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.
Read more about this topic: Piecewise Linear Manifold
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