Triangulation (topology)

In mathematics, topology generalizes the notion of triangulation in a natural way as follows:

A triangulation of a topological space $X$ is a simplicial complex "K", homeomorphic to "X", together with a homeomorphism "h":"K"$o$ "X".

Triangulation is useful in determining the properties of a topological space. For example, one can compute homology and cohomology groups of a triangulated space using simplicial homology and cohomology theories instead of more complicated homology and cohomology theories.

For topological manifolds, there is a slightly stronger notion of triangulation: a piecewise-linear triangulation (sometimes just called a triangulation) is a triangulation with the extra property that the link of any simplex is a piecewise-linear sphere. For manifolds of dimension at most 4 this extra property automatically holds, but in dimension "n" ≥ 5 the ("n"−3)-fold suspension of the Poincaré sphere is a topological manifold (homeomorphic to the "n"-sphere) with a triangulation that is not piecewise-linear: it has a simplex whose link is the suspension of the Poincaré sphere, which is not a manifold (though it is a homology manifold).

Differentiable manifolds (Stewart Cairns, harvs|txt=yes|authorlink=J.H.C. Whitehead|first=J.H.C.|last=Whitehead|year=1940, L.E.J. Brouwer, Hans Freudenthal, harvnb|Munkres|1966) and subanalytic sets (Heisuke Hironaka and Robert Hardt) admit a piecewise-linear triangulation.

Topological manifolds of dimensions 2 and 3 are always triangulable by an essentially unique triangulation (up to piecewise-linear equivalence); this was proved for surfaces by Tibor Radó in the 1920s and for three-manifolds by Edwin E. Moise and RH Bing in the 1950s, with later simplifications by Peter Shalen (harvnb|Moise|1977, harvnb|Thurston|1997). As shown independently by James Munkres, Steve Smale andharvs|txt=yes|authorlink=J.H.C. Whitehead|first=J.H.C.|last=Whitehead|year=1961, each of these manifolds admitsa smooth structure, unique up to diffeomorphism (see harvnb|Milnor|2007, harvnb|Thurston|1997).

In dimension 4, however, the E8 manifold does not admit a triangulation, and some compact 4-manifolds have an infinite number of triangulations, all piecewise-linear inequivalent. In dimension greater than 4, the question of whether all topological manifolds have triangulations is an open problem, though it is known that some do not have piecewise-linear triangulations (see Hauptvermutung).

Explicit methods of triangulation

An important special case of topological triangulation is that of two-dimensional surfaces, or closed 2-manifolds. There is a standard proof that smooth closed surfaces can be triangulated (see Jost 1997). Indeed, if the surface is given a Riemannian metric, each point "x" is contained inside a small convex geodesic triangle lying inside a normal ball with centre "x". The interiors of finitely many of the triangles will coverthe surface; since edges of different triangles either coincide or intersect transversally, this finite set of triangles can be used iteratively to construct a triangulation.

Another simple procedure for triangulating differentiable manifolds was given by Hassler Whitney in 1957, based on his embedding theorem. In fact, if "X" is a closed "n"-submanifold of "R"^m, subdivide a cubical lattice in "R"^m into simplices to give a triangulation of "R"^m. By taking the mesh of the lattice small enough and slightly moving finitely many of the vertices, the triangulation will be in "general position" with respect to "X": thus no simplices of dimension < "s"="m"-"n"intersect "X" and each "s"-simplex intersecting "X"
* does so in exactly one interior point;
* makes a strictly positive angle with the tangent plane;
* lies wholly inside some tubular neighbourhood of "X". These points of intersection and their barycentres (corresponding to higher dimensional simplices intersecting "X") generate an "n"-dimensional simplicial subcomplex in "R"^m, lying wholly inside the tubular neighbourhood. The triangulation is given by the projection of this simplicial complex onto "X".

Graphs on surfaces

A "Whitney triangulation" or "clean triangulation" of a surface is an embedding of a graph onto the surface in such a way that the faces of the embedding are exactly the cliques of the graph (Hartsfeld and Gerhard Ringel 1981; Larrión et al 2002; Malnič and Mohar 1992). Equivalently, every face is a triangle, every triangle is a face, and the graph is not itself a clique. The 1-skeletons of Whitney triangulations are exactly the locally cyclic graphs other than "K"₄.

Plantri and Fullgen are programs for generation of certain types of planar graphs; they were developed by Gunnar Brinkmann and Brendan McKay. [http://cs.anu.edu.au/~bdm/plantri/]

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