Geometrization of 3-manifolds

The idea of geometrization as a concept of mathematics was introduced in 1980 by William Thurston as a program for the classification of closed three-dimensional manifolds . The goal of the geometrization is to find a characteristic geometric structure on each of these building blocks after breaking down a 3-manifold into basic building blocks. The conjecture established by Thurston and now confirmed that this is always possible represents a generalization of the Poincaré conjecture . The proof was provided by Grigori Perelman with his work on the Ricci River in 2002.

Three-dimensional manifolds

A three-dimensional manifold (or 3-manifold for short) is a topological space that can be described locally by three-dimensional "maps", i.e. looks like normal three-dimensional Euclidean space in small areas . A whole 3-manifold, on the other hand, cannot generally be imagined as a subset of three-dimensional space. This becomes clear by looking at two-dimensional manifolds: A two-dimensional sphere (e.g. the surface of the earth) can be described locally by two-dimensional maps (every ordinary atlas is such a collection of maps). Nevertheless, the whole 2-sphere cannot be represented at once in a two-dimensional Euclidean plane. Analogous to the two-dimensional example, the map changes (now between the three-dimensional maps) determine the structure of the 3-manifold.

This shows a special property of 3-manifolds: While in even higher dimensions it depends on what kind of map change one allows (should they only be continuous , or differentiable , infinitely differentiable, etc.), this differentiation does not play a role up to dimension 3 Role. Mathematically precise, this means that there is exactly one differentiable structure on every topological 3-manifold . As a result, when examining 3-manifolds, topological methods and differential geometric methods can be combined. The branch of mathematics that deals with this is therefore also called three-dimensional geometry and topology .

The aim of three-dimensional geometry and topology is to understand and classify all possible closed (i.e. compact , without boundary ) 3-manifolds . This is a very difficult problem because - in contrast to 2-manifolds, for example - there is a vast number of closed 3-manifolds.

The program for geometrization proposed by William Thurston leads to such a classification by assigning (after a suitable decomposition of the 3-manifold) to each part a certain geometry, which in turn characterizes the topological structure of this part.

Breakdown into components

To “break down” a 3-manifold into components means to first cut it into two components along an embedded two-dimensional sphere . A three-dimensional ball is glued into each of the resulting edges (two spheres) so that the resulting components are again without an edge.

Through this decomposition along 2 spheres one can achieve that the resulting components are irreducible . This means that every embedded 2-sphere borders a 3-ball on one side and a further decomposition would therefore only result in the splitting off of an additional one . One can show that the decomposition into irreducible components is unambiguous except for the order and additional -es. ${\displaystyle S^{3}}$ ${\displaystyle S^{3}}$

If an irreducible component obtained in this way is of the form or has a finite fundamental group , then this component is not broken down further. All other components can now be further broken down along certain tori until one again obtains a clear breakdown, the components of which are all either atoroidal or Seifert-fibered . This decomposition is called the Jaco-Shalen-Johannson decomposition or JSJ decomposition for short . ${\displaystyle S^{2}\times S^{1}}$

In this way, building blocks are obtained from which all 3-manifolds can be reassembled through the reverse process of decomposition (“ connected sum ” and gluing edge gates). To classify the 3-manifolds, it is therefore sufficient to understand the building blocks of the JSJ decomposition, i.e. irreducible manifolds with finite fundamental groups as well as Seifert-fibered and atoroidal manifolds.

Model geometries

→ Main article : (G, X) structure

Thurston understands model geometry to be an abstract space, which looks the same for a resident and which should also be as simple as possible in its topological shape. Precisely this is a complete , simply connected Riemannian manifold with a transitive isometric group . Since the geometry of closed manifolds is to be described, it is also required that there is at least one compact manifold with this geometry, i. that is, there is a subgroup such that is compact. ${\displaystyle X}$ ${\displaystyle {\mathcal {G}}=\mathrm {Isom} (X)}$ ${\displaystyle H\subset {\mathcal {G}}}$ ${\displaystyle X/H}$

Two-dimensional models

Examples of such a model geometry are the Euclidean plane in dimension two (with the 2- torus as the compact quotient) or the two-dimensional sphere , i.e. the surface of a three-dimensional sphere that is already compact itself. Less known is the hyperbolic plane , which represents a third model geometry. All areas of gender can be represented as compact quotients of the hyperbolic level. ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle S^{2}}$ ${\displaystyle \mathbb {H} ^{2}}$ ${\displaystyle \geq 2}$

If the room is to look the same everywhere, it has to be equally curved at every point. In dimension two there is only one curvature quantity , namely the scalar curvature (or Gaussian curvature). It follows that the model geometries are already defined by their constant scalar curvature (except for scaling 0, 1, or −1) and there are no other two-dimensional model geometries apart from the three mentioned.

Three-dimensional models

In dimension three there are also the corresponding models with constant curvature , here these are

• the Euclidean space ${\displaystyle \mathbb {R} ^{3}}$
• the three-dimensional sphere (surface of a four-dimensional sphere) ${\displaystyle S^{3}}$
• the hyperbolic space ${\displaystyle \mathbb {H} ^{3}}$

Product geometries

In addition, there are other three-dimensional model geometries. This is due to the fact that the scalar curvature alone no longer specifies the local shape of the space and the curvature at one point depends on the plane through this point. This can be illustrated with a further three-dimensional model, namely

• the product of 2-sphere and straight . ${\displaystyle S^{2}\times \mathbb {R} }$

This space cannot be represented in three-dimensional Euclidean space, but it can be imagined as follows: The three-dimensional space can be understood like an onion through nested 2 spheres with an increasing diameter. If you now imagine that the diameter of the nested spheres does not grow, but remains constant at 1, if you go from the inside to the outside, you get the desired space. Alternatively, one can imagine 2 spheres lined up along a straight line, but they do not intersect.

If you are at a point in this space, you can either move on a (cross-sectional) sphere, or perpendicular to it along the straight line. In a plane tangential to a sphere, the curvature is 1, but if the plane contains the direction of the straight line, the curvature is 0.

The same construction can be used to form the product with a straight line from the hyperbolic plane:

• ${\displaystyle \mathbb {H} ^{2}\times \mathbb {R} }$

Here the curvature lies between −1 and 0, depending on the direction of the plane under consideration.

A metric like the one in the two product geometries is called homogeneous , but not isotropic : All points are "equal", but in a fixed point there are planes that differ from other planes through this point. Mathematically this means that the isometric group is transitive on the points , but not transitive on the frame (triples of orthonormal tangential vectors in a point).

Geometries with Lie group structure

Finally, there are three other model geometries that have the structure of a Lie group . these are

• the geometry of , the universal superposition of the special linear group ${\displaystyle {\tilde {\mathrm {SL} }}(2,\mathbb {R} )}$ ${\displaystyle \mathrm {SL} (2,\mathbb {R} )}$
• the Nile geometry
• the sol geometry

All three can be described as metrics on matrix groups . While the group of invertible 2 × 2 matrices is with determinant 1, the Nil geometry is on the nilpotent group of the upper 3 × 3 triangular matrices with diagonal 1 (also called Heisenberg group ) and the Sol geometry is on the solvable ( English solvable ) group of all upper 2 × 2 triangular matrices defined. As Lie groups, these groups each have a metric that is invariant under the link operation and is therefore homogeneous. ${\displaystyle \mathrm {SL} _{2}\mathbb {R} }$

Because the group is not simply connected as required , one moves on to its universal superposition. Since this makes no difference for local properties, it is sometimes also referred to as model geometry. ${\displaystyle \mathrm {SL} (2,\mathbb {R} )}$ ${\displaystyle \mathrm {SL} (2,\mathbb {R} )}$

The metric on can also be described as follows: is the group of real Möbiustr transformations and thus the isometrics of the hyperbolic plane . Since an isometry of one-to-one is determined by the image of a selected unit tangential vector, the following applies . , the space of tangential vectors of length 1, now has a metric induced by. The metric constructed in this way induces a metric on the universal overlay . This consideration provides examples for 3-manifolds with -geometry, namely unit tangential bundles of closed hyperbolic surfaces (surfaces of gender at least 2). ${\displaystyle {\tilde {\mathrm {SL} }}(2,\mathbb {R} )}$ ${\displaystyle \mathrm {PSL} (2,\mathbb {R} )}$ ${\displaystyle \mathbb {H} ^{2}}$ ${\displaystyle \mathbb {H} ^{2}}$ ${\displaystyle \mathrm {PSL} (2,\mathbb {R} )\cong UT\mathbb {H} ^{2}}$ ${\displaystyle UT\mathbb {H} ^{2}}$ ${\displaystyle \mathbb {H} ^{2}}$ ${\displaystyle \mathrm {PSL} (2,\mathbb {R} )}$ ${\displaystyle {\tilde {\mathrm {SL} }}(2,\mathbb {R} )}$ ${\displaystyle {\tilde {\mathrm {SL} }}(2,\mathbb {R} )}$

classification

The proof that the models described here are all possibilities of three-dimensional model geometries uses the stabilizer of the isometric group. This is the group of all those isometrics of a model that fix a certain point. In the case of Euclidean space, for example, it consists of the entire orthogonal group O (3) and is therefore three-dimensional, while in the case of product geometries, the direction must be obtained from an isometry, and thus the stabilizer only from the one-dimensional subgroup SO (2) consists. The size of the stabilizer is a measure of the symmetry of the model. ${\displaystyle \mathbb {R} }$

A further distinction can be made by finding a fiber that is invariant under the isometric group and whose leaves are mapped onto themselves by the stabilizer. In the case of product geometries, such a fiber is simply given by the cross-sections or . In any case, such a fiber must again be a two-dimensional model geometry so that the following overview results: ${\displaystyle S^{2}\times \{p\}}$ ${\displaystyle \mathbb {H} ^{2}\times \{p\}}$

Model geometry	stabilizer	structure	(Cut) curvature
Euclidean space ${\displaystyle \mathbb {R} ^{3}}$	three dimensional	isotropic	0 (flat)
3 sphere ${\displaystyle S^{3}}$	three dimensional	isotropic	1 (positive)
hyperbolic space ${\displaystyle \mathbb {H} ^{3}}$	three dimensional	isotropic	−1 (negative)
${\displaystyle S^{2}\times \mathbb {R} }$	one-dimensional	frays over ${\displaystyle S^{2}}$	in the fiber: 1, orthogonal: 0
${\displaystyle \mathbb {H} ^{2}\times \mathbb {R} }$	one-dimensional	frays over ${\displaystyle \mathbb {H} ^{2}}$	in the fiber: −1, orthogonal to it: 0
Nile geometry	one-dimensional	frays over ${\displaystyle \mathbb {R} ^{2}}$	in the fiber: 0, orthogonal to it: 1
${\displaystyle {\tilde {\mathrm {SL} }}(2,\mathbb {R} )}$	one-dimensional	frays over ${\displaystyle \mathbb {H} ^{2}}$	in the fiber: −1, orthogonal to it: 1
Sol geometry	zero dimensional	frays over ${\displaystyle \mathbb {R} }$	orthogonal to the fiber: 0

Thurston's Geometrization Conjecture

If a metric can be selected from a manifold resulting from the decomposition described above, which locally corresponds to one of the eight model geometries, then this manifold is called geometrizable . For example, a torus can be composed of flat, Euclidean maps and can therefore be geometrized.

Thurston has worked intensively on the study of 3-manifolds and found that a large class of them can be geometrized in this sense.

Among other things, he demonstrated this for hook manifolds and received the Fields Medal for it in 1982 . Based on this research, he has proposed that all closed 3-manifolds can be geometrized. This is known as Thurston's Geometrization Conjecture.

Significance of the geometry presumption

If a 3-manifold allows one of the eight model geometries, this provides conclusions about its topology: If the model geometry is not hyperbolic or spherical, it follows that the manifold has a Seifert grain . Since the topology of Seifert manifolds is known, it is considered to be well understood. Since their fundamental group z. As always a subgroup isomorphic to the fundamental group of the 2- torus , possesses, can the Geometrization also be formulated as: ${\displaystyle \mathbb {Z} \times \mathbb {Z} }$

Every irreducible closed 3-manifold satisfies exactly one of the following conditions:

1. It has a spherical metric.
2. It carries a hyperbolic metric.
3. Your fundamental group has a subgroup isomorphic to . ${\displaystyle \mathbb {Z} \times \mathbb {Z} }$

There are considerably more possibilities for spherical and hyperbolic manifolds and these are also not fully classified. Nevertheless, many of their properties are known and the classification represents a purely group-theoretical problem (namely to determine all free discrete subgroups of isometric groups from or , i.e. from or ). ${\displaystyle S^{3}}$ ${\displaystyle \mathbb {H} ^{3}}$ ${\displaystyle \mathrm {SO} (3,\mathbb {R} )}$ ${\displaystyle \mathrm {PSL} (2,\mathbb {C} )}$

From the reformulation of the geometry conjecture, the elliptization conjecture or spherical space conjecture follows

Each closed 3-manifold with finite fundamental group has a spherical metric and therefore is a quotient of the 3-sphere . ${\displaystyle S^{3}/\Gamma }$

and the hyperbolization presumption

Every closed irreducible 3-manifold with an infinite fundamental group is either hyperbolic or its fundamental group contains a subgroup isomorphic to . ${\displaystyle \mathbb {Z} \times \mathbb {Z} }$

Another special case of the geometry conjecture is the well-known Poincaré conjecture :

Every closed 3-manifold with a trivial fundamental group is homeomorphic to the 3-sphere . ${\displaystyle S^{3}}$

State of the presumption

Geometrization has long been known for two-dimensional closed manifolds. From the classification of the surfaces, together with the Gauss-Bonnet formula , it follows that the 2-sphere is the only surface with a spherical geometry, the 2-torus a Euclidean geometry and all surfaces of the higher sex are hyperbolic. ${\displaystyle S^{2}}$ ${\displaystyle T^{2}}$

Richard S. Hamilton was one of the first to try in the 1980s to prove geometrization with the help of the Ricci flow . He succeeded for manifolds with positive Ricci curvature as well as for manifolds on which the Ricci flow does not become singular.

With his work from 2002 and 2003, Grigori Perelman delivered the decisive step in the proof of the geometrization by finding methods that control the flow even when singularities occur . Perelman's work has not yet appeared in a refereed journal, but many mathematicians have dealt with them intensively without finding any significant errors or gaps. For this, Perelman was to be awarded the Fields Medal in 2006, but he turned it down.

Literature and web links

General overview of the geometry conjecture and the Ricci flow

• Bernhard Leeb : Geometrization of 3-dimensional manifolds and Ricci flow: On Perelman's proof of the conjectures of Poincaré and Thurston . ( Memento of April 19, 2008 in the Internet Archive ) (PDF; 437 kB) In: DMV-Mitteilungen , 14 (4), 2006, pp. 213–221.
• John W. Morgan : Recent progress on the Poincaré conjecture and the classification of 3-manifolds . (PDF; 293 kB) In: Bulletin of the AMS , 42 (1), 2005, pp. 57-78.

Topological basics and JSJ decomposition

• Allen Hatcher : Notes on Basic 3-Manifold Topology (PDF; 385 kB) 2000 (English).

Model geometries and Thurston's program

• Peter Scott : The Geometries of 3-Manifolds . (PDF; 7.8 MB) In: Bull. London Math. Soc. , 15, 1983, pp. 401-487.
• William Thurston: Three-Dimensional Geometry and Topology . Princeton University Press, 1997, ISBN 0-691-08304-5 .
• William Thurston: The Geometry and Topology of Three-Manifolds . (PDF) 1980 (English); Preparation of a seminar held by Thurston at Princeton in 1978/79.

Perelman's proof using the Ricci River

• Michael T. Anderson : Geometrization of 3-Manifolds via the Ricci Flow . (PDF; 146 kB) In: Notices of the AMS , 2004 (English). Overview of Perelman's evidence and the Ricci River.
• Grisha Perelman: The entropy formula for the Ricci flow and its geometric applications . Preprint 2002 (English).
• Grisha Perelman: Ricci flow with surgery on three manifolds . Preprint 2003 (English).
• Bruce Kleiner , John Lott : Notes on Perelman's papers . Detailed elaboration of Perelman's proof.
• Laurent Bessières , Gérard Besson , Michel Boileau , Sylvain Maillot , Joan Porti : Geometrization of 3-Manifolds . European Mathematical Society (EMS), 2010, ISBN 978-3-03719-082-1 , ujf-grenoble.fr (PDF)

This version was added to the list of articles worth reading on October 15, 2005 .