Simplicial complex

From Academic Kids

In mathematics, a simplicial complex is a topological space of a particular kind, built up of points, line segments, triangles, and their n-dimensional counterparts. Informally, a simplicial complex is made of a set of simplices that intersect with each other only by their common faces. In algebraic topology these spaces are found to be the easiest to deal with, in terms of concrete calculations. Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory.

Geometry

A simplicial complex C is a finite set of simplices that satisfies the following two conditions.

C1. Any face of a simplex from C is also in C.
C2. The intersection of any two simplices from C is either empty or is a face for both of them.

The constituting simplices are called the faces of the complex. See simplex for a taxonomy of faces. In addition, in the d-dimensional space, the d-faces are the cells of a simplicial complex.

A facet is a maximal (under inclusion) face of a simplicial complex. (Notice the difference from the "facet" of a simplex.)

The term cell is sometimes used in a broader sense to denote a polytope, possibly unbounded, leading to the definition of cell complex, whose faces are cells.

If the maximal dimension of the constituting simplices is k then the complex is called a k-complex. A k-complex is called homogeneous or pure, if its only faces are the faces of its k-faces. Alternatively, a complex is pure (homogeneous), if all its facets are of the same dimension. An example of a non-homogeneous complex is a triangle with a line segment attached to one of its vertices.

Algebraic topology

A simplicial complex is a combinatorial object telling one how to construct a topological space out of a number of simplices.

A simplex by itself of dimension k is represented by labels called 0-simplices (its set of vertices), 1-simplices (its set of edges), and so on up to a single k-simplex. The general finite simplicial complex is a set of instructions for joining a number of simplices of varying dimensions together, as a topological space in the abstract (not assumed to be a subset of Euclidean space). The joins are restricted to be at vertices, or along an edge, or by joining two faces.

To make this formal we have to define precisely within the r-simplices the relation with the r-1-simplices of being a face. Identifications between two simplicial complexes are allowed only if they respect the face relations. That is, identifications in one dimension must match up in lower dimensions. Then the general concept may be defined inductively starting with the simplex case.

For the definition of homology groups of a simplicial complex, one can read the corresponding chain complex directly - provided consistent orientations are made of all simplices.

The requirements of homotopy theory lead to rather more general spaces, the CW complexes, being used. 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.

See also

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