Simplicial complex
From Wikipedia, the free encyclopedia
In mathematics, a simplicial complex is a topological space of a particular kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory.
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[edit] Definitions
A simplicial complex 
is a set of simplices that satisfies the following conditions:
- 1. Any face of a simplex from is also in .
- 2. The intersection of any two simplices
is a face of both σ1 and σ2.
Note that the empty set is a face of every simplex. See also the definition of an abstract simplicial complex, which loosely speaking is a simplicial complex without an associated geometry.
A simplicial k-complex is a simplicial complex where the largest dimension of any simplex in equals k. For instance, a simplicial 2-complex must contain at least one triangle, and must not contain any tetrahedra or higher-dimension simplices.
A pure or homogeneous simplicial k-complex is a simplicial complex where every simplex of dimension less than k is the face of some simplex 
of dimension exactly k. Informally, a pure 1-complex "looks" like it's made of a bunch of lines, a 2-complex "looks" like it's made of a bunch of triangles, etc. An example of a non-homogeneous complex is a triangle with a line segment attached to one of its vertices.
A facet is any simplex in a complex that is not the face of any larger simplex. (Note the difference from the "facet" of a simplex.) A pure simplicial complex can be thought of as a complex where all facets have the same dimension.
Sometimes the term face is used to refer to a simplex of a complex, not to be confused with the face of a simplex.
For a simplicial complex embedded in a k-dimensional space, the k-faces are sometimes referred to as its cells. The term cell is sometimes used in a broader sense to denote a set homeomorphic to a simplex, leading to the definition of cell complex.
The underlying space, sometimes called the carrier of a simplicial complex is the union of its simplices.
[edit] Closure, Star, and Link
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The closure of a set of simplices S (denoted Cl S) is the smallest simplicial complex containing all the simplices in S. In other words, Cl S is the set containing all faces of every simplex in S.
The star of a set of simplices S (denoted St S) with respect to a simplicial complex K is the set of all simplices in K which have simplices in S as faces. (Note that the star is not necessarily a simplicial complex.)
The link of a set of simplices S (denoted Lk S) with respect to a simplicial complex K equals Cl St S - St Cl S. The link of S is in a sense the "boundary" of S with respect to K.
[edit] Algebraic topology
In algebraic topology simplicial complexes are often useful for concrete calculations. For the definition of homology groups of a simplicial complex, one can read the corresponding chain complex directly, provided that consistent orientations are made of all simplices. The requirements of homotopy theory lead to the use of more general spaces, the CW complexes. Infinite complexes are a technical tool basic in algebraic topology. See also the discussion at polytope of simplicial complexes as subspaces of Euclidean space, made up of subsets each of which is a simplex. That somewhat more concrete concept is there attributed to Alexandrov. Any finite simplicial complex in the sense talked about here can be embedded as a polytope in that sense, in some large number of dimensions.
