Чиприян Манолеску что-то охуенное доказал. Что существуют нетриангулируемые многообразия размерности больше 4.

http://arxiv.org/abs/1303.2354

Pin(2)-equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture
Ciprian Manolescu
(Submitted on 10 Mar 2013)

We define Pin(2)-equivariant Seiberg-Witten Floer homology for rational homology 3-spheres equipped with a spin structure. The analogue of Froyshov's correction term in this setting is an integer-valued invariant of homology cobordism whose mod 2 reduction is the Rokhlin invariant. As an application, we show that there are no homology 3-spheres Y of Rokhlin invariant one such that Y # Y bounds an acyclic smooth 4-manifold. By previous work of Galewski-Stern and Matumoto, this implies the existence of non-triangulable high-dimensional manifolds.