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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]].
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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&ndash;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]].
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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.
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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''').
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


A [[homeomorphism]] {{nowrap|1=ƒ : ''N'' → ''M''}} of ''m''-dimensional [[piecewise linear manifold]]s has an invariant {{nowrap|1=&kappa;(ƒ) ∈ H<sup>3</sup>(''M'';'''Z'''/2'''Z''')}} such that for {{nowrap|1=''m'' ≥ 4}}, ƒ is [[Isotopy|isotopic]] to a piecewise linear (PL) homeomorphism if and only if {{nowrap|1=&kappa;(ƒ) = 0}}. In the simply-connected case and with {{nowrap|1=''m'' ≥ 4}}, ƒ is [[homotopic]] to a PL homeomorphism if and only if
<!--'''PNG''' (currently default in production)
{{nowrap|1=[&kappa;(ƒ)] = 0 ∈ [''M'',''G''/PL]}}
:<math forcemathmode="png">E=mc^2</math>


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''  
'''source'''
:<math>\kappa(M)\in H^4(M;\mathbb{Z}/2\mathbb{Z})</math>
:<math forcemathmode="source">E=mc^2</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=&kappa;(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 is 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.


== References ==
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
{{Reflist}}


== External links ==
==Demos==
*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]]
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
[[Category:Disproved conjectures]]
 
[[Category:Geometric topology]]
 
[[Category:Structures on manifolds]]
* accessibility:
[[Category:Surgery theory]]
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
[[Category:German words and phrases]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.
 
==Test pages ==
 
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
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*[[Inputtypes|Inputtypes (private Wikis only)]]
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==Bug reporting==
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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