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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.

The question is: do there exist piecewise-smooth manifolds? Are they equivalent to something in some dimensions?

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    $\begingroup$ There are objects called "manifolds with corners", arxiv.org/PS_cache/arxiv/pdf/0910/0910.3518v1.pdf but I'm not sure this is somehow related to having an atlas made of piecewise-smooth changes of chart (the latter being -I suppose- the definition of "piecewise-smooth manifold"). $\endgroup$
    – Qfwfq
    Commented Jun 10, 2010 at 8:46
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    $\begingroup$ Charts on a manifold with corners are homeomorphisms from the manifold to open sets in the upper Corner space. An atlas on a manifold with corners is a maximal collection of charts, whose domain's cover the manifold and whose change of coordinates are smooth in the ordinary euclidean sense. I don't see how this is related to the notion of piecewise smoothness $\endgroup$
    – Daniel Barter
    Commented Jun 10, 2010 at 8:55

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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.

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    $\begingroup$ Hm, interesting... And is there a definition of something like your PD, but remaining in the same class under compositions? For instance, all compositions of your PD mappings? $\endgroup$
    – John Vrem
    Commented Jun 10, 2010 at 10:13
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    $\begingroup$ Some kind of Lipschitz condition was once studied by Siebenmann, wasn't it? I think the point was that then both smooth manifolds and PL manifolds have canonical Lipschitz structures and a Lipschitz manifold always arises from some PL structure which is unique up to some natural kind of equivalence. $\endgroup$
    – Tom Goodwillie
    Commented Jun 10, 2010 at 16:51
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    $\begingroup$ Wikipedia (en.wikipedia.org/wiki/PDIFF) claims that you can make a category out of piecewise-smooth manifolds and piecewise-smooth maps. Is Wikipedia wrong on this point? $\endgroup$
    – arsmath
    Commented Oct 30, 2010 at 21:53
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    $\begingroup$ It sounds wrong to me. I haven't looked up Wikipedia's McMullen reference, or McMullen's Thurston reference. $\endgroup$
    – Tom Goodwillie
    Commented Mar 23, 2011 at 23:01

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