Cell Complexes| 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 (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 ∈ 2A
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.
main page
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