PLmanifold

A PL-manifold, or piecewise-linear manifold, is a type of manifold equipped with a piecewise-linear (PL) structure. It is a topological manifold together with a triangulation such that the manifold is homeomorphic to the underlying polyhedron of a simplicial complex, and the changes of charts between simplices are given by linear maps on overlaps. Equivalently, a PL-manifold has an atlas of PL charts whose overlaps are PL homeomorphisms.

A defining feature of PL-manifolds is that their local neighborhoods can be modeled by Euclidean space in a combinatorial way. Each point has a neighborhood that is PL-homeomorphic to an open ball in R^n, and the surrounding structure is determined by how simplices fit together. PL maps between PL-manifolds preserve the piecewise-linear structure, meaning they restrict to linear maps on each simplex in appropriate triangulations.

Existence and examples are nuanced. Many smooth manifolds admit compatible PL structures and triangulations, making them PL-manifolds in a natural way. However, not every topological manifold is triangulable, and in higher dimensions there are obstructions to both triangulation and the existence of a PL structure. A classical obstruction is captured by the Kirby–Siebenmann class, which lies in a cohomology group and governs the existence of a PL structure on a topological manifold.

Relation to other categories is a defining feature: the PL category sits between the topological and smooth categories. PL-manifolds provide a bridge between combinatorial topology and differential topology, and they play a central role in areas such as triangulation theory, surgery, and the study of manifolds via simplicial methods.

See also: PL topology, triangulation, smooth manifold, Kirby–Siebenmann invariant.