HOME

TheInfoList



OR:

A Seifert fiber space is a
3-manifold In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...
together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle (
circle bundle In mathematics, a circle bundle is a fiber bundle where the fiber is the circle S^1. Oriented circle bundles are also known as principal ''U''(1)-bundles. In physics, circle bundles are the natural geometric setting for electromagnetism. A circ ...
) over a 2-dimensional
orbifold In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. D ...
. Many 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 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-dimens ...
.


Definition

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 In mathematics, a surface bundle is a bundle in which the fiber is a surface. When the base space is a circle the total space is three-dimensional and is often called a surface bundle over the circle. See also *Mapping torus In mathematics, the ...
of the automorphism of a disk given by rotation by an angle of 2\pi b/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 In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. D ...
, denoted by ''B'' and called the base —also called the orbit surface— of the fibration. It has an underlying 2-dimensional surface B_0, 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 Herbert Karl Johannes Seifert (; 27 May 1897, Bernstadt – 1 October 1996, Heidelberg) was a German mathematician known for his work in topology. Biography Seifert was born in Bernstadt auf dem Eigen, but soon moved to Bautzen, where he attend ...
classified all closed Seifert fibrations in terms of the following invariants. Seifert manifolds are denoted by symbols :\\, where: \varepsilon is one of the 6 symbols: o_1,o_2,n_1,n_2,n_3,n_4\,, (or Oo, No, NnI, On, NnII, NnIII in Seifert's original notation) meaning: *o_1 if ''B'' is
orientable In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
and ''M'' is orientable. *o_2 if ''B'' is orientable and ''M'' is not orientable. *n_1 if ''B'' is not orientable and ''M'' is not orientable and all generators of \pi_1(B) preserve orientation of the fiber. *n_2 if ''B'' is not orientable and ''M'' is orientable, so all generators of \pi_1(B) reverse orientation of the fiber. *n_3 if ''B'' is not orientable and ''M'' is not orientable and g\ge 2 and exactly one generator of \pi_1(B) preserves orientation of the fiber. *n_4 if ''B'' is not orientable and ''M'' is not orientable and g\ge 3 and exactly two generators of \pi_1(B) preserve orientation of the fiber. Here *''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. *(a_1,b_1),\ldots,(a_r,b_r) are the pairs of numbers determining the type of each of the ''r'' exceptional orbits. They are normalized so that 0< b_i < a_i when ''M'' is orientable, and 0< b_i \le a_i/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 b_i/a_i. If we drop the normalization conditions then the symbol can be changed as follows: *Changing the sign of both a_i and b_i has no effect. *Adding 1 to ''b'' and subtracting a_i from b_i has no effect. (In other words, we can add integers to each of the rational numbers (b,b_1/a_1, \ldots ,b_r/a_r provided that their sum remains constant.) *If the manifold is not orientable, changing the sign of b_i 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 ''b''s changed, which after normalization gives it the symbol :\ and it is homeomorphic to this as an unoriented manifold. The sum b+\sum b_i/a_i 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 In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
\chi(B) of the orbifold ''B'' is given by :\chi(B) = \chi(B_0) - \sum(1-1/a_i), where \chi(B_0) is the usual Euler characteristic of the underlying topological surface B_0 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 In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
of ''M'' fits into the exact sequence :\pi_1(S^1)\rightarrow\pi_1(M)\rightarrow\pi_1(B)\rightarrow1 where \pi_1(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 \pi_1(S^1) is cyclic, normal, and generated by the element ''h'' represented by any regular fiber, but the map from π1(''S''1) to π1(''M'') is not always injective. The fundamental group of ''M'' has the following
presentation A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...
by generators and relations: ''B'' orientable: :\langle u_1,v_1,...u_g,v_g,q_1,...q_r,h, u_ih=h^u_i, v_ih=h^v_i,q_ih=hq_i, q_j^h^=1, q_1...q_r _1,v_1.. _g,v_gh^b\rangle where ε is 1 for type ''o''1, and is −1 for type ''o''2. ''B'' non-orientable: :\langle v_1,...,v_g,q_1,...q_r,h, v_ih=h^v_i,q_ih=hq_i, q_j^h^=1, q_1...q_rv_1^2...v_g^2=h^b\rangle where ε''i'' is 1 or −1 depending on whether the corresponding generator ''v''''i'' preserves or reverses orientation of the fiber. (So ε''i'' are all 1 for type ''n''1, all −1 for type ''n''2, just the first one is one for type ''n''3, and just the first two are one for type ''n''4.)


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 ''S''2×R Thurston geometry if the fundamental group is infinite. Equivalently, the geometry is ''S''2×R if the manifold is non-orientable or if ''b'' + Σ''b''''i''/''a''''i''= 0, and spherical geometry otherwise. (''b'' integral) is ''S''2×''S''1 for ''b''=0, otherwise a
lens space A lens space is an example of a topological space, considered in mathematics. The term often refers to a specific class of 3-manifolds, but in general can be defined for higher dimensions. In the 3-manifold case, a lens space can be visualize ...
''L''(''b'',1). In particular, =''L''(1,1) is the 3-sphere. (''b'' integral) is the lens space ''L''(''ba''1+''b''1,''a''1). (''b'' integral) is ''S''2×''S''1 if ''ba''1''a''2+''a''1''b''2+''a''2''b''1 = 0, otherwise the lens space ''L''(''ba''1''a''2+''a''1''b''2+''a''2''b''1, ''ma''2+''nb''2) where ''ma''1 − ''n''(''ba''1 +''b''1) = 1. (''b'' integral) This is the prism manifold with fundamental group of order 4''a''3, (''b''+1)''a''3+''b''3, and first
homology group In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
of order 4, (''b''+1)''a''3+''b''3, . (''b'' integral) The fundamental group is a central extension of the tetrahedral group of order 12 by a cyclic group. (''b'' integral) The fundamental group is the product of a cyclic group of order , 12''b''+6+4''b''2 + 3''b''3, and a double cover of order 48 of the
octahedral group A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedr ...
of order 24. (''b'' integral) The fundamental group is the product of a cyclic group of order ''m''=, 30''b''+15+10''b''2 +6''b''3, and the order 120 perfect double cover of the icosahedral group. The manifolds are quotients of the
Poincaré homology sphere Poincaré is a French surname. Notable people with the surname include: * Henri Poincaré (1854–1912), French physicist, mathematician and philosopher of science * Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré * Luci ...
by cyclic groups of order ''m''. In particular, is the Poincaré sphere. (''b'' is 0 or 1.) These are the non-orientable 3-manifolds with ''S''2×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'' is 0 or 1.) These are the non-orientable 3-manifolds with ''S''2×R geometry. If ''ba''1+''b''1 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'' 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'' integral.) This is the (unique) prism manifold with fundamental group of order 4''a''1, ''ba''1 + ''b''1, and first homology group of order 4''a''1.


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'' + Σ''b''''i''/''a''''i''= 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 Anosov (masculine, russian: Аносов) or Anosova (feminine, russian: Аносова) is a Russian surname. Notable people with the surname include: * Anosov Pavel Petrovich (russian: Аносов Павел Петрович) () - Russian scienti ...
) automorphisms are not Seifert fiber spaces, but have
sol geometry 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-dimens ...
). 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'' integral, ''b''''i'' is 1 or 2) For ''b'' + Σ''b''''i''/''a''''i''= 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'' integral, ''b''''i'' is 1 or 3) For ''b'' + Σ''b''''i''/''a''''i''= 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'' integral, ''b''''2'' is 1 or 2, ''b''''3'' is 1 or 5) For ''b'' + Σ''b''''i''/''a''''i''= 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'' 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 .    (''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'' is 0 or 1) Two non-orientable Euclidean
Klein bottle In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a o ...
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 ''S''1 and other is the surface bundle associated to a
Dehn twist In geometric topology, a branch of mathematics, a Dehn twist is a certain type of self-homeomorphism of a surface (two-dimensional manifold). Definition Suppose that ''c'' is a simple closed curve in a closed, orientable surface ''S''. Let ' ...
of the
Klein bottle In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a o ...
. They are homeomorphic to the torus bundles .    Homeomorphic to the non-orientable Euclidean Klein bottle bundle , with first homology Z + Z/4Z.    (''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'' integral) For ''b''=−1 this is oriented Euclidean.    (''b'' integral) For ''b''=0 this is an oriented Euclidean manifold, homeomorphic to the 2-torus bundle over the cicle associated to an order 2 rotation of the 2-torus.    (''b'' is 0 or 1) The other two non-orientable Euclidean Klein bottle bundles. The one with ''b'' = 1 is homeomorphic to . 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 In mathematics, the y-homeomorphism, or crosscap slide, is a special type of auto-homeomorphism in non-orientable surfaces. It can be constructed by sliding a Möbius band included on the surface around an essential 1-sided closed curve until ...
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 ''SL''2(R), unless some finite cover splits as a product, in which case they have Thurston geometries of type ''H''2×R. This happens if the manifold is non-orientable or ''b'' + Σ''b''''i''/''a''''i''= 0.


References

* *
Herbert Seifert Herbert Karl Johannes Seifert (; 27 May 1897, Bernstadt – 1 October 1996, Heidelberg) was a German mathematician known for his work in topology. Biography Seifert was born in Bernstadt auf dem Eigen, but soon moved to Bautzen, where he attend ...
, ''Topologie dreidimensionaler gefaserter Räume'',
Acta Mathematica ''Acta Mathematica'' is a peer-reviewed open-access scientific journal covering research in all fields of mathematics. According to Cédric Villani, this journal is "considered by many to be the most prestigious of all mathematical research journ ...
60 (1933) 147–238 (There is a translation by W. Heil, published by Florida State University in 1976 and found in:
Herbert Seifert Herbert Karl Johannes Seifert (; 27 May 1897, Bernstadt – 1 October 1996, Heidelberg) was a German mathematician known for his work in topology. Biography Seifert was born in Bernstadt auf dem Eigen, but soon moved to Bautzen, where he attend ...
,
William Threlfall William Richard Maximilian Hugo Threlfall (25 June 1888, in Dresden – 4 April 1949, in Oberwolfach) was a British-born Germany, German mathematician who worked on algebraic topology. He was a coauthor of the standard textbook Lehrbuch der Topol ...
, ''Seifert and Threllfall: a textbook of topology'', Pure and Applied Mathematics, Academic Press Inc (1980), vol. 89.) *
Peter Orlik Peter Paul Nikolas Orlik (born 12 November 1938, in Budapest) is an American mathematician, known for his research on topology, algebra, and combinatorics. Orlik earned in 1961 his bachelor's degree from the Norwegian Institute of Technology in Tr ...
, ''Seifert manifolds'',
Lecture Notes in Mathematics ''Lecture Notes in Mathematics'' is a book series in the field of mathematics, including articles related to both research and teaching. It was established in 1964 and was edited by A. Dold, Heidelberg and B. Eckmann, Zürich. Its publisher is Spr ...
291,
Springer Springer or springers may refer to: Publishers * Springer Science+Business Media, aka Springer International Publishing, a worldwide publishing group founded in 1842 in Germany formerly known as Springer-Verlag. ** Springer Nature, a multinationa ...
(1972). * Frank Raymond, ''Classification of the actions of the circle on 3-manifolds'',
Transactions of the American Mathematical Society The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 p ...
31, (1968) 51–87. * William H. Jaco, ''Lectures on 3-manifold topology'' * William H. Jaco, Peter B. Shalen, ''Seifert Fibered Spaces in Three Manifolds:'' Memoirs Series No. 220 (
Memoirs of the American Mathematical Society ''Memoirs of the American Mathematical Society'' is a mathematical journal published in six volumes per year, totalling approximately 33 individually bound numbers, by the American Mathematical Society. It is intended to carry papers on new mathema ...
; v. 21, no. 220) * *John Hempel, ''3-manifolds'', American Mathematical Society, {{isbn, 0-8218-3695-1 *
Peter Scott Sir Peter Markham Scott, (14 September 1909 – 29 August 1989) was a British ornithologist, conservationist, painter, naval officer, broadcaster and sportsman. The only child of Antarctic explorer Robert Falcon Scott, he took an interest in ...

''The geometries of 3-manifolds.''errata
, Bull. London Math. Soc. 15 (1983), no. 5, 401–487. Fiber bundles 3-manifolds Geometric topology