serves to relate DIFF and PL, and it is equivalent to PL. 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 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. ==See also==