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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 functions. This is slightly stronger than the topological notion of a triangulation.
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 (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.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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