# 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)