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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]].

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]].

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=&kappa;(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=&kappa;(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=[&kappa;(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=&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 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'']

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[[Category:Disproved conjectures]]
[[Category:Geometric topology]]
[[Category:Structures on manifolds]]
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