Are PL-homeomorphic manifolds diffeomorphic?

Question

Take two smooth manifolds. Since they are smooth, they both possess triangulations. Now assume that the triangulations are related by Pachner moves, that is, the triangulated manifolds are PL-homeomorphic. Are the manifolds smooth then?

I guess I'm asking whether the inclusion of smooth manifolds into PL-manifolds is full.

Answer 1

Consider for example one of Milnor's exotic $7-$spheres and the standard $7-$sphere: they are PL-homeomorphic since the generalized Poincaré conjecture in the PL category is true in dimensions different than 4 (i.e. any two PL-manifold which is a homotopy sphere is PL-homeomorphic to a sphere), but they are not diffeomorphic as shown by Milnor.

Moreover there are PL manifolds which do not admit a compatible differentiable structure (e.g. Kervaire provided such an example).

References:

• M. Kervaire, A manifold which does not admit any differentiable structure, Comm. Math. Helv. 34 (1960), 257–270.
• J. W. Milnor, "On manifolds homeomorphic to the 7-sphere", Annals of Mathematics 64 (2) (1956), 399–405