Triangulation conjecture

Remarks

The question was raised by Hellmuth Kneser in 1926. The conjecture holds in dimensions $n \le 3$. For $n \le 1$ this is elementary, for $n = 2$ it is Radó's theorem, and for $n = 3$ it follows from Moise's theorem. It fails in dimension $4$: the closed $E_8$ manifold of Michael Freedman is not triangulable, which follows from properties of the Casson invariant. It fails in every dimension $n \ge 5$. Galewski and Stern, and independently Matumoto, reduced this case to a question about homology 3-spheres and homology cobordism. Ciprian Manolescu answered that question in 2013 with $\mathrm{Pin}(2)$-equivariant Seiberg–Witten Floer homology, which gives closed non-triangulable manifolds of each dimension $n \ge 5$. The conjecture asks for any simplicial triangulation, not one that is combinatorial. The stronger question whether every manifold has a piecewise linear structure was settled negatively earlier, by Kirby and Siebenmann in 1969, through the Kirby–Siebenmann class. Every smooth manifold is triangulable, so the counterexamples are manifolds without a smooth structure.