Triangulation (topology)

In topology, a branch of mathematics, a triangulation, or triangulation is a decomposition of a space into simplices (triangles, tetrahedra, or their higher -dimensional generalizations ).

Definition

A triangulation of a topological space is given by a ( abstract ) simplicial complex and a homeomorphism

The geometric realization on.

Triangulierbarkeit of manifolds

Manifolds up to the third dimension are always triangulated, this was proved by Tibor Radó 1925 for land and 1952 by Edwin Moise for 3-manifolds. Also in higher dimensions are differentiable manifolds always triangulated by the theorem of Whitehead. A simpler proof was Hassler Whitney with the help of his embedding theorem.

Differentiable All and all PL -manifolds are triangulated. Robion Kirby and Laurence Siebenmann showed that not all topological manifolds possess a PL- structure. But it also showed that there is no PL triangulable manifolds structure.

Andrew Casson showed using the Casson invariant named after him, that 4- manifolds with straight cut shape and signature 8 can not be triangulated. From Freedman's work, we know that there is such a 4 -manifold, it is called. Michael Davis and Tadeusz Januszkiewicz proven that you can get by hyperbolization of a non- triangulable aspherical 4 -manifold.

Late 70s constructed Galewski David and Ronald John Stern a manifold, which can then be triangulated exactly when each manifold of dimension can be triangulated. 2013 proved Ciprian Manolescu that Galewski Star -manifold can not be triangulated. ( Reason is that the Rokhlin - homomorphism does not split. ) Means hyperbolization showed Michael Davis, Jim Fowler and Jean -François Lafont then, that there are non- aspherical manifolds triangulable in dimension.

Main conjecture

The question of the uniqueness of triangulations was as so-called " main conjecture " known: if the geometric realizations and two simplicial complexes are homeomorphic, then there are combinatorially isomorphic subdivisions of simplicial complexes and? The main conjecture is false in general. First counterexamples found John Milnor 1961. Milnor's examples were no manifolds, from the work of Kirby - Siebenmann but also resulted manifolds as counter-examples.

Original motivation for the main conjecture was the proof of the topological invariance of combinatorially defined as invariants of simplicial homology. Despite the failure of the main conjecture such questions can often be answered with the simplicial approximation theorem.

Number of triangulations

The number of triangulations of a manifold can grow exponentially with the number of vertices. For 3- sphere which has been proved by Nevo and Wilson.