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In topology, an area of mathematics, the virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is virtually Haken. That is, it has a finite cover (a covering space with a finite-to-one covering map) that is a Haken manifold. After the proof of the geometrization conjecture by Perelman, the conjecture was only open for hyperbolic 3-manifolds.

Attributes	Values
rdf:type	yago:WikicatConjectures yago:Abstraction100002137 yago:Cognition100023271 yago:Concept105835747 yago:Content105809192 yago:Hypothesis105888929 yago:Idea105833840 yago:PsychologicalFeature100023100 yago:Speculation105891783
rdfs:label	Virtually Haken conjecture (en) Virtual哈肯猜想 (zh)
rdfs:comment	拓撲學中的virtual哈肯猜想（英語：Virtually Haken conjecture），是指緊緻可定向3維流形，若有無限基本群，就是virtual哈肯（virtually Haken）的，即有一個有限覆蓋（有限對一的覆蓋空間）是。通常認為這個猜想是Friedhelm Waldhausen在一篇1968年的論文最先提到，雖然他未在文中正式寫出。卡比的問題集，將這個猜想正式寫出為問題3.2。幾何化猜想由格里戈里·佩雷爾曼證明了之後，virtual哈肯猜想只剩下待證。 2012年3月12日，在的學術報告講課中提出了這個猜想的證明。隨後在該研究所的3-流形中的浸入曲面工作坊中，他在3月26和28日講了三堂課描述證明大綱。他已發出所宣稱的證明的預印本。他的證明是基於Kahn和Markovic的的證明，證明Malnormal Special Quotient定理時得到的結果，以及Bergeron和Wise的群的cubulation結果。 (zh) In topology, an area of mathematics, the virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is virtually Haken. That is, it has a finite cover (a covering space with a finite-to-one covering map) that is a Haken manifold. After the proof of the geometrization conjecture by Perelman, the conjecture was only open for hyperbolic 3-manifolds. (en)
dcterms:subject	3-manifolds Theorems in topology Conjectures that have been proved
Wikipage page ID	4692183 (xsd:integer)
Wikipage revision ID	1058865938 (xsd:integer)
Link from a Wikipage to another Wikipage	Robion Kirby Vladimir Markovic Institut Henri Poincaré 3-manifolds Theorems in topology Mathematics Geometrization conjecture Friedhelm Waldhausen Fundamental group Haken manifold Documenta Mathematica Topology 3-manifold Daniel Wise (mathematician) Nicolas Bergeron Quanta Magazine Grigori Perelman Covering space Hyperbolic 3-manifold Conjectures that have been proved Jeremy Kahn Ehrenpreis conjecture Ian Agol Orientable manifold Surface subgroup conjecture Virtually fibered conjecture Geometry and Topology Compact manifold Irreducible manifold
Link from a Wikipage to an external page	https://books.google.com/books%3Fid=Ve5uy_dx35cC&dq=kirby%20problem%20list&pg=PA273%7Ctitle=Problems https://www.quantamagazine.org/from-hyperbolic-geometry-to-cube-complexes-and-back-20121002/%7Cwebsite=
sameAs	Virtually Haken conjecture Virtually Haken conjecture Virtually Haken conjecture Virtually Haken conjecture Virtually Haken conjecture
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Cite_web dbt:Reflist dbt:Short_description dbt:Topology-stub
has abstract	In topology, an area of mathematics, the virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is virtually Haken. That is, it has a finite cover (a covering space with a finite-to-one covering map) that is a Haken manifold. After the proof of the geometrization conjecture by Perelman, the conjecture was only open for hyperbolic 3-manifolds. The conjecture is usually attributed to Friedhelm Waldhausen in a paper from 1968, although he did not formally state it. This problem is formally stated as Problem 3.2 in Kirby's problem list. A proof of the conjecture was announced on March 12, 2012 by Ian Agol in a seminar lecture he gave at the Institut Henri Poincaré. The proof appeared shortly thereafter in a preprint which was eventually published in Documenta Mathematica. The proof was obtained via a strategy by previous work of Daniel Wise and collaborators, relying on actions of the fundamental group on certain auxiliary spaces (CAT(0) cube complexes)It used as an essential ingredient the freshly-obtained solution to the surface subgroup conjecture by Jeremy Kahn and Vladimir Markovic. Other results which are directly used in Agol's proof include the Malnormal Special Quotient Theorem of Wise and a criterion of Nicolas Bergeron and Wise for the cubulation of groups. In 2018 related results were obtained by Piotr Przytycki and Daniel Wise proving that mixed 3-manifolds are also virtually special, that is they can be cubulated into a cube complex with a finite cover where all the hyperplanes are embedded which by the previous mentioned work can be made virtually Hanken (en) 拓撲學中的virtual哈肯猜想（英語：Virtually Haken conjecture），是指緊緻可定向3維流形，若有無限基本群，就是virtual哈肯（virtually Haken）的，即有一個有限覆蓋（有限對一的覆蓋空間）是。通常認為這個猜想是Friedhelm Waldhausen在一篇1968年的論文最先提到，雖然他未在文中正式寫出。卡比的問題集，將這個猜想正式寫出為問題3.2。幾何化猜想由格里戈里·佩雷爾曼證明了之後，virtual哈肯猜想只剩下待證。 2012年3月12日，在的學術報告講課中提出了這個猜想的證明。隨後在該研究所的3-流形中的浸入曲面工作坊中，他在3月26和28日講了三堂課描述證明大綱。他已發出所宣稱的證明的預印本。他的證明是基於Kahn和Markovic的的證明，證明Malnormal Special Quotient定理時得到的結果，以及Bergeron和Wise的群的cubulation結果。 (zh)
prov:wasDerivedFrom	wikipedia-en:Virtually_Haken_conjecture?oldid=1058865938&ns=0
page length (characters) of wiki page	5804 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Virtually_Haken_conjecture
is Link from a Wikipage to another Wikipage of	Incompressible surface Arithmetic hyperbolic 3-manifold Friedhelm Waldhausen Haken manifold 3-manifold Daniel Wise (mathematician) Hyperbolic 3-manifold Ehrenpreis conjecture Howson property Ian Agol Surface subgroup conjecture Virtually fibered conjecture Yair Minsky Virtual Haken conjecture
is Wikipage redirect of	Virtual Haken conjecture
is known for of	Ian Agol
is known for of	Ian Agol
is foaf:primaryTopic of	wikipedia-en:Virtually_Haken_conjecture

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