### Geometrisation of 3-Manifolds (EMS Tracts in Mathematics)

The first reference book on the subject originated in the annotated notes that the young student Seifert took during the courses of algebraic topology given by Threlfall. In , Seifert introduces a particular class of 3-manifolds, known as Seifert manifolds or Seifert fiber spaces. They have been since widely studied, well understood, and have given a great impact on the modern understanding of 3-manifolds.

They suit many nice properties, most of them being already known since the deep work of Seifert. Nevertheless one of their main properties, the so-called Seifert fiber space theorem, has been a long standing conjecture before its proof was completed by a huge collective work involving Waldhausen, Gordon and Heil, Jaco and Shalen, Scott, Mess, Tukia, Casson and Jungreis, and Gabai, for about twenty years. It has become another example of the characteristic meaning of the n1 for 3-manifolds.

The Seifert fiber space conjecture characterizes the Seifert fiber spaces with infinite n1 in the class of orientable irreducible 3-manifolds in terms of a property of their fundamental groups: they contain an infinite cyclic normal subgroup. It has now become a theorem of major importance in the understanding of compact 3-manifolds as we further explain.

We review here the motivations and applications for the understanding of 3-manifolds of the Seifert fiber space theorem, its generalizations for nonorientable 3-manifolds and PD 3 groups, and the various steps in its proof. Reviews on Seifert Fiber Space. Seifert fibered spaces originally appeared in a paper of Seifert ; they constitute a large class of 3-manifolds and are totally classified by means of a finite set of invariants.

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They have since widely appeared in the literature for playing a central key role in the topology of compact 3-manifolds and being nowadays and since the original paper of Seifert very well known and understood. They have allowed the development of central concepts in 3-manifold topology such as the JSJ-decomposition and Thurston's geometrization conjecture. Seifert Fiber Spaces. Let M and N be two 3-manifolds, each being a disjoint union of a collection of simple closed curves called fibers.

A fiber-preserving homeomorphism from M to N is a homeomorphism which sends each fiber of M onto a fiber of N; in such case M and N are said to be fiber-preserving homeomorphic. In the original definition of Seifert the English version of the original paper of H. Seifert introducing Seifert fibered spaces translated from German by W.

Heil , among other things he classifies them and computes their fundamental group  , a fibration by circles of a closed 3-manifold M is a decomposition into a disjoint union of simple closed curves such that each fiber has a neighborhood fiber-preserving homeomorphic to a fibered solid torus. For a fiber, being regular or exceptional does not depend neither on the fiber-preserving homeomorphism nor on the neighborhood involved.

The definition naturally extends to compact 3-manifolds with nonempty boundary. It requires that the boundary be a disjoint union of fibers; therefore all components in the boundary are closed with Euler characteristic 0 and can only consist of tori and Klein bottles. A Seifert fibration is a foliation by circles; the converse is true for orientable 3-manifolds: it is a result of Epstein  that the orientable Seifert fiber spaces are characterized among all compact orientable 3-manifolds as those that admit a foliation by circles.

For nonorientable 3-manifolds the two notions do not agree because of the possibility for a foliation by circles to be locally fiber-preserving homeomorphic to a fibered solid Klein bottle. The manifold obtained is nonorientable, with boundary a Klein bottle, and is naturally foliated by circles.

Figure 1.

Modern considerations have pointed out a need to enlarge the original definition of Seifert in order to englobe this phenomenon for nonorientable 3-manifolds. In a more modern terminology, a compact 3-manifold is a Seifert fiber space whenever it admits a foliation by circles. We will talk of a Seifert bundle to emphasize the nuance. It is a consequence of Epstein's result that such manifolds are those that decompose into disjoint unions of fibers with all fibers having a neighborhood fiber-preserving homeomorphic to a fibered solid torus or to a fibered solid Klein bottle, the latter appearing only for nonorientable manifolds.

As above, given a Seifert fibration of 3-manifold, all fibers fall in two parts, depending only on the fibration: the regular and the exceptional fibers; the latter can be either isolated or nonisolated. Let M be a Seifert bundle; given a Seifert fibration of M, the space of fibers, obtained by identifying each fiber to a point, is a surface B, called the base of the fibration.

This surface is naturally endowed with a structure of orbifold. Its singularities consists only of a finite number of cone points which are the images of the isolated exceptional fibers and of a finite number of reflector circles and lines which are the images of the nonisolated exceptional fibers. This base orbifold is an invariant of the Seifert fibration. In a modern terminology involving orbifolds, Seifert bundles are those 3-manifolds in the category of 3-orbifolds which are circle bundles over a 2-dimensional orbifold without corner reflectors singularity.

This yields a second invariant e, the Euler number associated to the bundle cf. A Seifert bundle M is a circle bundle over a 2-dimensional orbifold, and it follows that the fibration lifts to the universal covering space M of M into a fibration by circles or lines over an orbifold B without proper covering. When the fibers are circles, it can only be S2 with at most 2 cone points. In such case M is obtained by gluing on their boundary two fibered solid tori, and therefore it is a lens space, so that its universal covering space can only be S3 or S2 x R.

Proposition 1. Moreover, the Seifert fibration lifts in the universal cover to a foliation by lines in both two former cases or circles in the latter case. Recall that a 3-manifold M is irreducible when every sphere embedded in M bounds a ball. An irreducible 3-manifold which does not contain any embedded 2-sided P2 is said to be P2-irreducible. Now by Alexander's theorems S3 and R 3 are irreducible, and it follows that any 3-manifold covered by S3 or R3 is P2-irreducible. So that the only.

Theorem 2. The only irreducible and non-P -irreducible Seifert fiber bundle is P xS. Consider a two-sided simple curve in the base orbifold that consists only of nonsingular points and is closed or has its both extremities lying in the boundary. Whenever that curve does not bound on one of its sides a disk with at most one cone point, its preimage yields a properly embedded two-sided incompressible surface. Theorem 3. Existence of a Seifert fibration on a manifold M has a strong consequence on its fundamental group.

We have seen that the fibration lifts to the universal cover space M into a foliation by circles or lines with basis B. The kernel ker 0 consists of. Therefore ker 0 acts freely on each fiber. In case M is fibered by lines, ker 0 is infinite cyclic. In case M is S3, ker 0 is finite cyclic and nx M is finite.

Finally, since the covering restricts onto the fibers of M to covers of the fibers of M, ker 0 is generated by any regular fiber. Theorem 4.

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Let M be a Seifert bundle with base orbifold B. Then one has the short exact sequence:. Moreover N is infinite whenever nx M is infinite. Statement of the Seifert Fiber Space Theorem. As seen above an infinite fundamental group of a Seifert bundle contains a normal infinite cyclic subgroup. The Seifert fiber space theorem or S. It can be stated as follows. Theorem 5 Seifert Fiber Space Theorem, s. Let M be an orientable irreducible 3-manifold whose nx is infinite and contains a nontrivial normal cyclic subgroup. Then M is a Seifert fiber space.

It has become a theorem with the common work of a large number of topologists: for the Haken case: Waldhausen Haken manifolds whose group has a nontrivial center. Theorem 6 Seifert Bundle Theorem, S.

Then M is a Seifert bundle. Further Heil and Whitten have generalized the theorem to the nonorientable irreducible 3-manifolds cf. They deduce the torus theorem and geometrization in that case modulo fake P2 xS, that is, modulo the Poincare conjecture. We can also note that those two theorems can be generalized in several ways: for open 3-manifolds and 3-orbifolds , and for PD 3 groups , or by weakening the condition of existence of a normal Z by the existence of a nontrivial finite conjugacy class the condition in S.

It applies also to PD 3 groups . Explanations of the Hypotheses Involved. Let us now discuss the hypotheses of the theorems. So that in both conjectures one can replace:. With the Poincare conjecture now stated by Perelman M is obtained from S1 x S2, P P or from an irreducible 3-manifold by removing a finite number of balls. Therefore the result becomes as follows. Theorem 7. Then the manifold obtained by filling all spheres in dM with balls is a Seifert fiber space.

We will see that nevertheless the result generalizes also in that case by considering what Heil and Whitten call a Seifert bundle mod P. According to the Kneser-Milnor decomposition, such manifolds, after eventually filling all spheres in the boundary with balls and replacing all homotopy balls with balls, become irreducible.

## Geometrisation of 3-Manifolds (EMS Tracts in Mathematics)

Definitions Saddle tower minimal surface. While any small change of the surface increases its area, there exist other surfaces with. In mathematics, hyperbolic Dehn surgery is an operation by which one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold. Hyperbolic Dehn surgery exists only in dimension three and is one which distinguishes hyperbolic geometry in three dimensions from other dimensions.

Such an operation is often also called hyperbolic Dehn filling, as Dehn surgery proper refers to a "drill and fill" operation on a link which consists of drilling out a neighborhood of the link and then filling back in with solid tori. Hyperbolic Dehn surgery actually only involves "filling". We will generally assume that a hyperbolic 3-manifold is complete. Suppose M is a cusped hyperbolic 3-manifold with n cusps. M can be thought of, topologically, as the interior of a compact manifold with toral boundary. Suppose we have chosen a meridian and longitude for each boundary torus, i.

Figure-eight knot of practical knot-tying, with ends joined In knot theory, a figure-eight knot also called Listing's knot or a Cavendish knot is the unique knot with a crossing number of four. This makes it the knot with the third-smallest possible crossing number, after the unknot and the trefoil knot. The figure-eight knot is a prime knot.

Origin of name The name is given because tying a normal figure-eight knot in a rope and then joining the ends together, in the most natural way, gives a model of the mathematical knot. Each volume in the series offers an exposition of an active area of current research, provided by a group of mathematicians. List of books Morgan, John; Gang Tian The Geometrization Conjecture.

CIMM 5. Aspinwall, Paul S. Wilson Dirichlet Branes and Mirror Symmetry. CIMM 4. Morgan, John; Gang Tian CIMM 3. Lecture Notes on Motivic cohomology. CIMM 2. Mirror Symmetry. CIMM 1. In mathematics, a classification theorem answers the classification problem "What are the objects of a given type, up to some equivalence? It gives a non-redundant enumeration: each object is equivalent to exactly one class. A few related issues to classification are the following. The equivalence problem is "given two objects, determine if they are equivalent". A complete set of invariants, together with which invariants are realizable, solves the classification problem, and is often a step in solving it.

A computable complete set of invariants together with which invariants are realizable solves both the classification problem and the equivalence problem. A canonical form solves the classification problem, and is more data: it not only classifies every class, but provides a distinguished canonical element of each class. There exist many classification theorems in mathematics, as described below. Geometry Classification of Euclidean plane isometries Classification theorem of surfaces Clas. In mathematics, Thurston's classification theorem characterizes homeomorphisms of a compact orientable surface.

The case where S is a torus i. In what follows, we assume S has genus at least two, as this is the case Thurston considered. Note, however, that the cases where S has boundary or is not orientable are definitely still of interest. The three types in this classification are not mutually excl. This list of Russian mathematicians includes the famous mathematicians from the Russian Empire, the Soviet Union and the Russian Federation.

Alphabetical list A Arnold Georgy Adelson-Velsky, inventor of AVL tree algorithm, developer of Kaissa, the first world computer chess champion Aleksandr Aleksandrov, developer of CAT k space and Alexandrov's uniqueness theorem in geometry Pavel Alexandrov, author of the Alexandroff compactification and the Alexandrov topology Dmitri Anosov, developed Anosov diffeomorphism Vladimir Arnold, an author of the Kolmogorov—Arnold—Moser theorem in dynamical systems, solved Hilbert's 13th problem, raised the ADE classification and Arnold's rouble problems B Sergey Bernstein, developed the Bernstein polynomial, Bernstein's theorem and Bernstein inequalities in probability theory Nikolay Bogolyubov, mathematician and theoretical physicist, author of the edge-of-the-wedge theorem, Krylov—Bogolyubov theorem, describing function and multiple important contr.

This higher-dimensional Koch curve is a topological 2-manifold. In topology, a branch of mathematics, a topological manifold is a topological space which may also be a separated space which locally resembles real n-dimensional space in a sense defined below. Topological manifolds form an important class of topological spaces with applications throughout mathematics. All manifolds are topological manifolds by definition, but many manifolds may be equipped with additional structure e.

Every manifold has an "underlying" topological manifold, gotten by simply "forgetting" any additional structure the manifold has. It is common to place additional requireme. His advisors there were Vladimir Fock, a physicist, and Boris Delaunay, a mathematician. He completed his Ph. In he became a member of.

A drawing of a butterfly with bilateral symmetry, with left and right sides as mirror images of each other. A circle is said to be symmetric under rotation or to have rotational symmetry. Because the composition o. James W. Cannon born January 30, is an American mathematician working in the areas of low-dimensional topology and geometric group theory. Biographical data James W. Cannon was born on January 30, , in Bellefonte, Pennsylvania. Edmund Burgess. He was a Professor at the University of Wisconsin, Madison from to A perspective projection of a dodecahedral tessellation in H3. Note the recursive structure: each pentagon contains smaller pentagons, which contain smaller pentagons.

This is an example of a subdivision rule arising from a finite universe i. In mathematics, a finite subdivision rule is a recursive way of dividing a polygon or other two-dimensional shape into smaller and smaller pieces. Subdivision rules in a sense are generalizations of fractals. Instead of repeating exactly the same design over and over, they have slight variations in each stage, allowing a richer structure while maintaining the elegant style of fractals.

Substitution tilings are a well-studied type of subdivision rule. Definition A subdivision rule takes a tiling of the plane by polygons and turns it into a new tiling by subdividing each polygon into smaller polygons. It is finite. All such manifolds are prime, orientable, and closed. Spherical 3-manifolds are sometimes called elliptic 3-manifolds or Clifford-Klein manifolds.

The elliptization conjecture, proved by Grigori Perelman, states that conversely all 3-manifolds with finite fundamental group are spherical manifolds.

### 3-manifolds

The fundamental group is either cyclic, or is a central extension of a dihedral, tetrahedral, octahedral, or icosahedral group by a cyclic group of even order. This divides the set of such manifolds into 5 classes, described in the following sections. The spherical manifolds are exactly the manifolds with spherical geometry, one of the 8 geometries of T.

In mathematics, specifically geometry and topology, the classification of manifolds is a basic question, about which much is known, and many open questions remain. Main themes Overview Low-dimensional manifolds are classified by geometric structure; high-dimensional manifolds are classified algebraically, by surgery theory. The case of dimension 4 is somehow a boundary case, as it manifests "low dimensional" behaviour smoothly but not topologically ; see discussion of "low" versus "high" dimension.

Different categories of manifolds yield different classifications; these are related by the notion of "structure", and more general categories have neater theories. Positive curvature is constrained, negative curvature is generic. The abstract classification of high-dimensional manifolds is ineffective: given two manifolds presented as CW complexes, for instance , there is no algorithm to determine if they are isomorphi. The history of group theory, a mathematical domain studying groups in their various forms, has evolved in various parallel threads. There are three historical roots of group theory: the theory of algebraic equations, number theory and geometry.

Early 19th century The earliest study of groups as such probably goes back to the work of Lagrange in the late 18th century. However, this work was somewhat isolated, and publications of Augustin Louis Cauchy and Galois are more commonly referred to as the beginning of group theory. The theory did not develop in a vacuum, and so three important threads in its pre-history are developed here.

Development of permutation groups One foundational root of group theory was the quest of solutions of polynomial equations of degree higher than 4. An early source occurs in the problem of forming an equation of degree m having as its roots m of the ro. Geometrization conjecture. Fintushel, Ronald Pacific Journal of Mathematics. Shioya, T. Kapovitch, V. International Press. Paris: Soc. Kleiner, B. Bessieres, L. Bibcode : InMat. Otal, J.

Surveys in differential geometry. Cambridge, MA: Int. Gromov, M. Hautes Etudes Sci. Bessieres, G. Besson, M. Boileau, S. Maillot, J. European Mathematical Society, Zurich, Bibcode : arXiv Cao, J. Geometrization conjecture topic In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it.

Thurston announced a proof in the s and since then several complete Folders related to Geometrization conjecture: 3-manifolds Revolvy Brain revolvybrain Geometric topology Revolvy Brain revolvybrain Conjectures Revolvy Brain revolvybrain. Mathematical theory of 3-manifolds The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-ma Folders related to 3-manifold: Geometric topology Revolvy Brain revolvybrain 3-manifolds Revolvy Brain revolvybrain.

Mather, he gave a proof that the cohomology of the group of homeomorphisms of a man Folders related to William Thurston: Mathematicians from Washington, D. Virtually fibered conjecture topic In the mathematical subfield of 3-manifolds, the virtually fibered conjecture, formulated by American mathematician William Thurston, states that every closed, irreducible, atoroidal 3-manifold with infinite fundamental group has a finite cover which is a surface bundle over the circle.

The original interest in the virtually fibered conjecture as well as its weaker cousins, such as the virtually Haken conjecture stemmed from the fact that any of these conjectures, combined with Thurston's hyperbolization theor Folders related to Virtually fibered conjecture: 3-manifolds Revolvy Brain revolvybrain Conjectures Revolvy Brain revolvybrain. Thurston elliptization conjecture topic William Thurston's elliptization conjecture states that a closed 3-manifold with finite fundamental group is spherical, i.

References For the proof of the conjectures, see the references in the articles on geometrization con Folders related to Thurston elliptization conjecture: Conjectures that have been proved Revolvy Brain revolvybrain 3-manifolds Revolvy Brain revolvybrain Conjectures Revolvy Brain revolvybrain.

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• Low-dimensional topology topic A three-dimensional depiction of a thickened trefoil knot, the simplest non-trivial knot. Spherical space form conjecture topic In geometric topology, the spherical space form conjecture states that a finite group acting on the 3-sphere is conjugate to a group of isometries of the 3-sphere. List of conjectures topic This is a list of mathematical conjectures. Ricci flow topic Several stages of Ricci flow on a 2D manifold. List of unsolved problems in mathematics topic The Riemann zeta function, subject of the celebrated and influential unsolved problem known as the Riemann hypothesis.

Lists of unsolved problems in mathematics Over the course of time, several lists of unsolved mathematical problems have ap Folders related to List of unsolved problems in mathematics: Pages with DOIs inactive as of August Revolvy Brain revolvybrain Unsolved problems in mathematics Revolvy Brain revolvybrain Mathematics-related lists Revolvy Brain revolvybrain. Virtually Haken conjecture topic In topology, an area of mathematics, the virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is virtually Haken.

Geometrization theorem topic In geometry, geometrization theorem may refer to Thurstons's hyperbolization theorem for Haken 3-manifolds Thurston's geometrization conjecture proved by Perelman, a generalization of the hyperbolization theorem to all compact 3-manifolds. Folders related to Geometrization theorem: Mathematics disambiguation pages Revolvy Brain revolvybrain Mathematical disambiguation Revolvy Brain revolvybrain. Seifert fiber space topic A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. Hyperbolization theorem topic In geometry, Thurston's geometrization theorem or hyperbolization theorem implies that closed atoroidal Haken manifolds are hyperbolic, and in particular satisfy the Thurston conjecture.

Manifol Folders related to Hyperbolization theorem: 3-manifolds Revolvy Brain revolvybrain Theorems in geometry Revolvy Brain revolvybrain Hyperbolic geometry Revolvy Brain revolvybrain. Richard S. Bruce Kleiner topic Bruce Alan Kleiner is an American mathematician, working in differential geometry and topology and geometric group theory.

Hyperbolic volume topic The hyperbolic volume of the figure-eight knot is 2. By Mostow rigidity, when a link complement has a hyperbolic structure, this structure is uniquely determined, and Folders related to Hyperbolic volume: Knot theory Revolvy Brain revolvybrain Hyperbolic geometry Revolvy Brain revolvybrain. Surface subgroup conjecture topic Jeremy Kahn and Vladimir Markovic who first proved the conjecture, Aarhus, Folders related to Surface subgroup conjecture: 3-manifolds Revolvy Brain revolvybrain Conjectures Revolvy Brain revolvybrain.

Hyperbolic geometry topic Lines through a given point P and asymptotic to line R A triangle immersed in a saddle-shape plane a hyperbolic paraboloid , along with two diverging ultra-parallel lines In mathematics, hyperbolic geometry also called Bolyai—Lobachevskian geometry or Lobachevskian geometry is a non-Euclidean geometry.

When geometers first realised they were working with something other than the standard Euclidean geometry they Folders related to Hyperbolic geometry: Hyperbolic geometry Revolvy Brain revolvybrain modern geometry emmanuelbatulan. And NEW members were loved a mathematical name in this file, that they piled writing quality of these hostilities. I give it has a book geometrisation of 3 manifolds ems tracts in mathematics of F that I know not little, because these owners who had in the jS, they was who they reduced.

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