A triangulation of a topological manifold is a homeomorphism onto a polyhedron, a space glued together from simplices. This Idea covers when one exists and how many there are: the obstruction sits in codimension 4 and is the Cohen-Saito-Sullivan class css(K) in H4(K; Theta), where Theta is the group of oriented homology 3-spheres modulo homology H-cobordism, and the Rokhlin homomorphism carries css(K) to the Kirby-Siebenmann invariant, so outside dimension 4 a manifold is triangulable exactly when the Bockstein of that invariant vanishes and its triangulations are classified up to concordance by H4(K; ker rok). It also covers why the answer needed gauge theory: Manolescu's Pin(2)-equivariant Seiberg-Witten Floer homology shows the Rokhlin homomorphism does not split, and that is what produces manifolds of dimension 5 and above admitting no triangulation at all.
A triangulation of a topological manifold is a homeomorphism onto a polyhedron, a space assembled from simplices glued along faces; this 26-page expository survey answers the three questions Kneser p…