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$$\to$$ 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.

Piecewise linear structures
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. The link of a simplex s in a simplicial complex K is a subcomplex of K consisting of the simplices t that are disjoint from s and such that both s and t are faces of some higher-dimensional simplex in K. For instance, in a two-dimensional piecewise-linear manifold formed by a set of vertices, edges, and triangles, the link of a vertex s consists of the cycle of vertices and edges surrounding s: if t is a vertex in this cycle, it and s are both endpoints of an edge of K, and if t is an edge in this cycle, it and s are both faces of a triangle of K. This cycle is homeomorphic to a circle, which is a 1-dimensional sphere. For manifolds of dimension at most 4 this extra property automatically holds: in any simplicial complex homeomorphic to a manifold, the link of any simplex can only be homeomorphic to a sphere. But in dimension n ≥ 5 the (n &minus; 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 Poincaré homology sphere, a three-dimensional manifold that is not homeomorphic to a sphere.

The question of which manifolds have piecewise-linear triangulations has led to much research in topology. Differentiable manifolds (Stewart Cairns,, L.E.J. Brouwer, Hans Freudenthal, ) 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. As shown independently by James Munkres, Steve Smale and , each of these manifolds admits a smooth structure, unique up to diffeomorphism (see, ). 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 cover the 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 Rm, subdivide a cubical lattice in Rm into simplices to give a triangulation of Rm. 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 These points of intersection and their barycentres (corresponding to higher dimensional simplices intersecting X) generate an n-dimensional simplicial subcomplex in Rm, lying wholly inside the tubular neighbourhood. The triangulation is given by the projection of this simplicial complex onto 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.

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; Malniĝ and Mohar 1992). Equivalently, every face is a triangle, every triangle is a face, and the graph is not itself a clique. The clique complex of the graph is then homeomorphic to the surface. The 1-skeletons of Whitney triangulations are exactly the locally cyclic graphs other than K4.