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Seifert fiber space
A Seifert fiber space is a 3manifold together with a "nice" decomposition as a disjoint union of circles. In other words it is a S^1bundle (circle bundle) over a 2dimensional orbifold. Most "small" 3manifolds are Seifert fiber spaces, and they account for all compact oriented manifolds in 6 of the 8 Thurston geometries of the geometrization conjecture.
Definition
A standard fibered torus corresponding to (5,2) is obtained by gluing the top of the cylinder to the bottom by a 3/5 rotation counterclockwise.
A Seifert manifold is a closed 3manifold 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 2dimensional orbifold, denoted by B and called the base also called the orbit surface of the fibration. It has an underlying 2dimensional 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 nonorientable 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 oneparameter families of exceptional curves. In both of these cases, the base B of the fibration usually has a nonempty boundary.
Classification
Seifert classified all closed Seifert fibrations in terms of the following invariants. Seifert manifolds are denoted by symbols
\( \{b, (\varepsilon,g);(a_1,b_1),\dots,(a_r,b_r)\}\, \)
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 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 π_{1}(B) preserve orientation of the fiber.
 n_{2} if B is not orientable and M is orientable, so all generators of π_{1}(B) reverse orientation of the fiber.
 n_{3} if B is not orientable and M is not orientable and g≥ 2 and exactly one generator of π_{1}(B) preserves orientation of the fiber.
 n_{4} if B is not orientable and M is not orientable and g≥ 3 and exactly two generators of π_{1}(B) preserve orientation of the fiber.
 g is the genus of the underlying 2manifold 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}),...,(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}≤a_{i}/2 when M is not orientable.
The Seifert fibration of the symbol
\( \{b, (\epsilon,g);(a_1,b_1),...,(a_r,b_r)\} \)
can be constructed from that of symbol
\(\{0, (\epsilon,g);\} \)
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}, ..., 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 nonorientable fibrations are isomorphic as oriented or nonorientable 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
\( \{br, (\epsilon,g);(a_1,a_1b_1),...,(a_r,a_rb_r)\} \)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.
and it is homeomorphic to this as an unoriented manifold.
The sum b + Σb_{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 χ(B) of the orbifold B is given by
 χ(B) = χ(B_{0}) − Σ(1−1/a_{i})
where χ(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 of M fits into the exact sequence
\(\pi_1(S^1)\rightarrow\pi_1(M)\rightarrow\pi_1(B)\rightarrow1 \)
where π_{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 π_{1}(S^{1}) is cyclic, normal, and generated by the element h represented by any regular fiber, but the map from π_{1}(S^{1}) to π_{1}(M) is not always injective.
The fundamental group of M has the following presentation by generators and relations:
B orientable:
\( \langle u_1,v_1,...u_g,v_g,q_1,...q_r,hu_ih=h^{\epsilon}u_i, v_ih=h^{\epsilon}v_i,q_ih=hq_i, q_j^{a_j}h^{b_j}=1, q_1...q_r[u_1,v_1]...[u_g,v_g]=h^b\rangle \)
where ε is 1 for type o1, and is −1 for type o2.
B nonorientable:
\( \langle v_1,...,v_g,q_1,...q_r,h v_ih=h^{\epsilon_i}v_i,q_ih=hq_i, q_j^{a_j}h^{b_j}=1, q_1...q_rv_1^2...v_g^2=h^b\rangle \)
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 nonorientable or if b + Σb_{i}/a_{i}= 0, and spherical geometry otherwise.
{b; (o_{1}, 0);} (b integral) is S^{2}×S^{1} for b=0, otherwise a lens space L(b,1). ({1; (o_{1}, 0);} =L(1,1) is the 3sphere.)
{b; (o_{1}, 0);(a_{1}, b_{1})} (b integral) is the Lens space L(ba_{1}+b_{1},a_{1}).
{b; (o_{1}, 0);(a_{1}, b_{1}), (a_{2}, b_{2})} (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; (o_{1}, 0);(2, 1), (2, 1), (a_{3}, b_{3})} (b integral) This is the Prism manifold with fundamental group of order 4a_{3}(b+1)a_{3}+b_{3} and first homology group of order 4(b+1)a_{3}+b_{3}.
{b; (o_{1}, 0);(2, 1), (3, b_{2}), (3, b_{3})} (b integral) The fundamental group is a central extension of the tetrahedral group of order 12 by a cyclic group.
{b; (o_{1}, 0);(2, 1), (3, b_{2}), (4, b_{3})} (b integral) The fundamental group is the product of a cyclic group of order 12b+6+4b_{2} + 3b_{3} and a double cover of order 48 of the octahedral group of order 24.
{b; (o_{1}, 0);(2, 1), (3, b_{2}), (5, b_{3})} (b integral) The fundamental group is the product of a cyclic group of order m=30b+15+10b_{2} +6b_{3} and the order 120 perfect double cover of the icosahedral group. The manifolds are quotients of the Poincaré sphere by cyclic groups of order m. In particular {−1; (o_{1}, 0);(2, 1), (3, 1), (5, 1)} is the Poincaré sphere.
{b; (n_{1}, 1);} (b is 0 or 1.) These are the nonorientable 3manifolds 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 2sphere.
{b; (n_{1}, 1);(a_{1}, b_{1})} (b is 0 or 1.) These are the nonorientable 3manifolds 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 2sphere.
{b; (n_{2}, 1);} (b integral.) This is the Prism manifold with fundamental group of order 4b 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; (n_{2}, 1);(a_{1}, b_{1})} (b integral.) This is the (unique) Prism manifold with fundamental group of order 4a_{1}ba_{1} + b_{1} and first homology group of order 4a_{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 nonorientable or if b + Σb_{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 2torus 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; (o_{1}, 0); (3, b_{1}), (3, b_{2}), (3, b_{3})} (b integral, b_{i} is 1 or 2) For b + Σb_{i}/a_{i}= 0 this is an oriented Euclidean 2torus bundle over the circle, and is the surface bundle associated to an order 3 (trace −1) rotation of the 2torus.
{b; (o_{1}, 0); (2,1), (4, b_{2}), (4, b_{3})} (b integral, b_{i} is 1 or 3) For b + Σb_{i}/a_{i}= 0 this is an oriented Euclidean 2torus bundle over the circle, and is the surface bundle associated to an order 4 (trace 0) rotation of the 2torus.
{b; (o_{1}, 0); (2, 1), (3, b_{2}), (6, b_{3})} (b integral, b_{2} is 1 or 2, b_{3} is 1 or 5) For b + Σb_{i}/a_{i}= 0 this is an oriented Euclidean 2torus bundle over the circle, and is the surface bundle associated to an order 6 (trace 1) rotation of the 2torus.
{b; (o_{1}, 0); (2, 1), (2, 1), (2, 1), (2, 1)} (b integral) These are oriented 2torus bundles for trace −2 automorphisms of the 2torus. For b=−2 this is an oriented Euclidean 2torus bundle over the circle (the surface bundle associated to an order 2 rotation of the 2torus) and is homeomorphic to {0; (n_{2}, 2);}.
{b; (o_{1}, 1); } (b integral) This is an oriented 2torus bundle over the circle, given as the surface bundle associated to a trace 2 automorphism of the 2torus. For b=0 this is Euclidean, and is the 3torus (the surface bundle associated to the identity map of the 2torus).
{b; (o_{2}, 1); } (b is 0 or 1) Two nonorientable 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 S^{1} and other is the surface bundle associated to a Dehn twist of the Klein bottle. They are homeomorphic to the torus bundles {b; (n_{1}, 2);}.
{0; (n_{1}, 1); (2, 1), (2, 1)} Homeomorphic to the nonorientable Euclidean Klein bottle bundle {1; (n_{3}, 2);}, with first homology Z + Z/4Z.
{b; (n_{1}, 2); } (b is 0 or 1) These are the nonorientable Euclidean surface bundles associated with orientation reversing order 2 automorphisms of a 2torus 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; (o_{2}, 1);}.
{b; (n_{2}, 1); (2, 1), (2, 1)} (b integral) For b=−1 this is oriented Euclidean.
{b; (n_{2}, 2); } (b integral) For b=0 this is an oriented Euclidean manifold, homeomorphic to the 2torus bundle {−2; (o_{1}, 0); (2, 1), (2, 1), (2, 1), (2, 1)} over the cicle associated to an order 2 rotation of the 2torus.
{b; (n_{3}, 2); } (b is 0 or 1) The other two nonorientable Euclidean Klein bottle bundles. The one with b = 1 is homeomorphic to {0; (n_{1}, 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 yhomeomorphism 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 nonorientable or b + Σb_{i}/a_{i}= 0.
References
A.V. Chernavskii (2001), "Seifert fibration", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 9781556080104
Herbert Seifert, Topologie dreidimensionaler gefaserter Räume, Acta Math. 60 (1933) 147238 (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 3manifolds, Trans. Amer.Math. Soc 31, (1968) 5187.
William H. Jaco, Lectures on 3manifold topology ISBN 0821816934
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 0821822209
Matthew G. Brin Seifert fibered 3manifolds. Course notes.
John Hempel, 3manifolds, American Mathematical Society, ISBN 0821836951
Peter Scott The geometries of 3manifolds. (errata) Bull. London Math. Soc. 15 (1983), no. 5, 401487.
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