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Concept
Piecewise linear manifold
Summary
In mathematics, a piecewise linear (PL) manifold is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas, such that one can pass from chart to chart in it by piecewise linear functions. This is slightly stronger than the topological notion of a triangulation.
An isomorphism of PL manifolds is called a PL homeomorphism.
PL, or more precisely PDIFF, sits between DIFF (the category of smooth manifolds) and TOP (the category of topological manifolds): it is categorically "better behaved" than DIFF — for example, the Generalized Poincaré conjecture is true in PL (with the possible exception of dimension 4, where it is equivalent to DIFF), but is false generally in DIFF — but is "worse behaved" than TOP, as elaborated in surgery theory.
Smooth manifolds have canonical PL structures — they are uniquely triangulizable, by Whitehead's theorem on triangulation — but PL manifolds do not always have smooth structures — they are not always smoothable. This relation can be elaborated by introducing the category PDIFF, which contains both DIFF and PL, and is equivalent to PL.
One way in which PL is better behaved than DIFF is that one can take cones in PL, but not in DIFF — the cone point is acceptable in PL.
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.
Hauptvermutung
Not every topological manifold admits a PL structure, and of those that do, the PL structure need not be unique—it can have infinitely many. This is elaborated at Hauptvermutung.
The obstruction to placing a PL structure on a topological manifold is the Kirby–Siebenmann class. To be precise, the Kirby-Siebenmann class is the obstruction to placing a PL-structure on M x R and in dimensions n > 4, the KS class vanishes if and only if M has at least one PL-structure.
Official source
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