Thurston’s geometrization conjecture

Thurston’s geometrization conjecture

Thurston’s

geometrization conjecture

, also known simply as the

geometrization conjecture

, states that compact

3-manifolds

can be decomposed into pieces with geometric structures. The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for

surfaces

. It was proposed by

William Thurston

in the late 1970s, and implies several other

conjectures

, such as the PoincarÃÂ© conjecture and Thurston’s elliptization conjecture.

Grigori Perelman

sketched a proof of the geometrization conjecture in 2003 using

Ricci flow

with surgery, which (as of 2006) appears to be essentially correct.

1

The conjecture

A 3-manifold is called closed if it is compact and has no boundary.

Every closed 3-manifold has a prime decomposition: this means it is the

connected sum

of an

essentially unique

collection of prime three-manifolds. This reduces much of the study of 3-manifolds to the case of prime 3-manifolds: those that cannot be written as a non-trivial connected sum.

Every prime closed 3-manifold can be cut along tori, so that the interior of each of the resulting

manifolds

has a geometric structure with finite volume.

There are 8 possible geometric structures in 3

dimensions

, described in the next

section

.
Cutting a prime 3-manifold along tori into pieces that are Seifert manifolds or atoroidal is called the JSJ decomposition: there is a minimal way of doing this, which is essentially unique.

There are

similar

statements for

compact manifolds

with boundary without

S

2

boundary

components

.

In 2 dimensions the analogous statement says that every surface (without boundary) has a geometric structure consisting of a metric with constant curvature; it is not necessary to cut the manifold up first.

2

The eight Thurston geometries

A model

geometry

is a

simply connected

smooth manifold

M

acted on by a Lie group

G

, such that

G

is

maximal

among groups acting smoothly and transitively on

M

with compact

stabilizers

, and there is at least one compact manifold modeled on

M

.

A geometric structure on a manifold is an

isomorphism

of the manifold with

M

⁢

Γ

for some model geometry

M

where

Γ

is a discrete subgroup of

G

acting freely on

M

.

S

2

is the round 2-sphere and

ℍ

2

is the hyperbolic plane.

Seven of the eight geometries (all except hyperbolic) are now clearly understood and known to correspond to Seifert manifolds and torus bundles. Using information about Seifert manifolds, we can restate the conjecture more tersely as:

Every prime, compact 3-manifold falls into exactly one of the following categories:

•

It has a

spherical geometry

.

•

It has a hyperbolic geometry,

•

The

fundamental group

contains a

free abelian group

on two

generators

(the fundamental group of a torus).

•

It has

S

2

×

ℝ

geometry.

3

History

If Thurston’s conjecture is correct, then so is the PoincarÃÂ© conjecture (via Thurston’s elliptization conjecture). The

Fields Medal

was awarded to Thurston in 1982 partially for his proof of the conjecture for Haken manifolds.

The case of 3-manifolds that should be spherical has been slower, but provided the spark needed for Richard Hamilton to develop his Ricci flow. In 1982, Hamilton showed that given a closed 3-manifold with a metric of

positive

Ricci curvature, the Ricci flow would collapse the manifold to a point in finite time, which proves the geometrization conjecture for this case as the metric becomes ”almost round” just before the collapse. He later developed a program to prove the Geometrization Conjecture by Ricci flow.

In 2003 Grigori Perelman sketched a proof of the geometrization conjecture by extending Hamilton’s Ricci flow program to include surgery whenever the Ricci flow produces singularities. As of 2006 the consensus among those who have checked his work is that it is essentially correct, and the details have now been filled in. According to the Clay Mathematics Institute, he may be eligible for the Institute’s Millennium Prize Problems, although he has not submitted his work to a peer-reviewed journal.

This entry was adapted from the

Wikipedia

article

http://en.wikipedia.org/wiki/Thurston's_geometrization_conjecture

Geometrization conjecture as of November 10, 2006.

References

1

G. Perelman,

The entropy formula for the Ricci flow and its geometric applications

, 2002

2

G. Perelman,

Ricci flow with surgery on three-manifolds

, 2003

3

B. Kleiner and J. Lott,

Notes on Perelman’s Papers

, 2006

4

W. Thurston,

Three-dimensional geometry and topology,

Vol. 1. Edited by S. Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997.

Title

Thurston’s geometrization conjecture

Canonical

name

ThurstonsGeometrizationConjecture

Date of creation

2013-03-22 16:23:28

Last modified on

2013-03-22 16:23:28

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

12

Author

PrimeFan (13766)

Entry type

Conjecture

Classification

msc 57R60

Synonym

geometrization conjecture

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