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en>Yobot: Tagging using AWB (10703)

2015-01-07T10:13:28Z

Tagging using AWB (10703)

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	The '''Hauptvermutung''' (German for main conjecture) of [[geometric topology]] is the [[conjecture]] that any two [[Triangulation (topology)\|triangulations]] of a [[triangulable space]] have a common refinement, a single triangulation that is ~~a subdivision of both of them. It was originally formulated in 1908, by [[Ernst Steinitz\|Steinitz]]~~ and ~~[[Heinrich Franz Friedrich Tietze\|Tietze]].~~		Ed is what people call me and my spouse doesn't like it at all. The preferred hobby for him and his kids is to perform lacross and he would by no means give it up. Distributing production has been his profession for some time. Alaska is exactly where he's always been residing.<br><br>my homepage ... psychic love readings ([http://alfredarbouw.breda.nl/users/kourtneycardinp this site])

	~~This conjecture is now known to be false~~. The non-manifold version was disproved by [[John Milnor]]<ref>{{Cite journal\|first=John W.\|last= Milnor \|title=Two complexes which are homeomorphic but combinatorially distinct\|journal= [[Annals of Mathematics]]\|volume=74\|year=1961\|issue= 2 \|pages=575–590\|id= {{MathSciNet\|id=133127}}\|doi=10.2307/1970299\|postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for ~~the cite~~ to ~~end in a ".", as necessary. --> \|jstor=1970299}}</ref> in 1961 using [[Analytic torsion\|Reidemeister torsion]].~~

	~~The [[manifold]] version is true in [[dimension]]s {{nowrap\|1=''m'' ≤ 3}}. The cases {{nowrap\|1=''m'' = 2~~ and ~~3}} were proved~~ by [[Tibor Radó]] and [[Edwin E. Moise]]<ref>{{cite book \| last = Moise \| first = Edwin E. \| title = Geometric Topology in Dimensions 2 and 3 \| publisher = New York : Springer-Verlag \| location = New York \| year = 1977 \| isbn = 978-0-387-90220-3 }}</ref> in the 1920s and 1950s, respectively.

	An obstruction to the manifold version was formulated by [[Andrew Casson]] and [[Dennis Sullivan]] in 1967–9 (originally in the [[Simply connected space\|simply-connected]] case), using the [[Rochlin invariant]] and the [[cohomology group]] H<sup>3</sup>(''M'';'''Z'''/2'''Z''').

	~~A [[homeomorphism]] {{nowrap\|1=ƒ : ''N'' → ''M''}} of ''m''-dimensional [[piecewise linear manifold]]s~~ has ~~an invariant {{nowrap\|1=κ(ƒ) ∈ H<sup>3</sup>(''M'';'''Z'''/2'''Z''')}} such that~~ for ~~{{nowrap\|1=''m'' ≥ 5}}, ƒ is [[Homotopy#Isotopy\|isotopic]] to a piecewise linear (PL) homeomorphism if and only if {{nowrap\|1=κ(ƒ) = 0}}~~. ~~In the simply-connected case and with {{nowrap\|1=''m'' ≥ 5}}, ƒ~~ is ~~[[homotopic]] to a PL homeomorphism if and only if~~
	~~{{nowrap\|1=[κ(ƒ)] = 0 ∈ [~~'~~'M'',''G''/PL]}}~~

	~~The obstruction to the manifold Hauptvermutung is now seen as a relative version of the triangulation obstruction of [[Robion Kirby\|Rob Kirby]] and [[Larry Siebenmann]], obtained in 1970~~. ~~The [[Kirby–Siebenmann obstruction]] is defined for any [[Compact space\|compact]] ''m''-dimensional topological manifold ''M''~~
	:<~~math~~>~~\kappa(M)\in H^4(M;\mathbb{Z}/2\mathbb{Z})~~<~~/math~~>
	~~again using the Rochlin invariant~~. ~~For {{nowrap\|1=''m'' ≥ 5}}, ''M'' has a PL structure (i~~.e. ~~can be triangulated by a PL manifold) if and only if {{nowrap\|1=κ~~(ƒ) = 0}}, and if this obstruction is 0 the PL structures are parametrized by H<sup>3</sup>(''M'';'''Z'''/2'''Z'''). In particular there are only a finite number of essentially distinct PL structures on ''M''.

	~~For compact simply-connected manifolds of dimension 4 [[Simon Donaldson]] found examples with an infinite number of inequivalent [~~[PL structure]]s, and [[Michael Freedman]] found the [[E8 manifold]] which not only has no PL structure, but is not even homeomorphic to a simplicial complex. In dimensions greater than 4 the question of whether all compact manifolds are homeomorphic to simplicial complexes is an important open question that may have recently been solved in the negative. On March 13, 2013, Ciprian Manolescu posted a preprint on the ArXiv claiming to show that there are 5-manifolds, and hence in every dimension greater than or equal to 5, that are not homeomorphic to a simplicial complex, and thus that do not admit a simplicial triangulation.

	~~== References ==~~
	~~{{Reflist}}~~

	~~== External links ==~~
	*http://~~www.maths.ed~~.ac.uk/~~~aar~~/~~haupt Additional material, including original sources~~
	*{{cite arXiv\|title=Piecewise linear structures on topological manifolds\|year=2001\|version=\|eprint=math.AT/0105047\|last1=Rudyak \| first1=Yuli B.\|class=math.AT}}
	*Andrew Ranicki (ed.) ~~[http://www.maths.ed.ac.uk/~aar/books/haupt.pdf ''The Hauptvermutung Book''] ISBN 0-7923-4174-0~~
	*Andrew Ranicki [http://www.maths.ed.ac.uk/~aar/slides/orsay.pdf ''High-dimensional manifolds then and now'']

	~~[[Category:Articles with inconsistent citation formats]]~~
	~~[[Category:Disproved conjectures]]~~
	~~[[Category:Geometric topology]]~~
	~~[[Category:Structures on manifolds]]~~
	~~[[Category:Surgery theory]]~~
	~~[[Category:German words and phrases]]~~

en>Gilderien: Disambiguated: non-Abelian → Non-abelian group

2013-07-30T00:22:31Z

Disambiguated: non-Abelian → Non-abelian group

← Older revision		Revision as of 02:22, 30 July 2013
Line 1:		Line 1:
	~~cocina Marche comprende una mezcla abundante de comida rústica y mariscos. El vino de primera necesidad en la zona es verdicghdo~~, ~~un juvenil, fresco consumen que se complementa perfectamente con pescados y mariscos. Cuando haya tomado la decisión de la Marca podría ser el lugar para usted personalmente, ghd le encantarán~~ a ~~entender que solo hay una amplia variedad de soluciones de alojamiento~~. ~~Estos consisten en habitaciones~~, ~~residencias, villas , haciendas, e incluso castlellas. Y Marche auténtica finca está ahora un precio más razonable que en otros componentes de la Italia~~.~~<br><br>~~		The '''Hauptvermutung''' (German for main conjecture) of [[geometric topology]] is the [[conjecture]] that any two [[Triangulation (topology)\|triangulations]] of a [[triangulable space]] have a common refinement, a single triangulation that is a subdivision of both of them. It was originally formulated in 1908, by [[Ernst Steinitz\|Steinitz]] and [[Heinrich Franz Friedrich Tietze\|Tietze]].

	~~If you~~ have ~~any inquiries relating~~ to ~~wherever~~ and ~~how~~ to ~~use~~ [http://~~tinyurl~~.~~com~~/~~qhycvg3 baratas ghd planchas~~]~~, you can contact us at our own web page~~.		This conjecture is now known to be false. The non-manifold version was disproved by [[John Milnor]]<ref>{{Cite journal\|first=John W.\|last= Milnor \|title=Two complexes which are homeomorphic but combinatorially distinct\|journal= [[Annals of Mathematics]]\|volume=74\|year=1961\|issue= 2 \|pages=575–590\|id= {{MathSciNet\|id=133127}}\|doi=10.2307/1970299\|postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. --> \|jstor=1970299}}</ref> in 1961 using [[Analytic torsion\|Reidemeister torsion]].

			The [[manifold]] version is true in [[dimension]]s {{nowrap\|1=''m'' ≤ 3}}. The cases {{nowrap\|1=''m'' = 2 and 3}} were proved by [[Tibor Radó]] and [[Edwin E. Moise]]<ref>{{cite book \| last = Moise \| first = Edwin E. \| title = Geometric Topology in Dimensions 2 and 3 \| publisher = New York : Springer-Verlag \| location = New York \| year = 1977 \| isbn = 978-0-387-90220-3 }}</ref> in the 1920s and 1950s, respectively.

			An obstruction to the manifold version was formulated by [[Andrew Casson]] and [[Dennis Sullivan]] in 1967–9 (originally in the [[Simply connected space\|simply-connected]] case), using the [[Rochlin invariant]] and the [[cohomology group]] H<sup>3</sup>(''M'';'''Z'''/2'''Z''').

			A [[homeomorphism]] {{nowrap\|1=ƒ : ''N'' → ''M''}} of ''m''-dimensional [[piecewise linear manifold]]s has an invariant {{nowrap\|1=κ(ƒ) ∈ H<sup>3</sup>(''M'';'''Z'''/2'''Z''')}} such that for {{nowrap\|1=''m'' ≥ 5}}, ƒ is [[Homotopy#Isotopy\|isotopic]] to a piecewise linear (PL) homeomorphism if and only if {{nowrap\|1=κ(ƒ) = 0}}. In the simply-connected case and with {{nowrap\|1=''m'' ≥ 5}}, ƒ is [[homotopic]] to a PL homeomorphism if and only if
			{{nowrap\|1=[κ(ƒ)] = 0 ∈ [''M'',''G''/PL]}}

			The obstruction to the manifold Hauptvermutung is now seen as a relative version of the triangulation obstruction of [[Robion Kirby\|Rob Kirby]] and [[Larry Siebenmann]], obtained in 1970. The [[Kirby–Siebenmann obstruction]] is defined for any [[Compact space\|compact]] ''m''-dimensional topological manifold ''M''
			:<math>\kappa(M)\in H^4(M;\mathbb{Z}/2\mathbb{Z})</math>
			again using the Rochlin invariant. For {{nowrap\|1=''m'' ≥ 5}}, ''M'' has a PL structure (i.e. can be triangulated by a PL manifold) if and only if {{nowrap\|1=κ(ƒ) = 0}}, and if this obstruction is 0 the PL structures are parametrized by H<sup>3</sup>(''M'';'''Z'''/2'''Z'''). In particular there are only a finite number of essentially distinct PL structures on ''M''.

			For compact simply-connected manifolds of dimension 4 [[Simon Donaldson]] found examples with an infinite number of inequivalent [[PL structure]]s, and [[Michael Freedman]] found the [[E8 manifold]] which not only has no PL structure, but is not even homeomorphic to a simplicial complex. In dimensions greater than 4 the question of whether all compact manifolds are homeomorphic to simplicial complexes is an important open question that may have recently been solved in the negative. On March 13, 2013, Ciprian Manolescu posted a preprint on the ArXiv claiming to show that there are 5-manifolds, and hence in every dimension greater than or equal to 5, that are not homeomorphic to a simplicial complex, and thus that do not admit a simplicial triangulation.

			== References ==
			{{Reflist}}

			== External links ==
			*http://www.maths.ed.ac.uk/~aar/haupt Additional material, including original sources
			*{{cite arXiv\|title=Piecewise linear structures on topological manifolds\|year=2001\|version=\|eprint=math.AT/0105047\|last1=Rudyak \| first1=Yuli B.\|class=math.AT}}
			*Andrew Ranicki (ed.) [http://www.maths.ed.ac.uk/~aar/books/haupt.pdf ''The Hauptvermutung Book''] ISBN 0-7923-4174-0
			*Andrew Ranicki [http://www.maths.ed.ac.uk/~aar/slides/orsay.pdf ''High-dimensional manifolds then and now'']

			[[Category:Articles with inconsistent citation formats]]
			[[Category:Disproved conjectures]]
			[[Category:Geometric topology]]
			[[Category:Structures on manifolds]]
			[[Category:Surgery theory]]
			[[Category:German words and phrases]]

en>Bibcode Bot: Adding 0 arxiv eprint(s), 2 bibcode(s) and 0 doi(s). Did it miss something? Report bugs, errors, and suggestions at User talk:Bibcode Bot

2012-07-30T01:19:50Z

Adding 0 arxiv eprint(s), 2 bibcode(s) and 0 doi(s). Did it miss something? Report bugs, errors, and suggestions at User talk:Bibcode Bot

New page

cocina Marche comprende una mezcla abundante de comida rústica y mariscos. El vino de primera necesidad en la zona es verdicghdo, un juvenil, fresco consumen que se complementa perfectamente con pescados y mariscos. Cuando haya tomado la decisión de la Marca podría ser el lugar para usted personalmente, ghd le encantarán a entender que solo hay una amplia variedad de soluciones de alojamiento. Estos consisten en habitaciones, residencias, villas , haciendas, e incluso castlellas. Y Marche auténtica finca está ahora un precio más razonable que en otros componentes de la Italia.<br><br>

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