The Triangulation Conjecture is false in all dimensions n >= 5

You need to know: Polytope, vertex of a polytope, face of a polytope, n-dimensional topological manifold, closed manifold, homeomorphism.

Background: A k-simplex is a k-dimensional polytope which is the convex hull of its vertices. A simplicial complex K is a set of simplices such that (i) every face of a simplex from K is also in K, and (ii) the non-empty intersection of any two simplices in K is a face of both of them. A simplicial complex K is called locally finite if each vertex of K belongs only to finitely many simplices in K. A simplicial triangulation is a homeomorphism to a locally finite simplicial complex.

The Theorem: On 10th March 2013, Ciprian Manolescu submitted to arxiv a paper in which he proved that for every , there exists a closed n-dimensional topological manifold that does not admit a simplicial triangulation.

Short context: It is known that any two-dimensional surface can be approximated by gluing together triangles. The Triangulation Conjecture (in dimension n) states that every n-dimensional topological manifold has a simplicial triangulation. The conjecture is true for , was known to be false in dimension four, and was open for . The Theorem states that the conjecture is false for every .

Links: Free arxiv version of the original paper is here, journal version is here.

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