Simplicial complex

A simplicial complex is a term of algebraic topology. For a simplicial complex is a purely algebraic writable object, by means of which the critical properties can be characterized algebraically by certain designated as a triangulated topological spaces. In particular, simplicial complexes are used to define invariants of the underlying topological space.

The idea of ​​the simplicial complex is to investigate a topological space by the fact that - if possible - is constructed by combining a lot of simplices in d-dimensional Euclidean space which is homeomorphic to the given topological space. The " Assembly Instructions " of the simplices, ie the information about how the simplices are joined together, is then characterized purely algebraically in the form of a sequence of group homomorphisms.

Definition

An abstract simplex is a finite non-empty set. An element of an abstract simplex is called a corner of a non-empty subset of is again an abstract simplex and facet ( or page ) called.

An ( abstract ) simplicial complex is a set of simplices with the property that every facet of a simplex to part again, so. The union set of all vertices of simplices of the simplicial complex is called the vertex set or Eckpunktbereich and designated.

The dimension of an abstract simplex containing corners is defined as the dimension of the simplicial complex, and is defined as the maximum dimension of all the simplices. If the dimension of the simplices is not limited, it means infinite dimensional.

The simplicial complex is called finite if it is a finite set, and locally finite if every vertex belongs to only finitely many simplices.

The skeleton of a simplicial complex is the union of all its simplices of dimension.

Geometric simplicial

A geometric simplicial complex is a set of simplices in some Euclidean space with the property that every facet of a simplex again heard and that for all simplices, the average is either empty or a common facet of and.

The geometric simplicial complex is called geometric realization of the abstract simplicial complex. All geometric realization of an abstract simplicial complex homeomorphic to each other.

The geometric simplicial complex carries the weak topology: A set is accurately completed if their intersection is completed with each simplex.

A topological space is called triangulated if it is homeomorphic to a geometric simplicial complex.

Simplicial pictures

A simplicial map is a mapping between the vertex sets are mapped in each simplex of the corners of the image on the vertices of a simplex in.

Conversely, a continuous map after finitely many barycentric subdivisions by a simplicial map approximate, see simplicial approximation theorem.

The simplicial chain complex than

Be a finite simplicial complex. The p-th simplicial group is the free abelian group generated by the set of simplices with dimension, it is listed with. The elements of the group are called simplicial p- chains. If you choose a total order for all vertices that lie in any simplex of, we obtain by restriction also an order for each p- simplex. An edge operator is then defined by

Where the group element generated from the corners says. For the boundary operator applies to all simplicial p- chains. Therefore, a chain complex and can be explained in the usual way on this homology. This homology is called simplicial homology.

History

Triangulations and a formulated in matrix notation equivalent to the chain complex formed therefrom were examined by Henri Poincaré in the late nineteenth century. Simplizale pictures was first used in 1912 by Brouwer. In the 1920s, then came the vision that led to the concept of the chain complex.