Seifert fibration

A class of fibrations of three-dimensional manifolds by circles; defined by H. Seifert [1]. Every fibre of a Seifert fibration has a neighbourhood in the manifold $ M ^ {3} $ with standard fibration by circles, arising from the product $ D ^ {2} \times [ 0 , 1 ] $ of a disc and a closed interval, each point $ ( x , 0 ) $ being identified with the point $ ( g ( x) , 1 ) $, where $ g $ is the rotation of $ D ^ {2} $ through the angle $ 2 \mu \nu / \mu $( $ \mu $ and $ \nu $ are coprime integers, $ 0 \leq \nu < \mu $). The images of the intervals $ x \times [ 0 , 1 ] $ in the resulting solid torus $ P $ constitute fibres: each fibre, except the central one, consists of $ \mu $ intervals if $ \nu \neq 0 $; the central fibre is said to be singular if $ \nu > 0 $. The invariants $ ( \mu , \nu ) $ are usually replaced by the Seifert invariants $ ( \alpha , \beta ) $, where $ \alpha = \mu $ and $ \beta $ is defined by the condition

$$ 0 < \beta < \alpha ,\ \beta \nu \equiv 1 \mathop{\rm mod} \alpha . $$

The invariants $ \alpha $ and $ \beta $ admit a geometric interpretation: In the fibration induced on the boundary of $ P $, consider a meridian $ m $( a curve contractible in $ P $ but not in $ \partial P $) and a parallel $ l $( cutting $ m $ transversally just once), and also any fibre $ f $ and a secant $ g $( all four curves are simple and closed); then, subject to a suitable orientation,

$$ m = \alpha g + \beta f ,\ l = - \nu g - \mu f . $$

Moreover $ \beta \nu - \alpha \mu = 1 $.

The first problem concerning Seifert fibrations is to classify them up to fibrewise homeomorphisms. It turns out [1] that if $ M ^ {3} $ admits a Seifert fibration, then there exists a mapping $ \pi : M ^ {3} \rightarrow B ^ {2} $, where $ B ^ {2} $ is a two-dimensional manifold, and the fibres are $ \pi ^ {-1} ( x) $, $ x \in B ^ {2} $. There are six types of Seifert fibrations: the types $ O _ {1} $ and $ O _ {2} $, in which $ B ^ {2} $ is orientable and $ M ^ {3} $ is orientable in the case $ O _ {1} $ and non-orientable in the case $ O _ {2} $, with the genus of $ B ^ {2} $ in this case at least $ 1 $; and the types $ n _ {i} $, $ i = 1 , 2 , 3 , 4 $, in which $ B ^ {2} $ is non-orientable. In the case $ n _ {1} $, transport of a fibre along a path in $ B ^ {2} $ does not change the orientation of the fibre; in the case $ n _ {2} $ there is a system of generators and transport along each of them reverses the orientation; in the case $ n _ {3} $ only one of the generators does not reverse orientation; and in the case $ n _ {4} $ only two of the generators do not reverse orientation; the genus of $ B ^ {2} $ is at least $ 2 $ for $ n _ {3} $, and at least $ 3 $ for $ n _ {4} $. The manifold $ M ^ {3} $ is orientable only for the type $ n _ {4} $. Each Seifert fibration is associated with a system of invariants

$$ \{ b ; ( \epsilon , p ) ; \ ( \alpha _ {1} , \beta _ {1} ) ; \dots ; \ ( \alpha _ {r} , \beta _ {r} ) \} , $$

so that, up to fibrewise homeomorphisms, there is exactly one Seifert fibration with a given such system. Here $ \epsilon = O _ {i} $ or $ n _ {i} $, $ p $ is the genus of $ B ^ {2} $, $ ( \alpha _ {i} , \beta _ {i} ) $ are the Seifert invariants for the singular fibres; $ r $ is the number of singular fibres; $ \beta _ {i} \leq \alpha / 2 $ in the cases $ \epsilon = O _ {2} , n _ {1} , n _ {3} , n _ {4} $; and, finally, $ b $ is an integer if $ \epsilon = O _ {1} $ or $ n _ {2} $ and a residue $ \mathop{\rm mod} 2 $ in the other cases, with $ b = 0 $ if $ \alpha _ {i} = 2 $ for at least one fibre. The geometric meaning of $ b $ is as follows: Choose a section on the boundary of a neighbourhood of each singular fibre, and extend the set of all these sections to a section in the whole complement to the singular fibres. This can be done up to one non-singular fibre; the boundary of the extended section approaches that fibre, twisting around it with degree $ b $. In the case $ O _ {1} $ and $ n _ {2} $, when the orientation of $ M ^ {3} $ is reversed, the number $ b $ is replaced by $ - b - r $, and $ \beta _ {i} $ by $ \alpha _ {i} - \beta _ {i} $.

The second point of interest in the theory of Seifert fibrations is to show that a closed manifold $ M ^ {3} $ admits at most one Seifert fibration up to fibrewise homeomorphisms. This has been proved for what are known as large Seifert fibrations, which are spaces of type $ K ( \pi , 1 ) $, i.e. their homotopy type is defined by the fundamental group. The fundamental group $ \pi _ {1} ( M ^ {3} ) $ of a manifold equipped with a Seifert fibration is conveniently described in terms of a special system of generators: sections $ g _ {j} $ on the boundaries of neighbourhoods of singular fibres, elements $ a _ {i} , b _ {i} $( or $ V _ {i} $, if $ B ^ {2} $ is non-orientable), whose images in $ \pi _ {1} ( B ^ {2} ) $ are canonical generators, and a non-singular fibre $ h $. The defining relations for the generators, in the cases $ O _ {1} $ and $ O _ {2} $, are

$$ a _ {i} ha _ {i} ^ {-1} = h ^ {\epsilon _ {i} } ,\ \ b _ {i} hb _ {i} ^ {-1} = h ^ {\epsilon _ {i} } ,\ \ g _ {j} hg _ {j} ^ {-1} = h ,\ g _ {j} ^ {\alpha _ {i} } h ^ {\beta _ {j} } = 1 , $$

$$ g _ {1} \dots g _ {r} [ a _ {1} , b _ {1} ] \dots [ a _ {p} , b _ {p} ] = h ^ {b} , $$

and in the cases $ n _ {i} $,

$$ v _ {i} hv _ {i} ^ {-1} = h ^ {\epsilon _ {i} } ,\ \ g _ {j} hg _ {j} ^ {-1} = h ,\ g _ {j} ^ {\alpha _ {j} } h ^ {\beta _ {j} } = 1 , $$

$$ g _ {1} \dots g _ {r} v _ {1} ^ {2} \dots v _ {p} ^ {2} = h ^ {b} , $$

where $ \epsilon _ {i} = \pm 1 $, depending on whether the fibre orientation is reversed under transport along the corresponding generator of $ \pi _ {1} ( B ^ {2} ) $. Among the manifolds with small Seifert fibrations are the following: for the type $ O _ {1} $, all fibrations with

$$ p = 0 ,\ r \leq 2 ; $$

$$ p = 0 ,\ r = 3 ,\ \frac{1}{\alpha _ {1} } + \frac{1}{\alpha _ {2} } + \frac{1}{\alpha _ {3} } > 1 ; $$

$$ p = 1 ,\ r = 0 ; $$

$$ \{ - 2 ; ( O _ {1} , 0 ) ; ( 2 , 1 ) ; ( 2 , 1 ) ; ( 2 , 1 ) ; ( 2 , 1 ) \} ; $$

for the type $ O _ {2} $— only fibrations with $ p = 1 $, $ r = 0 $; for the types $ n _ {1} $ and $ n _ {2} $, fibrations with $ p = 1 $, $ r \leq 1 $; $ p = 2 $, $ r = 0 $; for the type $ n _ {3} $, fibrations with $ p = 2 $, $ r = 0 $. All Seifert fibrations of type $ n _ {4} $ are large. All small Seifert fibrations have been listed; there are 10 types (see [3]).

The free actions of finite groups on the three-dimensional sphere commute with the natural action of the group $ \mathop{\rm SO} ( 2) $ on the sphere, and it therefore turns out that the orbit spaces of these actions are Seifert fibrations with finite fundamental groups. These are the only known examples to date (1990) of $ M ^ {3} $ with finite $ \pi _ {1} ( M ^ {3} ) $. Some Seifert fibrations arise as boundaries of spherical neighbourhoods of isolated singular points on algebraic surfaces that are invariant under the action of the multiplicative group of complex numbers. Namely, these are Seifert fibrations of type $ \{ b; ( O _ {1} , p ); ( \alpha _ {1} , \beta _ {1} ) ; \dots ; ( \alpha _ {r} , \beta _ {r} ) \} $ with $ b + r > 0 $. Identification of these manifolds makes it possible to construct an explicit resolution of singularities, with the action of $ \mathbf C ^ {*} $ taken into consideration (and also to present a full description of isolated singularities on surfaces in $ \mathbf C ^ {3} $ that admit the action of $ \mathbf C ^ {*} $). There are also Seifert fibrations on locally flat Riemannian manifolds obtained by factorization of Euclidean space by the free action of a discrete group of motions (there are 6 oriented and 4 non-oriented manifolds, all but one of which are different fibrations over the circle, the fibre being a torus or a Klein surface).

Seifert fibrations are important in the topology of three-dimensional manifolds (cf. Topology of manifolds; Three-dimensional manifold), for example, in order to identify manifolds whose fundamental groups have a centre [4]. There are also generalizations of the concept to other classes of fibrations with singular fibres.

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