# Triangulation (topology)

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

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–defined for dimensions 0, 1, 2, . . . inductively–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, *t* 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. But in this article the word "triangulation" is just used to mean homeomorphic to a simplicial complex.

For manifolds of dimension at most 4, any triangulation of a manifold is a piecewise linear triangulation: 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* − 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 sphere, a three-dimensional manifold that is not homeomorphic to a sphere. This is the double suspension theorem, due to James W. Cannon and R.D. Edwards in the 1970s.^{[1]}^{[2]} ^{[3]}^{[4]}^{[5]}

The question of which manifolds have piecewise-linear triangulations has led to much research in topology.
Differentiable manifolds (Stewart Cairns, J. H. C. Whitehead, L. E. J. Brouwer, Hans Freudenthal, James Munkres),^{[6]}^{[7]} and subanalytic sets (Heisuke Hironaka and Robert Hardt) admit a piecewise-linear triangulation, technically by passing via the PDIFF category.
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 R. H. Bing in the 1950s, with later simplifications by Peter Shalen.^{[8]}^{[9]} As shown independently by James Munkres, Steve Smale and J. H. C. Whitehead,^{[10]}^{[11]} each of these manifolds admits a smooth structure, unique up to diffeomorphism.^{[9]}^{[12]} 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, Rob Kirby and Larry Siebenmann constructed manifolds that do not have piecewise-linear triangulations (see Hauptvermutung). Further, Ciprian Manolescu proved that there exist compact manifolds of dimension 5 (and hence of every dimension greater than 5) that are not homeomorphic to a simplicial complex, i.e., that do not admit a triangulation.^{[13]}

An important special case of topological triangulation is that of two-dimensional surfaces, or closed 2-manifolds. There is a standard proof that smooth compact surfaces can be triangulated.^{[14]} 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,^{[15]} 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*

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*.

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.^{[16]}^{[17]}^{[18]} 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 *K*_{4}.