View source for Piecewise linear manifold

{{Short description|Topological manifold with a piecewise linear structure on it.}}
In [[mathematics]], a '''piecewise linear manifold''' ('''PL manifold''') is a [[topological manifold]] together with a '''piecewise linear structure''' on it. Such a structure can be defined by means of an [[atlas (topology)|atlas]], such that one can pass from [[chart (topology)|chart]] to chart in it by [[piecewise linear function]]s. This is slightly stronger than the topological notion of a [[Triangulation (topology)|triangulation]].{{efn|A PL structure also requires that the link of a simplex be a PL-sphere. An example of a topological triangulation of a manifold that is not a PL structure is, in dimension ''n''&nbsp;≥&nbsp;5, the (''n''&nbsp;&minus;&nbsp;3)-fold  [[Suspension (topology)|suspension]] of the [[Poincaré homology sphere|Poincaré sphere]] (with some fixed triangulation): it has a simplex whose link is the Poincaré sphere, a three-dimensional manifold that is not homeomorphic to a sphere, hence not a PL-sphere. See {{slink|Triangulation (topology)|Piecewise linear structures}} for details.}}
An [[isomorphism]] of PL manifolds is called a '''PL homeomorphism'''{{anchor|PL homeomorphism}}.
== Relation to other categories of manifolds ==
[[File:PDIFF.svg|135px|thumb|[[PDIFF]] serves to relate DIFF and PL, and it is equivalent to PL.]]
PL, or more precisely PDIFF, sits between DIFF (the category of [[smooth manifold]]s) 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 ===
Smooth manifolds have canonical PL structures — they are uniquely ''triangulizable,'' by Whitehead's theorem on [[triangulation (topology)|triangulation]] {{Harv|Whitehead|1940}}<ref>{{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 }}</ref><ref>{{springer|id=T/t093230|author=M.A. Shtan'ko|title=Topology of manifolds}}</ref> — but PL manifolds do not always have [[smooth structure]]s — they are not always ''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 [[Cone (topology)|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|''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 sphere]]s.
=== Topological manifolds ===
{{main|Hauptvermutung}}
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 is the [[Kirby–Siebenmann class]].  To be precise, the Kirby-Siebenmann class is the [[Obstruction theory|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.<ref>{{cite journal |first1=S. |last1=Akbulut |first2=L. |last2=Taylor |title=A topological resolution theorem |journal=[[Bulletin of the American Mathematical Society]] |series=(N.S.) |volume=2 |issue=1 |year=1980 |pages=174–176 |doi=10.1090/S0273-0979-1980-14709-6 |doi-access=free }}</ref><ref>{{cite journal |first1=S. |last1=Akbulut |first2=L. |last2=Taylor |title=A topological resolution theorem |journal=[[Publications Mathématiques de l'IHÉS]] |volume=53 |issue=1 |year=1981 |pages=163–196 |doi=10.1007/BF02698689 |s2cid=121566364 |url=http://www.numdam.org/item/PMIHES_1981__53__163_0/ }}</ref> Compact PL manifolds are homeomorphic to [[real algebraic set|real-algebraic set]]s.<ref>{{cite journal |first1=S. |last1=Akbulut |first2=H. C. |last2=King |title=A topological characterization of real algebraic varieties |journal=Bulletin of the American Mathematical Society |series=(N.S.) |volume=2 |issue=1 |year=1980 |pages=171–173 |doi=10.1090/S0273-0979-1980-14708-4 |doi-access=free }}</ref><ref>{{cite journal |first1=S. |last1=Akbulut |first2=H. C. |last2=King |title=Real algebraic structures on topological spaces |journal=Publications Mathématiques de l'IHÉS |volume=53 |issue=1 |year=1981 |pages=79–162 |doi=10.1007/BF02698688 |s2cid=13323578 |url=http://www.numdam.org/item/PMIHES_1981__53__79_0/ }}</ref>  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==
*[[Simplicial manifold]]
==Notes==
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==References==
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*{{cite journal | authorlink = J. H. C. Whitehead | first = J. H. C. | last = Whitehead | title = On ''C''<sup>1</sup>-Complexes | journal = [[The Annals of Mathematics]] | series = Second Series | volume = 41 | number = 4 | date=October 1940 | pages = 809–824 | doi= 10.2307/1968861 | jstor= 1968861}}
*{{cite arXiv | last=Rudyak | first=Yuli B. | year=2001 | title=Piecewise linear structures on topological manifolds | eprint=math.AT/0105047 }}
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[[Category:Structures on manifolds]]
[[Category:Geometric topology]]
[[Category:Manifolds]]