A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S 1 {\displaystyle S^{1}} -bundle (circle bundle) over a 2-dimensional orbifold. 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.
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 ) {\displaystyle (a,b)} with a > 0 {\displaystyle a>0} is the surface bundle of the automorphism of a disk given by rotation by an angle of 2 π b / a {\displaystyle 2\pi b/a} (with the natural fibering by circles). If a = 1 {\displaystyle a=1} the middle fiber is called ordinary, while if a > 1 {\displaystyle 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 B 0 {\displaystyle 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 classified all closed Seifert fibrations in terms of the following invariants. Seifert manifolds M {\displaystyle M} are denoted by symbols
{ b , ( ε , g ) ; ( a 1 , b 1 ) , … , ( a r , b r ) } {\displaystyle \{b,(\varepsilon ,g);(a_{1},b_{1}),\dots ,(a_{r},b_{r})\}\,}
where:
ε {\displaystyle \varepsilon } is one of the 6 symbols: o 1 , o 2 , n 1 , n 2 , n 3 , n 4 {\displaystyle 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 {\displaystyle o_{1}} if B is orientable and M is orientable.
o 2 {\displaystyle o_{2}} if B is orientable and M is not orientable.
n 1 {\displaystyle n_{1}} if B is not orientable and M is not orientable and all generators of π 1 ( B ) {\displaystyle \pi _{1}(B)} preserve orientation of the fiber.
n 2 {\displaystyle n_{2}} if B is not orientable and M is orientable, so all generators of π 1 ( B ) {\displaystyle \pi _{1}(B)} reverse orientation of the fiber.
n 3 {\displaystyle n_{3}} if B is not orientable and M is not orientable and g ≥ 2 {\displaystyle g\geq 2} and exactly one generator of π 1 ( B ) {\displaystyle \pi _{1}(B)} preserves orientation of the fiber.
n 4 {\displaystyle n_{4}} if B is not orientable and M is not orientable and g ≥ 3 {\displaystyle g\geq 3} and exactly two generators of π 1 ( B ) {\displaystyle \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 {\displaystyle a_{i}} is 2.
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