Piecewise-smooth manifolds? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T13:45:56Z http://mathoverflow.net/feeds/question/27656 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/27656/piecewise-smooth-manifolds Piecewise-smooth manifolds? John Vrem 2010-06-10T07:23:12Z 2010-06-12T04:33:21Z <p>We know about the existence of topological (Top), differentiable (Diff) and piecewise-linear (PL) manifolds, and such things that, say, in four dimensions PL=Diff, but $\ne$Top.</p> <p>The question is: do there exist <em>piecewise-smooth</em> manifolds? Are they equivalent to something in some dimensions?</p> http://mathoverflow.net/questions/27656/piecewise-smooth-manifolds/27673#27673 Answer by Tom Goodwillie for Piecewise-smooth manifolds? Tom Goodwillie 2010-06-10T09:46:41Z 2010-06-12T04:33:21Z <p>A homeomorphism $h:U\rightarrow V$ between open subsets of $\mathbb R^n$ is called piecewise differentiable (PD) -- you could also say piecewise smooth -- if there is a triangulation of $U$ by linear simplices such that the restriction of $h$ to each simplex is a smooth embedding. This class of maps is invariant under composition with diffeomorphisms on one side, and under composition with piecewise linear homeomorphisms on the other. They play an essential role in defining the good notion of compatiblility of a PL structure and a smooth structure. It is a nontrivial result of J H C Whitehead that for every smooth structure there is a compatible PL structure, unique up to a certain equivalence relation. The composition of PD maps is not in general PD, though, so this does not lead (by using "PD atlases") to a notion of PD manifold.</p>