Seifert fiber space
A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles.
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.
A standard fibered torus corresponding to a pair of coprime integers
is the surface bundle of the automorphism of a disk given by rotation by an angle of
, 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.
, (or Oo, No, NnI, On, NnII, NnIII in Seifert's original notation) meaning: Here The Seifert fibration of the symbol can be constructed from that of symbol by using surgery to add fibers of types b and
If we drop the normalization conditions then the symbol can be changed as follows: 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.
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.
is the usual Euler characteristic of the underlying topological surface
The behavior of M depends largely on the sign of the orbifold Euler characteristic of B.
The fundamental group of M fits into the exact sequence where
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.
B non-orientable: where εi is 1 or −1 depending on whether the corresponding generator vi preserves or reverses orientation of the fiber.
The normalized symbols of Seifert fibrations with positive orbifold Euler characteristic are given in the list below.
They have a spherical Thurston geometry if the fundamental group is finite, and an S2×R Thurston geometry if the fundamental group is infinite.
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.
The normalized symbols of Seifert fibrations with zero orbifold Euler characteristic are given in the list below.
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.
{b; (o2, 1); } (b is 0 or 1) Two non-orientable Euclidean Klein bottle bundles over the circle.
{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.
{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 circle associated to an order 2 rotation of the 2-torus.
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.
