Inmathematics, apiecewise linear manifold (PL manifold) is atopological manifold together with apiecewise linear structure on it. Such a structure can be defined by means of anatlas, such that one can pass fromchart to chart in it bypiecewise linear functions. This is slightly stronger than the topological notion of atriangulation.[a]Anisomorphism of PL manifolds is called aPL homeomorphism.
PL, or more precisely PDIFF, sits between DIFF (the category ofsmooth manifolds) and TOP (the category of topological manifolds): it is categorically "better behaved" than DIFF — for example, theGeneralized 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 insurgery theory.
Every smooth manifold has a canonical PL structure — it is uniquelytriangulizable, by Whitehead's theorem ontriangulation (Whitehead 1940)[1][2] — but a PL manifold might not have asmooth structure — it might not besmoothable. This relation can be elaborated by introducing the categoryPDIFF, 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 takecones in PL, but not in DIFF — the cone point is acceptable in PL.A consequence is that theGeneralized Poincaré conjecture is true in PL for dimensions greater than four — the proof is to take ahomotopy sphere, remove two balls, apply theh-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 toexotic spheres.
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 atHauptvermutung.
The obstruction to placing a PL structure on a topological manifoldM is theKirby–Siebenmann class; to be precise, it is theobstruction to placing a PL-structure onM xR and in dimensionsn > 4, the KS class vanishes if and only ifM has at least one PL-structure.
An A-structure on a PL manifold is a structure which gives an inductive way of resolving the PL manifold to a smooth manifold. Compact PL manifolds admit A-structures.[3][4] Compact PL manifolds are homeomorphic toreal-algebraic sets.[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.