Hauptvermutung
The Hauptvermutung (German for main conjecture) of geometric topology is the conjecture that any two triangulations of a triangulable space have a common refinement, a single triangulation that is a subdivision of both of them. It was originally formulated in 1908, by Steinitz and Tietze.
This conjecture is now known to be false. The non-manifold version was disproved by John Milnor in 1961 using Reidemeister torsion.
The manifold version is true in dimensions m ≤ 3. The cases m = 2 and 3 were proved by Tibor Radó and Edwin E. Moise in the 1920s and 1950s, respectively.
An obstruction to the manifold version was formulated by Andrew Casson and Dennis Sullivan in 1967–9 (originally in the simply-connected case), using the Rochlin invariant and the cohomology group H3(M;Z/2Z).
A homeomorphism ƒ : N → M of m-dimensional piecewise linear manifolds has an invariant κ(ƒ) ∈ H3(M;Z/2Z) such that for m ≥ 5, ƒ is isotopic to a piecewise linear (PL) homeomorphism if and only if κ(ƒ) = 0. In the simply-connected case and with m ≥ 5, ƒ is homotopic to a PL homeomorphism if and only if [κ(ƒ)] = 0 ∈ [M,G/PL]
...
Wikipedia