---
tag: B27HB43K86
kind: statement
role: theorem
name: Geometrization conjecture
msc: [57K30, 57K32, 57M50]
uses: [R20TB2KEQQ, 702SNFQG0W, FZH5T5340Q, TX4MSZ0SRP, 6N9GW98MR6, 7CBT4SM83Y]
about: [R20TB2KEQQ]
sources:
  - url: https://en.wikipedia.org/wiki/Geometrization_conjecture
same_as:
  wikidata: Q1503309
  wikipedia: Geometrization conjecture
---

## Statement

There are eight model geometries, the simply connected homogeneous [[Riemannian manifolds|7CBT4SM83Y]]
$$S^3, \quad \R^3, \quad \mathbb{H}^3, \quad S^2 \times \R, \quad \mathbb{H}^2 \times \R, \quad \widetilde{SL}(2, \R), \quad \mathrm{Nil}, \quad \mathrm{Sol},$$
each with its maximal compatible group of isometries.
A manifold has a *geometric structure* modelled on such a geometry $X$ if it is a quotient $X / \Gamma$.
Here $\Gamma$ is a discrete group of isometries of $X$ acting freely.

Let $M$ be a closed connected [[orientable|702SNFQG0W]] prime [[3-manifold|R20TB2KEQQ]].
Then $M$ contains a finite collection of disjoint embedded [[incompressible|6N9GW98MR6]] tori with the following property.
The interior of each component of $M$ 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|TX4MSZ0SRP]], applied after the [[prime decomposition|FZH5T5340Q]].
For closed manifolds with finite fundamental group the conclusion is that $M = S^3 / \Gamma$ for a finite group $\Gamma$ acting freely by isometries, which contains the [[Poincaré conjecture|3R064KFF68]] as the case $\Gamma = 1$.
Examples: [[lens spaces|AZBTMF196E]] are spherical, the $3$-torus is Euclidean, and the complement of the figure-eight knot is hyperbolic.

## Explanations

### L1 human

Every closed orientable $3$-dimensional space can be cut along spheres and tori into pieces, each of which carries one of eight standard geometries.
The most common geometry, by far, is hyperbolic geometry.
So topology in dimension $3$ is controlled by geometry.

### L2 human

The theorem reduces many questions about $3$-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 $S^3$, and, with later work of Agol and Wise, the virtually Haken conjecture.