Cell Complexes

In General > s.a. topology / CW-Complex; tiling [cellular decomposition of manifolds, including in physics, froth].
$Topological p-cell: A space homeomorphic to (the interior of) a p-disk.$ Affine p-cell: A bounded convex polygon.
$Finite, regular cell complex: A finite set of non-empty pairwise disjoint open cells, such that (a) The closure of each cell is homeomorphic to a ball and its boundary to a sphere in some dimension, and (b) The boundary of each cell is a union of cells. * Applications: Used in math to define homology theories (simplicial, cubical); For physics, see below. * Relationships: Cell complexes can be PL-manifolds, but it is not possible to check if a given 4D one is a PL-manifold or not. @ General references: Fritsch & Piccinini 90. @ Related topics: Forman Top(98) [Witten-Morse theory]. Properties and Operations on Cell Complexes > s.a. euler number. * Duality: An operation which produces a new cell complex Ω* starting from any given complex Ω, by associating with each k-dimensional cell ω in Ω a (nk)-dimensional dual cell ω*, whose boundary consists of the duals of all cells which have ω on their boundary. Simplicial Complex > s.a. Link of a Vertex; manifolds; simplex; Skeleton. * Idea: A cell complex in which the cells are simplices; One of the most widely used types.$ Def: A simplicial complex in $$\mathbb R$$n is a collection K of simplices, such that (1) for all σ in K every face of σ is also in K; (2) for all σ and τ in K, στ is a face of both σ and τ, unless στ = Ø.
* Abstract simplicial complex: A collection S of finite non-empty sets such that all the non-empty subsets of an element of S also belong to S; If AS then for all B ∈ 2A with B ≠ Ø, BS; Every abstract simplicial complex has a unique representation as a simplicial complex, up to a linear isomorphism.
* Duality: The dual of a simplicial complex is a Voronoi complex.
* And manifolds: The polyhedron of a simplicial complex is a topological manifold iff the link of each cell has the homology of a sphere, and the link of every vertex is simply connected [@ in Thurston 97, p121].
* Saturated: A shellable complex with maximal modular homology.
@ General references: in Sakai 13.
@ Discrete differential geometry: Kheyfets et al PRD(89); Ambjørn et al NPB(97)ht/96 [4D, integral invariants and curvature]; Korepanov n.SI/00 [moves, curvature]; Alsing et al CQG(11)-a1107 [Ricci tensor]; Thüringen MG13-a1302 [fields and discrete calculus]; > s.a. discrete geometry.
@ With constant-curvature simplices: Rovelli & Vidotto PRD(15)-a1502 [and quantization of geometry]; in Han JHEP(16)-a1509 [and approximation of smooth 4-geometries].
@ And physics: Reitz & Bianconi a2003 [diffusion processes and spectral dimension, renormalization group approach]; > s.a. random walk [quantum walk].
@ Related topics: Korepanov JNMP(01)m.GT/00 [invariants of PL-manifolds]; Brown et al DM(04) [k-polynomial and k-fractal]; Mnukhin & Siemons JCTA(05) [saturated]; Hachimori DM(08) [2D, decompositions]; Barmak & Minian a0907 [strong homotopy types]; Miller et al CMP(14) [simplicial Ricci flow]; > s.a. 3D manifolds [torsion invariant]; graph [neighborhood complex]; harmonic maps; morse functions; cover of a topological space [nerve]; types of topological spaces [finite].
> Examples: see Vietoris-Rips Complex.