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Piecewise linear manifold

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 function] s.

An isomorphism of PL manifolds is called a PL homeomorphism .

Relation to other categories of manifolds
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, but not in DIFF – but is "worse behaved" than TOP, as elaborated in [surgery theory] .

Smooth manifolds
[Smooth manifold] s have canonical PL structures – they are uniquely triangulizable, by Whitehead's theorem on triangulation – but PL manifolds do not always have [smooth structure] s – 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 4 – 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 sphere] s.

Topological manifolds

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