Cell
Complexes| Cell
Complexes |
In General > s.a. [topology]; CW-Complex.
$ 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.
Properties and Operations on Cell Complexes > s.a. euler
number.
* Duality: An operation which produces a new cell complex 
*
starting from any given complex ,
by associating with each k-dimensional
cell
in a (n–k)-dimensional
dual cell
*, whose boundary consists
of the duals of all cells which have 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 Rn is
a collection K of simplices, such that (1) for all 
in
K every face of is
also in K;
(2) for all and 
in K, 
is
a face of both and ,
unless =
Ø.
* Abstract: 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.
@ 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]; > s.a. discrete geometry.
@ 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]; > s.a. graph [neighborhood
complex], harmonic
maps, morse
functions.
@ And physics: Oriti gq/05-in
[quantum gravity]; > s.a. BF theories; computational
physics;
regge calculus; types
of quantum field theories [discrete]; types
of
spacetimes.
References > s.a. tiling [cellular
decomposition of manifolds, including in physics, froth].
@ General: Fritsch & Piccinini 90.
@ Related topics: Forman Top(98) [Witten-Morse theory].
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22 jun 2008