Piecewise linear manifold

From Wikipedia, the free encyclopedia

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.

Contents 1 Relation to other categories of manifolds 1.1 Smooth manifolds 1.2 Topological manifolds 2 See also 3 References

[edit] 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.

[edit] Smooth manifolds

Smooth manifolds have canonical PL structures – they are uniquely triangulizable, by Whitehead's theorem on triangulation^[1][2] – 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.

The obstruction to smoothing is the Kirby–Siebenmann class.

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 spheres.

[edit] Topological manifolds

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.

[edit] See also

• Kirby–Siebenmann class
• Hauptvermutung
• Simplicial manifold

[edit] References

• Rudyak, Yuli B. (2001). Piecewise linear structures on topological manifolds. arΧiv:math.AT/0105047. 
• SpringerLink: Topology of manifolds