Piecewise linear manifold

Smooth manifolds have canonical PL structures — they are uniquely triangulizable, by Whitehead's theorem on triangulation (Whitehead 1940)[1][2] — but PL manifolds do not always have smooth structures — they are not always smoothable.

A consequence is that the Generalized Poincaré conjecture is true in PL for dimensions greater than four — the proof is to take a homotopy sphere, remove two balls, apply the h-cobordism theorem to conclude that this is a cylinder, and then attach cones to recover a sphere.

This last step works in PL but not in DIFF, giving rise to exotic spheres.

The obstruction to placing a PL structure on a topological manifold is the Kirby–Siebenmann class.

[5][6] Put another way, A-category sits over the PL-category as a richer category with no obstruction to lifting, that is BA → BPL is a product fibration with BA = BPL × PL/A, and PL manifolds are real algebraic sets because A-manifolds are real algebraic sets.