanifold arrangement	Codim-1 submanifolds	Tangent bundle complement, Salvetti CW	Intersection poset, face cat.
Topoplane/topological hyperplane	Deformed balls/topoplanes	Cell complexes, $f$-polynomials	Lattice of flats, Möbius func.
Pseudohyperplanes/pseudospheres	(Pseudo)spheres, spheres	Microbundle complement, Salvetti complex	Oriented matroid face poset
Disk or general "ball" objects	Topological disks, spheres	Dual graphs, chamber structure	Overlap number, ply
Convex chain/valuation approach	Atoms in semiregular CW	Valuation/algebra of convex chains	Kernel of reciprocity/period sum

The domain of topological arrangements unifies combinatorial, topological, and algebraic methods to study a wide spectrum of arrangement-induced complexes beyond the classical linear case. The consequences for topology, geometry, and combinatorics are broad, and ongoing developments continue to reveal deeper connections between combinatorial data and the topology of partitioned spaces (Deshpande, 2011, Deshpande, 2012, Deshpande, 2012, Khovanskii et al., 6 Jan 2026, Randriamaro, 2020).