In mathematics, triangulation describes the replacement of topological spaces with simplicial complexes by the choice of an appropriate homeomorphism. A space that admits such a homeomorphism is called a triangulable space. Triangulations can also be used to define a piecewise linear structure for a space, if one exists. Triangulation has various applications both in and outside of mathematics, for instance in algebraic topology, in complex analysis, and in modeling.
Motivation 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 arising from their combinatorial pattern, for instance, the Euler characteristic. Triangulation allows one to assign such quantities to topological spaces. Investigations concerning the existence and uniqueness of triangulations established a new branch in topology, namely piecewise linear topology (or PL topology). Its main purpose is to study the topological properties of simplicial complexes and their generalizations, cell-complexes.
Simplicial complexes
Abstract simplicial complexes An abstract simplicial complex above a set V {\displaystyle V} is a system T ⊂ P ( V ) {\displaystyle {\mathcal {T}}\subset {\mathcal {P}}(V)} of non-empty subsets such that:
{ v 0 } ∈ T {\displaystyle \{v_{0}\}\in {\mathcal {T}}} for each v 0 ∈ V {\displaystyle v_{0}\in V} ; if E ∈ T {\displaystyle E\in {\mathcal {T}}} and ∅ ≠ F ⊂ E , {\displaystyle \emptyset \neq F\subset E,} then F ∈ T {\displaystyle F\in {\mathcal {T}}} . The elements of T {\displaystyle {\mathcal {T}}} are called simplices, the elements of V {\displaystyle V} are called vertices. A simplex with n + 1 {\displaystyle n+1} vertices has dimension n {\displaystyle n} by definition. The dimension of an abstract simplicial complex is defined as dim ( T ) = sup { dim ( F ) : F ∈ T } ∈ N ∪ ∞ {\displaystyle {\text{dim}}({\mathcal {T}})={\text{sup}}\;\{{\text{dim}}(F):F\in {\mathcal {T}}\}\in \mathbb {N} \cup \infty } . Abstract simplicial complexes can be realized as geometrical objects by associating each abstract simplex with a geometric simplex, defined below.
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