Triangulation

A triangulation of a polyhedron, or rectilinear triangulation, is a representation of the polyhedron (cf. Polyhedron, abstract) as the space of a geometric simplicial complex $ K $, that is, a decomposition of it into closed simplices such that any two simplices either do not intersect or intersect along a common face. Rectilinear triangulations of polyhedra serve as the main tool for studying them. Any polyhedron has a triangulation and any two triangulations of it have a common subdivision.

The closed star $ \mathop{\rm St} ( \sigma , T) $ of a simplex $ \sigma $ in a triangulation $ T $ is the union of the simplices of $ T $ containing $ \sigma $. There is a representation of the closed star of a simplex $ \sigma \in T $ as the union (or join, cf. Union of sets) of $ \sigma $ and its link: $ \mathop{\rm St} ( \sigma , T) = \sigma \star \mathop{\rm lk} ( \delta , T) $. In particular, the star of a vertex is a cone over its link. If a simplex $ \sigma \in T $ is represented as the join of two of its faces $ \delta $ and $ \gamma $, then $ \mathop{\rm lk} ( \sigma , T) = \mathop{\rm lk} ( \delta , \mathop{\rm lk} ( \gamma , T)) $. The link of a simplex does not depend on $ T $: If $ \sigma $ is a simplex in rectilinear triangulations $ T _ {1} $, $ T _ {2} $ of the same polyhedron, then the polyhedra $ | \mathop{\rm lk} ( \sigma , T _ {1} ) | $ and $ | \mathop{\rm lk} ( \sigma , T _ {2} ) | $ are PL-homeomorphic. The open star of a simplex $ \sigma \in T $ is defined as the union of the interiors of those simplices of $ T $ containing $ \sigma $ as a face. The open stars of the vertices of a triangulation of a polyhedron $ P $ form an open covering of $ P $. The nerve of this covering (cf. Nerve of a family of sets) is simplicially isomorphic to the triangulation. Two triangulations $ T _ {1} $ and $ T _ {2} $ of polyhedra $ P _ {1} $ and $ P _ {2} $ are combinatorially equivalent if certain subdivisions of them are simplicially isomorphic. In order that two triangulations $ T _ {1} $ and $ T _ {2} $ be combinatorially equivalent it is necessary and sufficient that $ P _ {1} $ and $ P _ {2} $ be PL-homeomorphic. A triangulation of a manifold is said to be combinatorial if the star of any of its vertices is combinatorially equivalent to a simplex. In this case the star of any simplex of the triangulation is also combinatorially equivalent to a simplex.

If $ P $ is a closed subpolyhedron of a polyhedron $ Q $, then any triangulation $ K $ of $ P $ can be extended to some triangulation $ L $ of $ Q $. In this case one says that the pair of geometric simplicial complexes $ ( L, K) $ triangulates the pair $ ( Q, P) $. A triangulation of the direct product $ \sigma \times \delta \in \mathbf R ^ {m} \times \mathbf R ^ {n} $ of two simplices $ \sigma \in \mathbf R ^ {m} $, $ \delta \in \mathbf R ^ {n} $ can be constructed as follows. The vertices of the triangulation are the points $ c _ {ij} = ( a _ {i} b _ {j} ) $, $ 0 \leq i \leq \mathop{\rm dim} \delta $, where $ a _ {i} $ are the vertices of $ \sigma $ and $ g _ {j} $ are the vertices of $ \delta $. The vertices $ c _ {i _ {0} j _ {0} } \dots c _ {i _ {p} j _ {p} } $, where $ i _ {0} \leq \dots \leq i _ {k} $, span a $ k $- dimensional simplex if and only if none of these coincide and $ j _ {0} \leq \dots \leq j _ {k} $. A triangulation of the direct product of two simplicial complexes with ordered vertices can be carried out in the same way.

A triangulation of a topological space, or curvilinear triangulation, is a pair $ ( K, f ) $, where $ K $ is a geometric simplicial complex and $ f: | K | \rightarrow X $ is a homeomorphism. Two triangulations $ ( K, f ) $ and $ ( L, g) $ of a space $ X $ coincide if $ g ^ {-} 1 f: | K | \rightarrow | L | $ is a simplicial isomorphism. If $ \sigma $ is a simplex of a complex $ K $ and $ ( K, f ) $ is a triangulation of $ X $, then the space $ f ( \sigma ) $ endowed with the homeomorphism $ f \ \mid _ \sigma : \sigma \rightarrow f ( \sigma ) $ is called a topological simplex. The star and the link of a topological simplex of a triangulated topological space are defined in the same way as in the case of rectilinear triangulations. If a point $ a \in X $ is a vertex of triangulations $ ( K, f ) $ and $ ( L, g) $ of $ X $, then its links in these triangulations are homotopy equivalent.

References

[1]	P.S. Aleksandrov, "Combinatorial topology" , Graylock , Rochester (1956) (Translated from Russian)
[2]	D.B. Fuks, V.A. Rokhlin, "Beginner's course in topology. Geometric chapters" , Springer (1981) (Translated from Russian)

References

[a1]	E.C. Zeeman, "Seminar on combinatorial topology" , IHES (1963)
[a2]	H. Seifert, W. Threlfall, "Lehrbuch der Topologie" , Chelsea, reprint (1970)
[a3]	I.M. Singer, J.A. Thorpe, "Lecture notes on elementary topology and geometry" , Springer (1967)
[a4]	L.C. Glaser, "Geometrical combinatorial topology" , 1–2 , v. Nostrand (1970)