Geometrization conjecture

The idea of ​​geometrization was introduced in 1980 by William Thurston as a program for the classification of closed three-dimensional manifolds. The aim of the geometrization is to find after the decomposition of a 3-manifold in the basic building blocks on each of these blocks a characteristic geometric structure. The team fielded by Thurston conjecture that this is always possible is a generalization of the Poincaré conjecture and was proved by Grigori Perelman with his work on the Ricci flow.

• 4.1 The importance of the geometrization
• 4.2 state of the presumption

Three-dimensional manifolds

A three-dimensional manifold (or short 3-manifold ) is a topological space, which can be described locally by three-dimensional " map ", ie on small areas so looks like the ordinary three-dimensional Euclidean space. A whole 3-manifold in contrast, can not generally thought of as a subset of the three-dimensional space. This is clear by inspection of two-dimensional manifolds: A two-dimensional sphere ( eg the surface) can be locally described by two-dimensional maps ( any ordinary Atlas is such a collection of maps). Nevertheless, one can not show on once in a two-dimensional Euclidean plane the whole 2-sphere. Analogous to the two-dimensional example, lay the cards change ( now between the three-dimensional maps) the structure of the 3-manifold fixed.

This shows a special characteristic of 3-manifolds: While it matters in even higher dimensions, which one allows for changing cards ( they should only be continuous, or differentiable, infinitely differentiable, etc.), this distinction plays up to dimension 3 no role. Mathematically precise, this means that there is exactly on every topological 3-manifold is a differentiable structure. This has the consequence that it is possible in the study of 3-manifolds combine topological methods and differential-geometric methods. The branch of mathematics that deals with this, are therefore also called three-dimensional geometry and topology.

The aim of the three-dimensional geometry and topology is ( compact without boundary that is ) to understand all sorts of closed 3-manifolds and classify. This is a very difficult problem, since - in contrast to 2 -manifolds - are a vast number of closed 3-manifolds.

Proposed by William Thurston program geometrisation leads to such a classification, by ( a suitable decomposition of the 3-manifold ) each section assigns a particular geometry, which, in turn, characterized by the topological structure of this portion.

Decomposition into components

A 3-manifold into components to " dismantle " means, they first cut open along an embedded two-dimensional sphere into two components. In the resulting edges (two spheres ) are now glued each one a three-dimensional ball, so that the resulting components are again without any border.

Through this decomposition along 2- spheres can be reached that the resulting components are irreducible. This means that every embedded 2-sphere bounded on one side of a 3- ball, and a further decomposition would therefore only the elimination of an additional result. One can show that the decomposition into irreducible components is unique up to order and extra -en.

If a resulting irreducible component of the shape or does it have a finite fundamental group, this component will not be further decomposed. All other components can be along certain Tori now further decompose, until a turn unique decomposition whose components are all either atoroidal Seifert fibered or. This decomposition is called Jaco - Shalen - Johannson decomposition or short JSJ decomposition.

In this way, building blocks that can be put together by the reverse process of decomposition ( " connected sum " and bonding Randtori ) all 3-manifolds again. Therefore, the classification of 3-manifolds, it is sufficient to understand the building blocks of the JSJ - decomposition, ie irreducible manifolds with finite fundamental group and Seifert - fibred manifolds and atoroidale.

Model geometries

Thurston understood as a model geometry spoken clearly an abstract space, everywhere looks the same for a resident, and also in its topological shape should be as simple as possible. Precisely this is a complete, simply connected Riemannian manifold with a transitive isometry group. Since the geometry of the closed manifolds will be described, is also required that there is at least one compact manifold with this geometry, ie is that there is a sub-group, so that compact.

Two-dimensional models

Examples of such a model geometry in two dimension Euclidean plane ( the two - torus as compact quotient ) or the two-dimensional sphere, that is the surface of a three-dimensional sphere, which is already itself compact. Less well known is the hyperbolic plane, which represents a third model geometry. All surfaces of genus can be represented as a compact quotient of the hyperbolic plane.

Now, if the space is to look the same everywhere, it must be curved on every point. In dimension two, there are only a curvature size, namely the scalar curvature (or Gaussian curvature ). It follows that the model geometries (up to scaling 0, 1, or -1) already defined by its constant scalar curvature and there are besides the three mentioned no other two-dimensional model geometries.

Three-dimensional models

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

• Euclidean space,
• The three-dimensional sphere ( surface of a four-dimensional sphere),
• Hyperbolic space.

Product geometries

In addition, however, there is another three-dimensional model geometries. This is because the scalar not itself dictates the shape of the local area and the curvature at a point on the plane in question is dependent through this point. Illustrate this can be at a further three-dimensional model, namely

• The product of the 2-sphere and straight,

This space can not be represented in three-dimensional Euclidean space, but one can imagine it as follows: The three-dimensional space can be conceived as an onion by nested 2- spheres of increasing diameter. If you imagine now requires that the diameter of the nested spheres is not growing, but remains constant at 1 when going from inside to outside, we obtain the desired space. Alternatively, one can imagine two spheres aligned along a straight line, but do not cut yourself.

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

With the same structure can be seen from the hyperbolic plane to form the product of a straight line:

• .

Here the curvature is between -1 and 0, depending on the direction of the considered one level.

A metric as in the two product geometries are known as homogeneous, but not isotropic: Although all the points "equal", but at a fixed point, there are planes which are different from the other layers by this point. Mathematically this means that the isometry group is transitive on the points, but not transitive on the frame ( triples of orthonormal tangent vectors at 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 covering of the special linear group
• The Nile geometry
• Sol geometry.

All three can be described as a metric to matrix groups. While the group of invertible 2 × 2 matrices with determinant 1, is the Nile geometry on the nilpotent group of upper triangular matrices with diagonal 3x3 - 1 (also called Heisenberg group ) and the sol- geometry on the resolvable (English solvable ) group of all upper triangular 2 × 2 defined. Lie groups as these groups carry one metric that is invariant under the operation links and thus homogeneous.

Because as required by the group is not simply connected, the user returns to its universal covering. Since this makes no difference to local properties, is sometimes spoken of as a model geometry.

The metric can also be described as follows: is the group of real Möbius transformations and thus of isometries of the hyperbolic plane. Since an isometry is uniquely determined by the image of a selected Einheitstangentialvektors, applies. , The space of tangent vectors of length 1, now wears a of induced metric. The thus constructed metric on finally induces a metric on the universal cover. This observation provides examples of 3-manifolds with geometry, namely Einheitstangentialbündel closed hyperbolic surfaces ( surfaces of genus at least 2).

Classification

The proof that the models described here are all the possibilities of three-dimensional model geometries, using the stabilizer of the isometry group. This is the group of all those isometries of a model who hold a certain point. In the case of Euclidean space it for example consists of the whole orthogonal group O (3 ) and is therefore in three dimensions, while in the case of product geometries, the direction must be obtained from an isometric, and thus the stabilizer, only the one-dimensional subset of SO (2) there. The size of the stabilizer is a measure of the symmetry of the model.

A further distinction can be made by finding a fibration which is invariant under the isometry group and whose leaves are mapped by the stabilizer itself. In the case of product geometries such a fibration is simply given by the cross-sections respectively. In any case, such a fiber has to be a two-dimensional model geometry again, so that the following overview gives:

Thurston's geometrization

May be mounted on a resulting from the decomposition described above manifold choose a metric that locally corresponds to one of eight model geometries, it is called this variety geometrisierbar. For example, can be a torus of flat, Euclidean cards together and is therefore geometrisierbar.

Thurston has been intensively engaged in the study of 3-manifolds and found that a large class of them is geometrisierbar in this sense.

Among others, he has proven this for hook -manifolds and for 1982 received the Fields Medal. Based on this research, he has established a presumption that can be geometricize all closed 3-manifolds. This is referred to as Thurston geometrization.

Importance of geometrization

Can a 3-manifold one of the eight model geometries, it provides conclusions about their topology: Is the model geometry is not hyperbolic or spherical, it follows that the manifold has a Seifert fibration. Since the topology of Seifert manifolds is known, these are understood as well. Since its fundamental group, for example, always a subgroup isomorphic to the fundamental group of the 2- torus, has, can the geometrization also be formulated as follows:

For spherical and hyperbolic manifolds, there are many more possibilities and these are also not completely classified. However, many of its properties are known and the classification is a purely group theoretic problem ( namely all the free discrete subgroups of the isometry groups of respectively, ie from or to be determined).

From the reformulation of the geometrization Elliptisierungsvermutung or spherical space form conjecture follows

And the Hyperbolisierungsvermutung

Another special case of the geometrization is the famous Poincaré conjecture:

State of the presumption

For two-dimensional closed manifolds the geometrization has been known for long. From the classification of surfaces follows the Gaussian Bonnet formula that the 2-sphere having a spherical surface as a single geometry, a torus - 2 Euclidean geometry and all of the surfaces are of higher genus hyperbolic.

Richard Hamilton has tried in the 1980s as one of the first to prove by means of the Ricci flow, the geometrization. He succeeded for manifolds of positive Ricci curvature, and for such manifolds on which the Ricci flow is not singular.

Grisha Perelman has delivered with his works of 2002 and 2003, the crucial step in the proof of the geometrization by finding methods that control the flow even in the presence of singularities. Perelman's work have not yet been published in a refereed journal, yet many mathematicians have dealt intensively with them, without finding significant errors or omissions. For this purpose, Perelman should be awarded the 2006 Fields Medal, but this he refused from.

Literature / Web Links

• Bernhard Leeb: geometrization of 3-dimensional manifolds and Ricci flow: At Perelman's proof of the Poincaré conjecture of Thurston and (PDF, 437 kB), DMV releases, 14 (4 ), 2006, pages 213-221.
• John W. Morgan: Recent progress on the Poincaré conjecture and the classification of 3 - manifolds (PDF, 293 kB), Bulletin of the AMS, 42 (1 ), 2005, pages 57-78.

• Peter Scott: The Geometries of 3 - Manifolds (PDF, 7.8 MB), 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, 1980 ( English), development of a 1978/79 held by Thurston at Princeton Seminary.

• Michael T. Anderson: geometrization of 3 - manifolds via the Ricci Flow ( PDF, 146 kB), Notices of the AMS, 2004 ( English ). Overview of Perelman's proof and the Ricci flow
• Bruce Kleiner, John Lott: Notes on Perelman 's papers ( English ) Detailed elaboration of Perelman's proof.
• Laurent Bessières, Gérard Besson, Michel Boileau, Sylvain Maillot, Joan Porti: Geometrisation of 3 - manifolds, European Mathematical Society (EMS ), 2010, ISBN 978-3-03719-082-1 ​​, pdf