Seifert fiber space

A Seifert fiber space is a 3-manifold together with a "nice" decomposition as a disjoint union of circles. In other words, it is a -bundle (circle bundle) over a 2-dimensional orbifold. Most "small" 3-manifolds are Seifert fiber spaces, and they account for all compact oriented manifolds in 6 of the 8 Thurston geometries of the geometrization conjecture.

Definition

Bild Seiferttorus
A standard fibered torus corresponding to (5,2) is obtained by gluing the top of the cylinder to the bottom by a 2/5 rotation counterclockwise.

A Seifert manifold is a closed 3-manifold together with a decomposition into a disjoint union of circles (called fibers) such that each fiber has a tubular neighborhood that forms a standard fibered torus.

A standard fibered torus corresponding to a pair of coprime integers (a,b) with a>0 is the surface bundle of the automorphism of a disk given by rotation by an angle of 2πb/a (with the natural fibering by circles). If a = 1 the middle fiber is called ordinary, while if a>1 the middle fiber is called exceptional. A compact Seifert fiber space has only a finite number of exceptional fibers.

The set of fibers forms a 2-dimensional orbifold, denoted by B and called the base -also called the orbit surface- of the fibration. It has an underlying 2-dimensional surface B0, but may have some special orbifold points corresponding to the exceptional fibers.

The definition of Seifert fibration can be generalized in several ways. The Seifert manifold is often allowed to have a boundary (also fibered by circles, so it is a union of tori). When studying non-orientable manifolds, it is sometimes useful to allow fibers to have neighborhoods that look like the surface bundle of a reflection (rather than a rotation) of a disk, so that some fibers have neighborhoods looking like fibered Klein bottles, in which case there may be one-parameter families of exceptional curves. In both of these cases, the base B of the fibration usually has a non-empty boundary.

Classification

Herbert Seifert classified all closed Seifert fibrations in terms of the following invariants. Seifert manifolds are denoted by symbols

where: is one of the 6 symbols: , (or Oo, No, NnI, On, NnII, NnIII in Seifert's original notation) meaning:

o1 if B is orientable and M is orientable.
o2 if B is orientable and M is not orientable.
n1 if B is not orientable and M is not orientable and all generators of π1(B) preserve orientation of the fiber.
n2 if B is not orientable and M is orientable, so all generators of π1(B) reverse orientation of the fiber.
n3 if B is not orientable and M is not orientable and g≥ 2 and exactly one generator of π1(B) preserves orientation of the fiber.
n4 if B is not orientable and M is not orientable and g≥ 3 and exactly two generators of π1(B) preserve orientation of the fiber.
g is the genus of the underlying 2-manifold of the orbit surface.
b is an integer, normalized to be 0 or 1 if M is not orientable and normalized to be 0 if in addition some a'i is 2.
(a1,b1),...,(ar,br) are the pairs of numbers determining the type of each of the r exceptional orbits. They are normalized so that 0<bi<ai when M is orientable, and 0<biai/2 when M is not orientable.

The Seifert fibration of the symbol

can be constructed from that of symbol

by using surgery to add fibers of types b and bi/ai.

If we drop the normalization conditions then the symbol can be changed as follows:

  • Changing the sign of both ai and bi has no effect.
  • Adding 1 to b and subtracting ai from bi has no effect. (In other words, we can add integers to each of the rational numbers (b, b1/a1, ..., br/ar provided that their sum remains constant.)
  • If the manifold is not orientable, changing the sign of bi' has no effect.
  • Adding a fiber of type (1,0) has no effect.

Every symbol is equivalent under these operations to a unique normalized symbol. When working with unnormalized symbols, the integer b can be set to zero by adding a fiber of type (1, b).

Two closed Seifert oriented or non-orientable fibrations are isomorphic as oriented or non-orientable fibrations if and only if they have the same normalized symbol. However, it is sometimes possible for two Seifert manifolds to be homeomorphic even if they have different normalized symbols, because a few manifolds (such as lens spaces) can have more than one sort of Seifert fibration. Also an oriented fibration under a change of orientation becomes the Seifert fibration whose symbol has the sign of all the bs changed, which after normalization gives it the symbol

and it is homeomorphic to this as an unoriented manifold.

The sum b + Σbi/ai is an invariant of oriented fibrations, which is zero if and only if the fibration becomes trivial after taking a finite cover of B.

The orbifold Euler characteristic χ(B) of the orbifold B is given by

χ(B) = χ(B0) − Σ(1−1/ai)

where χ(B0) is the usual Euler characteristic of the underlying topological surface B0 of the orbifold B. The behavior of M depends largely on the sign of the orbifold Euler characteristic of B.

Fundamental group

The fundamental group of M fits into the exact sequence

where π1(B) is the orbifold fundamental group of B (which is not the same as the fundamental group of the underlying topological manifold). The image of group π1(S1) is cyclic, normal, and generated by the element h represented by any regular fiber, but the map from π1(S1) to π1(M) is not always injective.

The fundamental group of M has the following presentation by generators and relations:

B orientable:

where ε is 1 for type o1, and is −1 for type o2.

B non-orientable:

where εi is 1 or −1 depending on whether the corresponding generator vi preserves or reverses orientation of the fiber. (So εi are all 1 for type n1, all −1 for type n2, just the first one is one for type n3, and just the first two are one for type n4.)

Positive orbifold Euler characteristic

The normalized symbols of Seifert fibrations with positive orbifold Euler characteristic are given in the list below. These Seifert manifolds often have many different Seifert fibrations. They have a spherical Thurston geometry if the fundamental group is finite, and an S2×R Thurston geometry if the fundamental group is infinite. Equivalently, the geometry is S2×R if the manifold is non-orientable or if b + Σbi/ai= 0, and spherical geometry otherwise.

{b; (o1, 0);} (b integral) is S2×S1 for b=0, otherwise a lens space L(b,1). In particular, {1; (o1, 0);} =L(1,1) is the 3-sphere.

{b; (o1, 0);(a1, b1)} (b integral) is the lens space L(ba1+b1,a1).

{b; (o1, 0);(a1, b1), (a2, b2)} (b integral) is S2×S1 if ba1a2+a1b2+a2b1 = 0, otherwise the lens space L(ba1a2+a1b2+a2b1, ma2+nb2) where ma1n(ba1 +b1) = 1.

{b; (o1, 0);(2, 1), (2, 1), (a3, b3)} (b integral) This is the prism manifold with fundamental group of order 4a3|(b+1)a3+b3| and first homology group of order 4|(b+1)a3+b3|.

{b; (o1, 0);(2, 1), (3, b2), (3, b3)} (b integral) The fundamental group is a central extension of the tetrahedral group of order 12 by a cyclic group.

{b; (o1, 0);(2, 1), (3, b2), (4, b3)} (b integral) The fundamental group is the product of a cyclic group of order |12b+6+4b2 + 3b3| and a double cover of order 48 of the octahedral group of order 24.

{b; (o1, 0);(2, 1), (3, b2), (5, b3)} (b integral) The fundamental group is the product of a cyclic group of order m=|30b+15+10b2 +6b3| and the order 120 perfect double cover of the icosahedral group. The manifolds are quotients of the Poincaré homology sphere by cyclic groups of order m. In particular, {−1; (o1, 0);(2, 1), (3, 1), (5, 1)} is the Poincaré sphere.

{b; (n1, 1);} (b is 0 or 1.) These are the non-orientable 3-manifolds with S2×R geometry. If b is even this is homeomorphic to the projective plane times the circle, otherwise it is homeomorphic to a surface bundle associated to an orientation reversing automorphism of the 2-sphere.

{b; (n1, 1);(a1, b1)} (b is 0 or 1.) These are the non-orientable 3-manifolds with S2×R geometry. If ba1+b1 is even this is homeomorphic to the projective plane times the circle, otherwise it is homeomorphic to a surface bundle associated to an orientation reversing automorphism of the 2-sphere.

{b; (n2, 1);} (b integral.) This is the prism manifold with fundamental group of order 4|b| and first homology group of order 4, except for b=0 when it is a sum of two copies of real projective space, and |b|=1 when it is the lens space with fundamental group of order 4.

{b; (n2, 1);(a1, b1)} (b integral.) This is the (unique) prism manifold with fundamental group of order 4a1|ba1 + b1| and first homology group of order 4a1.

Zero orbifold Euler characteristic

The normalized symbols of Seifert fibrations with zero orbifold Euler characteristic are given in the list below. The manifolds have Euclidean Thurston geometry if they are non-orientable or if b + Σbi/ai= 0, and nil geometry otherwise. Equivalently, the manifold has Euclidean geometry if and only if its fundamental group has an abelian group of finite index. There are 10 Euclidean manifolds, but four of them have two different Seifert fibrations. All surface bundles associated to automorphisms of the 2-torus of trace 2, 1, 0, −1, or −2 are Seifert fibrations with zero orbifold Euler characteristic (the ones for other (Anosov) automorphisms are not Seifert fiber spaces, but have sol geometry). The manifolds with nil geometry all have a unique Seifert fibration, and are characterized by their fundamental groups. The total spaces are all acyclic.

{b; (o1, 0); (3, b1), (3, b2), (3, b3)}    (b integral, bi is 1 or 2) For b + Σbi/ai= 0 this is an oriented Euclidean 2-torus bundle over the circle, and is the surface bundle associated to an order 3 (trace −1) rotation of the 2-torus.

{b; (o1, 0); (2,1), (4, b2), (4, b3)}    (b integral, bi is 1 or 3) For b + Σbi/ai= 0 this is an oriented Euclidean 2-torus bundle over the circle, and is the surface bundle associated to an order 4 (trace 0) rotation of the 2-torus.

{b; (o1, 0); (2, 1), (3, b2), (6, b3)}    (b integral, b2 is 1 or 2, b3 is 1 or 5) For b + Σbi/ai= 0 this is an oriented Euclidean 2-torus bundle over the circle, and is the surface bundle associated to an order 6 (trace 1) rotation of the 2-torus.

{b; (o1, 0); (2, 1), (2, 1), (2, 1), (2, 1)}    (b integral) These are oriented 2-torus bundles for trace −2 automorphisms of the 2-torus. For b=−2 this is an oriented Euclidean 2-torus bundle over the circle (the surface bundle associated to an order 2 rotation of the 2-torus) and is homeomorphic to {0; (n2, 2);}.

{b; (o1, 1); }   (b integral) This is an oriented 2-torus bundle over the circle, given as the surface bundle associated to a trace 2 automorphism of the 2-torus. For b=0 this is Euclidean, and is the 3-torus (the surface bundle associated to the identity map of the 2-torus).

{b; (o2, 1); }   (b is 0 or 1) Two non-orientable Euclidean Klein bottle bundles over the circle. The first homology is Z+Z+Z/2Z if b=0, and Z+Z if b=1. The first is the Klein bottle times S1 and other is the surface bundle associated to a Dehn twist of the Klein bottle. They are homeomorphic to the torus bundles {b; (n1, 2);}.

{0; (n1, 1); (2, 1), (2, 1)}   Homeomorphic to the non-orientable Euclidean Klein bottle bundle {1; (n3, 2);}, with first homology Z + Z/4Z.

{b; (n1, 2); }   (b is 0 or 1) These are the non-orientable Euclidean surface bundles associated with orientation reversing order 2 automorphisms of a 2-torus with no fixed points. The first homology is Z+Z+Z/2Z if b=0, and Z+Z if b=1. They are homeomorphic to the Klein bottle bundles {b; (o2, 1);}.

{b; (n2, 1); (2, 1), (2, 1)}   (b integral) For b=−1 this is oriented Euclidean.

{b; (n2, 2); }   (b integral) For b=0 this is an oriented Euclidean manifold, homeomorphic to the 2-torus bundle {−2; (o1, 0); (2, 1), (2, 1), (2, 1), (2, 1)} over the cicle associated to an order 2 rotation of the 2-torus.

{b; (n3, 2); }   (b is 0 or 1) The other two non-orientable Euclidean Klein bottle bundles. The one with b = 1 is homeomorphic to {0; (n1, 1); (2, 1), (2, 1)}. The first homology is Z+Z/2Z+Z/2Z if b=0, and Z+Z/4Z if b=1. These two Klein bottle bundle are surface bundles associated to the y-homeomorphism and the product of this and the twist.

Negative orbifold Euler characteristic

This is the general case. All such Seifert fibrations are determined up to isomorphism by their fundamental group. The total spaces are aspherical (in other words all higher homotopy groups vanish). They have Thurston geometries of type the universal cover of SL2(R), unless some finite cover splits as a product, in which case they have Thurston geometries of type H2×R. This happens if the manifold is non-orientable or b + Σbi/ai= 0.

References

  • A.V. Chernavskii (2001) [1994], "Seifert fibration", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  • Herbert Seifert, Topologie dreidimensionaler gefaserter Räume, Acta Math. 60 (1933) 147–238 (There is a translation by W. Heil, published by Florida state university in 1976 and found in: Herbert Seifert, William Threlfall, Seifert and Threllfall: a textbook of topology, Pure and Applied Mathematics, Academic Press Inc (1980), vol. 89.)
  • Peter Orlik Seifert manifolds, Lecture Notes in Mathematics 291, Springer (1972).
  • Frank Raymond Classification of the actions of the circle on 3-manifolds, Trans. Amer.Math. Soc 31, (1968) 51–87.
  • William H. Jaco, Lectures on 3-manifold topology ISBN 0-8218-1693-4
  • William H. Jaco, Peter B. Shalen Seifert Fibered Spaces in Three Manifolds: Memoirs Series No. 220 (Memoirs of the American Mathematical Society; v. 21, no. 220) ISBN 0-8218-2220-9
  • Matthew G. Brin Seifert fibered 3-manifolds. Course notes.
  • John Hempel, 3-manifolds, American Mathematical Society, ISBN 0-8218-3695-1
  • Peter Scott The geometries of 3-manifolds. (errata) Bull. London Math. Soc. 15 (1983), no. 5, 401–487.
David Gabai

David Gabai, a mathematician, is the Hughes-Rogers Professor of Mathematics at Princeton University. Focused on low-dimensional topology and hyperbolic geometry, he is a leading researcher in those subjects.

Fibered manifold

In differential geometry, in the category of differentiable manifolds, a fibered manifold is a surjective submersion

i.e. a surjective differentiable mapping such that at each point yE the tangent mapping

is surjective, or, equivalently, its rank equals dim B.

Flat manifold

In mathematics, a Riemannian manifold is said to be flat if its curvature is everywhere zero. Intuitively, a flat manifold is one that "locally looks like" Euclidean space in terms of distances and angles, e.g. the interior angles of a triangle add up to 180°.

The universal cover of a complete flat manifold is Euclidean space. This can be used to prove the theorem of

Bieberbach (1911, 1912) that all compact flat manifolds are finitely covered by tori; the 3-dimensional case was proved earlier by Schoenflies (1891).

Geometrization conjecture

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. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries (Euclidean, spherical, or hyperbolic).

In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by William Thurston (1982), and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture.

Thurston's hyperbolization theorem implies that Haken manifolds satisfy the geometrization conjecture. Thurston announced a proof in the 1980s and since then several complete proofs have appeared in print.

Grigori Perelman sketched a proof of the full geometrization conjecture in 2003 using Ricci flow with surgery.

There are now several different manuscripts (see below) with details of the proof. The Poincaré conjecture and the spherical space form conjecture are corollaries of the geometrization conjecture, although there are shorter proofs of the former that do not lead to the geometrization conjecture.

Heidelberg University Faculty of Mathematics and Computer Science

The Faculty of Mathematics and Computer Science is one of twelve faculties at the University of Heidelberg. It comprises the Institute of Mathematics, the Institute of Applied Mathematics, the School of Applied Sciences, and the Institute of Computer Science. The faculty maintains close relationships to the Interdisciplinary Center for Scientific Computing (IWR) and the Mathematics Center Heidelberg (MATCH). The first chair of mathematics was entrusted to the physician Jacob Curio in the year 1547.

Homology sphere

In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer n ≥ 1. That is,

H0(X,Z) = Z = Hn(X,Z)and

Hi(X,Z) = {0} for all other i.Therefore X is a connected space, with one non-zero higher Betti number: bn. It does not follow that X is simply connected, only that its fundamental group is perfect (see Hurewicz theorem).

A rational homology sphere is defined similarly but using homology with rational coefficients.

Jacques Feldbau

Jacques Feldbau was a French mathematician, born on 22 October 1914 in Strasbourg, of an Alsatian Jewish traditionalist family. He died on 22 April 1945 at the Ganacker Camp, annex of the concentration camp of Flossenbürg in Germany. As a mathematician he worked on differential geometry and topology. He was the very first student of Charles Ehresmann.

He is known as one of the founders of the theory of fiber bundles. He is the one who first proved that a fiber bundle over a simplex is trivializable and who used this to classify bundles over spheres.In a paper, written together with Ehresmann, he introduced the notion of an associated bundle and proved results known today as the exact homotopy sequence of a fibration.

List of Heidelberg University people

Alumni and faculty of the university include many founders and pioneers of academic disciplines, and a large number of internationally acclaimed philosophers, poets, jurisprudents, theologians, natural and social scientists. 56 Nobel Laureates, at least 18 Leibniz Laureates, and two "Oscar" winners have been associated with Heidelberg University. Nine Nobel Laureates received the award during their tenure at Heidelberg.Besides several Federal Ministers of Germany and Prime Ministers of German States, five Chancellors of Germany have attended the university, the latest being Helmut Kohl, the "Chancellor of the Reunification". Heads of State or Government of Belgium, Bulgaria, Greece, Nicaragua, Serbia, Thailand, a British Crown Prince, a Secretary General of NATO and a director of the International Peace Bureau have also been educated at Heidelberg; among them Nobel Peace Laureates Charles Albert Gobat and Auguste Beernaert. Former university affiliates in the field of religion include Pope Pius II, Cardinals, Bishops, and with Philipp Melanchthon and Zacharias Ursinus two key leaders of Protestant Reformation. Outstanding university affiliates in the legal profession include a President of the International Court of Justice, two Presidents of the European Court of Human Rights, a President of the International Tribunal for the Law of the Sea, a Vice President of the International Criminal Court, an Advocate General at the European Court of Justice, at least 16 Justices of the Federal Constitutional Court of Germany, a President of the Federal Court of Justice, a President of the Federal Court of Finance, a President of the Federal Labor Court, two Attorney Generals of Germany, and a British Law Lord. In business, Heidelberg alumni and faculty notably founded, co-founded or presided over ABB Group; Astor corporate enterprises; BASF; BDA; Daimler AG; Deutsche Bank; EADS; Krupp AG; Siemens AG; and Thyssen AG.

Alumni in the field of arts include classical composer Robert Schumann, philosophers Ludwig Feuerbach and Edmund Montgomery, poet Joseph Freiherr von Eichendorff and writers Christian Friedrich Hebbel, Gottfried Keller, Irene Frisch, Heinrich Hoffmann, Sir Muhammad Iqbal, José Rizal, W. Somerset Maugham, Jean Paul, and Literature Nobel Laureate Carl Spitteler. Amongst Heidelberg alumni in other disciplines are the "Father of Psychology" Wilhelm Wundt, the "Father of Physical Chemistry" J. Willard Gibbs, the "Father of American Anthropology" Franz Boas, Dmitri Mendeleev, who created the periodic table of elements, inventor of the two-wheeler principle Karl Drais, Alfred Wegener, who discovered the continental drift, as well as political theorist Hannah Arendt, political scientist Carl Joachim Friedrich, and sociologists Karl Mannheim, Robert E. Park and Talcott Parsons.

Philosophers Georg Wilhelm Friedrich Hegel, Karl Jaspers, Hans-Georg Gadamer, and Jürgen Habermas served as university professors, as did also the pioneering scientists Hermann von Helmholtz, Robert Wilhelm Bunsen, Gustav Robert Kirchhoff, Emil Kraepelin, the founder of scientific psychiatry, and outstanding social scientists such as Max Weber, the founding father of modern sociology.

Present faculty include Medicine Nobel Laureates Bert Sakmann (1991) and Harald zur Hausen (2008), Chemistry Nobel Laureate Stefan Hell (2014), 7 Leibniz Laureates, former Justice of the Federal Constitutional Court of Germany Paul Kirchhof, and Rüdiger Wolfrum, the former President of the International Tribunal for the Law of the Sea.

List of geometric topology topics

This is a list of geometric topology topics, by Wikipedia page. See also:

topology glossary

List of topology topics

List of general topology topics

List of algebraic topology topics

Publications in topology

SL2(R)

In mathematics, the special linear group SL(2,R) or SL2(R) is the group of 2 × 2 real matrices with determinant one:

It is a connected noncompact simple real Lie group with applications in geometry, topology, representation theory, and physics.

SL(2,R) acts on the complex upper half-plane by fractional linear transformations. The group action factors through the quotient PSL(2,R) (the 2 × 2 projective special linear group over R). More specifically,

PSL(2,R) = SL(2,R)/{±I},

where I denotes the 2 × 2 identity matrix. It contains the modular group PSL(2,Z).

Also closely related is the 2-fold covering group, Mp(2,R), a metaplectic group (thinking of SL(2,R) as a symplectic group).

Another related group is SL±(2,R) the group of real 2 × 2 matrices with determinant ±1; this is more commonly used in the context of the modular group, however.

Spherical 3-manifold

In mathematics, a spherical 3-manifold M is a 3-manifold of the form

where is a finite subgroup of SO(4) acting freely by rotations on the 3-sphere . All such manifolds are prime, orientable, and closed. Spherical 3-manifolds are sometimes called elliptic 3-manifolds or Clifford-Klein manifolds.

Virtually fibered conjecture

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.

A 3-manifold which has such a finite cover is said to virtually fiber. If M is a Seifert fiber space, then M virtually fibers if and only if the rational Euler number of the Seifert fibration or the (orbifold) Euler characteristic of the base space is zero.

The hypotheses of the conjecture are satisfied by hyperbolic 3-manifolds. In fact, given that the geometrization conjecture is now settled, the only case needed to be proven for the virtually fibered conjecture is that of hyperbolic 3-manifolds.

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 theorem, would imply the geometrization conjecture. However, in practice all known attacks on the "virtual" conjecture take geometrization as a hypothesis, and rely on the geometric and group-theoretic properties of hyperbolic 3-manifolds.

The virtually fibered conjecture was not actually conjectured by Thurston. Rather, he posed it as a question and has stated that it was intended as a challenge (and not meant to indicate he believed it). The conjecture was finally settled in the affirmative in a series of papers from 2009 to 2012.

In a posting on the ArXiv on 25 Aug 2009, Daniel Wise implicitly implied (by referring to a then unpublished longer manuscript) that he had proven the conjecture for the case where the 3-manifold is closed, hyperbolic, and Haken. This was followed by a survey article in Electronic Research Announcements in Mathematical Sciences.

Several more preprints have followed, including the aforementioned longer manuscript by Wise. In March 2012, during a conference at Institut Henri Poincaré in Paris, Ian Agol announced he could prove the virtually Haken conjecture for closed hyperbolic 3-manifolds

. Taken together with Daniel Wise's results, this implies the virtually fibered conjecture for all closed hyperbolic 3-manifolds.

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