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205:. 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.
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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. Compact PL manifolds are homeomorphic to
282:(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
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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
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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
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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
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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
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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.
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97:) and TOP (the category of topological manifolds): it is categorically "better behaved" than DIFF — for example, the
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is a kind of manifold which is discretization of a manifold. It usually means a piecewise linear manifold made by
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in PL, but not in DIFF — the cone point is acceptable in PL. A consequence is that the
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is a special kind of combinatorial manifold which is defined in digital space. See
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Rudyak, Yuli B. (2001). "Piecewise linear structures on topological manifolds".
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The obstruction to placing a PL structure on a topological manifold is the
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is true in PL for dimensions greater than four — the proof is to take a
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One way in which PL is better behaved than DIFF is that one can take
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Smooth manifolds have canonical PL structures — they are uniquely
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PL, or more precisely PDIFF, sits between DIFF (the category of
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62:. This is slightly stronger than the topological notion of a
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Topological manifold with a piecewise linear structure on it.
431:"A topological characterization of real algebraic varieties"
137:, which contains both DIFF and PL, and is equivalent to PL.
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This relation can be elaborated by introducing the category
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Triangulation (topology) § Piecewise linear structures
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serves to relate DIFF and PL, and it is equivalent to PL.
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on it. Such a structure can be defined by means of an
188:. To be precise, the Kirby-Siebenmann class is the
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360:Bulletin of the American Mathematical Society
209:Combinatorial manifolds and digital manifolds
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78:Relation to other categories of manifolds
125:) — but PL manifolds do not always have
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468:Publications Mathématiques de l'IHÉS
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313:Whitehead Triangulations (Lecture 3)
390:"A topological resolution theorem"
355:"A topological resolution theorem"
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462:Akbulut, S.; King, H. C. (1981).
429:Akbulut, S.; King, H. C. (1980).
388:Akbulut, S.; Taylor, L. (1981).
353:Akbulut, S.; Taylor, L. (1980).
448:10.1090/S0273-0979-1980-14708-4
374:10.1090/S0273-0979-1980-14709-6
146:Generalized Poincaré conjecture
99:Generalized Poincaré conjecture
152:, remove two balls, apply the
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274: − 3)-fold
70:of PL manifolds is called a
336:Encyclopedia of Mathematics
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117:by Whitehead's theorem on
60:piecewise linear functions
48:piecewise linear structure
18:Piecewise linear structure
511:The Annals of Mathematics
36:piecewise linear manifold
565:Structures on manifolds
331:"Topology of manifolds"
329:M.A. Shtan'ko (2001) ,
217:combinatorial manifold
186:Kirby–Siebenmann class
129:— they are not always
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310:(February 13, 2009),
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504:(October 1940). "On
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248:Simplicial manifold
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570:Geometric topology
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58:to chart in it by
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16:(Redirected from
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508:-Complexes".
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131:smoothable.
68:isomorphism
40:PL manifold
32:mathematics
559:Categories
437:. (N.S.).
363:. (N.S.).
294:References
276:suspension
157:-cobordism
575:Manifolds
416:121566364
341:EMS Press
488:13323578
242:See also
532:1968861
278:of the
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541:arXiv
528:JSTOR
484:S2CID
412:S2CID
317:(PDF)
254:Notes
142:cones
135:PDIFF
87:PDIFF
66:. An
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52:atlas
34:, a
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30:In
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