Piecewise linear manifold

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.

Smooth manifolds

Every smooth manifold has a canonical PL structure — it is uniquely triangulizable, by Whitehead's theorem on triangulation — but a PL manifold might not have a smooth structure — it might not be smoothable. This relation can be elaborated by introducing the category PDIFF, 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 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 four — 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.

Topological manifolds

Main article: Hauptvermutung

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.

The obstruction to placing a PL structure on a topological manifold M is the Kirby–Siebenmann class; to be precise, it is the obstruction to placing a PL-structure on M x R and in dimensions n 4, the KS class vanishes if and only if M has at least one PL-structure.

Real algebraic sets

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. Compact PL manifolds are homeomorphic to real-algebraic sets. 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.

Combinatorial manifolds and digital manifolds

• A combinatorial manifold is a kind of manifold which is discretization of a manifold. It usually means a piecewise linear manifold made by simplicial complexes.
• A digital manifold is a special kind of combinatorial manifold which is defined in digital space. See digital topology.

Notes

References

References

1. Lurie, Jacob. (February 13, 2009). "Whitehead Triangulations (Lecture 3)".
2. M.A. Shtan'ko. "Topology of manifolds".
3. (1980). "A topological resolution theorem". [[Bulletin of the American Mathematical Society]].
4. (1981). "A topological resolution theorem". [[Publications Mathématiques de l'IHÉS]].
5. (1980). "A topological characterization of real algebraic varieties". Bulletin of the American Mathematical Society.
6. (1981). "Real algebraic structures on topological spaces". Publications Mathématiques de l'IHÉS.