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

$

$

*

*

@

@

**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 (*n*−*k*)-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*, *σ*
∩ *τ* is a face of both *σ* and *τ*,
unless *σ* ∩ *τ* = 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 *A* ∈
*S* then for all *B* ∈ 2^{A}
with *B* ≠ Ø, *B* ∈ *S*; 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.

> __Online resources__:
see Wikipedia page.

**And Physics** > s.a. computational physics;
laplace operator; types of quantum field theories [discrete].

@ __Quantum-gravity motivated__:
Oriti in(07)gq/05;
Finkelstein a1108-conf [simplicial quantum dynamics];
Girelli et al a2105 [phase space using 2-groups];
> s.a. action for gravity; regge calculus;
types of spacetimes.

@ __Gauge theories__: Halvorsen & Sørensen a1107 [Yang-Mills-Higgs action];
> s.a. BF theories;
chern-simons theories; solutions
of gauge theories; yang-mills theories.

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send feedback and suggestions to bombelli at olemiss.edu – modified 24 may 2021