Math 2.0

Manifolds and cell complexes Three-manifolds Theorem

Geometrization conjecture

Builds on 3-manifoldOrientabilityPrime decomposition of 3-manifoldsJSJ decompositionIncompressible surfaceRiemannian manifold

Statement

There are eight model geometries, the simply connected homogeneous Riemannian manifolds S3,ℝ3,ℍ3,S2×ℝ,ℍ2×ℝ,SL~(2,ℝ),Nil,Sol,S^3, \quad \R^3, \quad \mathbb{H}^3, \quad S^2 \times \R, \quad \mathbb{H}^2 \times \R, \quad \widetilde{SL}(2, \R), \quad \mathrm{Nil}, \quad \mathrm{Sol}, each with its maximal compatible group of isometries. A manifold has a geometric structure modelled on such a geometry XX if it is a quotient X/ΓX / \Gamma. Here Γ\Gamma is a discrete group of isometries of XX acting freely.

Let MM be a closed connected orientable prime 3-manifold. Then MM contains a finite collection of disjoint embedded incompressible tori with the following property. The interior of each component of MM cut along these tori has a geometric structure of finite volume.

Remarks

Thurston stated the conjecture around 1982 and proved it for Haken manifolds. Perelman proved the general case in 2002 and 2003 with Hamilton's Ricci flow with surgery. Except for some manifolds with Sol geometry, the tori can be taken to be those of the JSJ decomposition, applied after the prime decomposition. For closed manifolds with finite fundamental group the conclusion is that M=S3/ΓM = S^3 / \Gamma for a finite group Γ\Gamma acting freely by isometries, which contains the Poincaré conjecture as the case Γ=1\Gamma = 1. Examples: lens spaces are spherical, the 33-torus is Euclidean, and the complement of the figure-eight knot is hyperbolic.

Going deeper

For a student

The theorem reduces many questions about 33-manifolds to questions about the eight geometries. Closed manifolds with six of the geometries are Seifert fiber spaces. Closed Sol manifolds are torus bundles over the circle up to a double cover. The hyperbolic pieces are the rich remaining case. Consequences include the Poincaré conjecture, the classification of finite groups acting freely on S3S^3, and, with later work of Agol and Wise, the virtually Haken conjecture.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

Show the path as a graph
3-manifold, concept, no evidence level3-manifoldconceptConnected sum, concept, no evidence levelConnected sumconceptIncompressible surface, concept, no evidence levelIncompressible surfaceconceptOrientability, concept, no evidence levelOrientabilityconceptRiemannian manifold, concept, no evidence levelRiemannian manifoldconceptSatellite knot, concept, no evidence levelSatellite knotconceptSeifert fiber space, concept, no evidence levelSeifert fiber spaceconceptn-sphere, object, no evidence leveln-sphereobjectPrime decomposition of 3-manifolds, statement, no evidence levelPrime decomposition o…statementJSJ decomposition, statement, no evidence levelJSJ decompositionstatementGeometrization conjecture, statement, no evidence levelGeometrization conjec…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. Connected sum concept
  3. Incompressible surface concept
  4. Orientability concept
  5. Riemannian manifold concept
  6. Satellite knot concept
  7. Seifert fiber space concept
  8. n-sphere object
  9. Prime decomposition of 3-manifolds statement
  10. JSJ decomposition statement
  11. Geometrization conjecture statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Geometrization_conjecture
About this page
TagB27HB43K86
URIm2://B27HB43K86
Kindstatement, theorem
WritingW0 stub
MSC57K30, 57K32, 57M50
StatusThis copy of the site has no build data. Run m2 status B27HB43K86.
About3-manifold
Uses3-manifold, Orientability, Prime decomposition of 3-manifolds, JSJ decomposition, Incompressible surface, Riemannian manifold
Same as
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.