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| [[File:NURBstatic.svg|thumb|[[Spline (mathematics)|Splines]] are piecewise-smooth, hence in PDIFF, but not globally smooth or piecewise-linear, hence not in DIFF or PL.]]
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| In [[geometric topology]], '''PDIFF,''' for ''p''iecewise ''diff''erentiable, is the [[Category (mathematics)|category]] of [[piecewise]]-[[smooth function|smooth]] [[manifolds]] and piecewise-smooth [[function (mathematics)|maps]] between them. It properly contains Diff – the category of [[smooth manifold]]s and [[smooth function]]s between them – and [[Piecewise linear manifold|PL]] – the category of piecewise linear manifolds and piecewise linear maps between them – and the reason it is defined is to allow one to relate these two categories. Further, piecewise functions such as [[Spline (mathematics)|splines]] and [[polygonal chain]]s are common in mathematics, and PDIFF provides a category for discussing them.
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| == Motivation ==
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| [[File:PDIFF.svg|135px|thumb|PDIFF serves to relate DIFF and PL, and it is equivalent to PL.]]
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| PDIFF is mostly a technical point: smooth maps are not piecewise linear (unless linear), and piecewise linear maps are not smooth (unless globally linear) – the intersection is [[linear map]]s, or more precisely [[affine map]]s (because not based) – so they cannot directly be related: they are separate generalizations of the notion of an affine map.
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| However, while a smooth manifold is not a PL manifold, it carries a canonical PL structure – it is uniquely triangularizable; conversely, not every PL manifold is smoothable. For a particular smooth manifold or smooth map between smooth manifolds, this can be shown by breaking up the manifold into small enough pieces, and then linearizing the manifold or map on each piece: for example, a circle in the plane can be approximated by a triangle, but not by a [[2-gon]], since this latter cannot be linearly embedded.
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| This relation between Diff and PL requires choices, however, and is more naturally shown and understood by including both categories in a larger category, and then showing that the inclusion of PL is an equivalence: every smooth manifold and every PL manifold ''is'' a PDiff manifold. Thus, going from Diff to PDiff and PL to PDiff are natural – they are just inclusion. The map PL to PDiff, while not an equality – not every piecewise smooth function is piecewise linear – is an equivalence: one can go backwards by linearize pieces. Thus it can for some purposes be inverted, or considered an isomorphism, which gives a map <math>\text{Diff} \to \text{PDiff} \to \text{PL}.</math> These categories all sit inside TOP, the category of topological manifold and continuous maps between them.
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| In summary, PDiff is more general than Diff because it allows pieces (corners), and one cannot in general smooth corners, while PL is no less general that PDiff because one can linearize pieces (more precisely, one may need to break them up into smaller pieces and then linearize, which is allowed in PDiff).
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| ==History==
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| That every smooth (indeed, ''C''<sup>1</sup>) manifold has a unique PL structure was originally proven in {{Harv|Whitehead|1940}}. A detailed expositionary proof is given in {{Harv|Munkres|1966}}. The result is elementary and rather technical to prove in detail, so it is generally only sketched in modern texts, as in the brief proof outline given in {{Harv|Thurston|1997}}. A very brief outline is given in {{Harv|McMullen|1997}}, while a short but detailed proof is given in {{Harv|Lurie|2009}}.
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| ==References==
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| {{refbegin}}
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| *{{Citation |first= Jacob |last= Lurie |authorlink= Jacob Lurie |url= http://www-math.mit.edu/~lurie/937notes/937Lecture3.pdf |title= Whitehead Triangulations (Lecture 3)| date= February 13, 2009 }}
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| *{{cite newsgroup |title= Re: PL and DIFF manifolds: a question |authorlink= Curtis T. McMullen |first= C. T. |last=McMullen |date= 21 Aug 1997 |newsgroup= sci.math.research |id= |url= http://www.math.niu.edu/~rusin/known-math/97/pl_diff |accessdate=May 10, 2012}}
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| *{{citation|authorlink= James R. Munkres |first=James R. |last= Munkres |title= Elementary Differential Topology |year= 1966 |series= Annals of Mathematics Studies |volume= 54 |publisher= [[Princeton University Press]] |isbn= 0-69109093-9 |postscript=, Chapter II}}
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| *{{citation|authorlink=William Thurston |first=William |last= Thurston |title= Three-Dimensional Geometry and Topology |publisher=[[Princeton University Press]] |year=1997 |isbn= 978-0-69108304-9 |pages= 194–195 |postscript=, PDIFF described as "piecewise smooth"}}
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| *{{cite jstor|1968861}}
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| {{refend}}
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| [[Category:Geometric topology]]
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