Triangulation (topology)

In mathematics, triangulation describes the replacement of topological spaces by piecewise linear spaces, i.e. the choice of a homeomorphism in a suitable simplicial complex. Spaces being homeomorphic to a simplicial complex are called triangulable. Triangulation has various uses in different branches of mathematics, for instance in algebraic topology, in complex analysis or in modeling. On the one hand, it is sometimes useful to forget about superfluous information of topological spaces: The replacement of the original spaces with simplicial complexes may help to recognize crucial properties and to gain a better understanding of the considered object. On the other hand, simplicial complexes are objects of combinatorial character and therefore one can assign them quantities rising from their combinatorial pattern, for instance, the Euler characteristic. Triangulation allows now to assign such quantities to topological spaces. Investigations concerning the existence and uniqueness of triangulations established a new branch in topology, namely the piecewise-linear-topology (short PL- topology). Its main purpose is topological properties of simplicial complexes and its generalization, cell-complexes. An abstract simplicial complex above a set is a system of non-empty subsets such that: for each ; if and . The elements of are called simplices, the elements of are called vertices. A simplex with vertices has dimension by definition. The dimension of an abstract simplicial complex is defined as . Abstract simplicial complexes can be thought of as geometrical objects too. This requires the term of geometric simplex. Let be affinely independent points in , i.e. the vectors are linearly independent. The set is said to be the simplex spanned by . It has dimension by definition. The points are called the vertices of , the simplices spanned by of the vertices are called faces and the boundary is defined to be the union of its faces.