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 → 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 – 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, 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. 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 R.D. Edwards in the 1970s.

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