Math 2.0

Manifolds and cell complexes Three-manifolds Theorem

Virtually Haken conjecture

Builds on 3-manifoldOrientabilityHaken manifoldCovering spaceFundamental group

Statement

Let MM be a compact connected orientable irreducible 3-manifold with infinite fundamental group, where irreducible means that every embedded 22-sphere bounds a 33-ball. Then MM has a finite covering space that is a Haken manifold.

Remarks

Agol completed the proof in 2012. By the geometrization theorem, the main case is that of a closed hyperbolic 33-manifold. Kahn and Markovic showed that its fundamental group contains many quasi-Fuchsian surface subgroups, and Bergeron and Wise deduced that it acts properly and cocompactly on a CAT(0) cube complex. Agol then proved that such a cubulated hyperbolic group is virtually special in the sense of Haglund and Wise. Wise's theory then gives a finite cover with positive first Betti number, which is Haken. For example, Burton, Rubinstein and Tillmann showed that the Seifert–Weber space is not Haken, so the theorem gives it a finite cover that is Haken.

Going deeper

For a student

Agol's work proves much more: every closed hyperbolic 33-manifold has a finite cover that fibers over the circle, and its fundamental group is large and linear over ℤ\Z. These results answer several of Thurston's questions on 33-manifolds from 1982. They also show that the fundamental groups of such manifolds are residually finite in a strong form, being virtually subgroups of right-angled Artin groups.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph
3-manifold, concept, no evidence level3-manifoldconceptCovering space, concept, no evidence levelCovering spaceconceptFundamental group, concept, no evidence levelFundamental groupconceptHaken manifold, concept, no evidence levelHaken manifoldconceptOrientability, concept, no evidence levelOrientabilityconceptVirtually Haken conjecture, statement, no evidence levelVirtually Haken conje…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. 3-manifold concept
  2. Covering space concept
  3. Fundamental group concept
  4. Haken manifold concept
  5. Orientability concept
  6. Virtually Haken conjecture statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Virtually_Haken_conjecture
About this page
TagGYB411M966
URIm2://GYB411M966
Kindstatement, theorem
WritingW0 stub
MSC57K30, 57K32, 20F65
StatusThis copy of the site has no build data. Run m2 status GYB411M966.
AboutHaken manifold
Uses3-manifold, Orientability, Haken manifold, Covering space, Fundamental group
Same as
Changed2 November 2026

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