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 , 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 of a simplex
in a triangulation
is the union of the simplices of
containing
. There is a representation of the closed star of a simplex
as the union (or join, cf. Union of sets) of
and its link:
. In particular, the star of a vertex is a cone over its link. If a simplex
is represented as the join of two of its faces
and
, then
. The link of a simplex does not depend on
: If
is a simplex in rectilinear triangulations
,
of the same polyhedron, then the polyhedra
and
are PL-homeomorphic. The open star of a simplex
is defined as the union of the interiors of those simplices of
containing
as a face. The open stars of the vertices of a triangulation of a polyhedron
form an open covering of
. The nerve of this covering (cf. Nerve of a family of sets) is simplicially isomorphic to the triangulation. Two triangulations
and
of polyhedra
and
are combinatorially equivalent if certain subdivisions of them are simplicially isomorphic. In order that two triangulations
and
be combinatorially equivalent it is necessary and sufficient that
and
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 is a closed subpolyhedron of a polyhedron
, then any triangulation
of
can be extended to some triangulation
of
. In this case one says that the pair of geometric simplicial complexes
triangulates the pair
. A triangulation of the direct product
of two simplices
,
can be constructed as follows. The vertices of the triangulation are the points
,
, where
are the vertices of
and
are the vertices of
. The vertices
, where
, span a
-dimensional simplex if and only if none of these coincide and
. 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 , where
is a geometric simplicial complex and
is a homeomorphism. Two triangulations
and
of a space
coincide if
is a simplicial isomorphism. If
is a simplex of a complex
and
is a triangulation of
, then the space
endowed with the homeomorphism
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
is a vertex of triangulations
and
of
, 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) |
Comments
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) |
Triangulation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Triangulation&oldid=17514