PDIFF
From Wikipedia, the free encyclopedia
PDIFF is the category of piecewise-smooth manifolds and piecewise-smooth maps between them. It contains Diff and PL. This is mostly a technical point: a smooth manifold is not a PL manifold, but carries a canonical PL structure (conversely, not every PL manifold is smoothable). However, every smooth manifold and every PL manifold is a PDIFF manifold.
That is, to go from Diff to PL requires linearizing the manifolds and maps between them, while Diff to PDiff and PL to PDiff are natural, and then going PDiff to PL is the step of linearizing the pieces.
[edit] See also
- McMullen on Known Math
- Thurston, "Three-Dimensional Geometry and Topology", PUP, 1997. (Discussion of PDiff)
