View source for Seifert fibration

A class of fibrations of three-dimensional manifolds by circles; defined by H. Seifert [[#References|[1]]]. Every fibre of a Seifert fibration has a neighbourhood in the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s0838201.png" /> with standard fibration by circles, arising from the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s0838202.png" /> of a disc and a closed interval, each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s0838203.png" /> being identified with the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s0838204.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s0838205.png" /> is the rotation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s0838206.png" /> through the angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s0838207.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s0838208.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s0838209.png" /> are coprime integers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382010.png" />). The images of the intervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382011.png" /> in the resulting solid torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382012.png" /> constitute fibres: each fibre, except the central one, consists of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382013.png" /> intervals if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382014.png" />; the central fibre is said to be singular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382015.png" />. The invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382016.png" /> are usually replaced by the Seifert invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382019.png" /> is defined by the condition

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The invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382022.png" /> admit a geometric interpretation: In the fibration induced on the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382023.png" />, consider a meridian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382024.png" /> (a curve contractible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382025.png" /> but not in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382026.png" />) and a parallel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382027.png" /> (cutting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382028.png" /> transversally just once), and also any fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382029.png" /> and a secant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382030.png" /> (all four curves are simple and closed); then, subject to a suitable orientation,

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Moreover <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382032.png" />.

The first problem concerning Seifert fibrations is to classify them up to fibrewise homeomorphisms. It turns out [[#References|[1]]] that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382033.png" /> admits a Seifert fibration, then there exists a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382034.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382035.png" /> is a two-dimensional manifold, and the fibres are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382037.png" />. There are six types of Seifert fibrations: the types <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382039.png" />, in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382040.png" /> is orientable and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382041.png" /> is orientable in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382042.png" /> and non-orientable in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382043.png" />, with the genus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382044.png" /> in this case at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382045.png" />; and the types <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382047.png" />, in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382048.png" /> is non-orientable. In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382049.png" />, transport of a fibre along a path in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382050.png" /> does not change the orientation of the fibre; in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382051.png" /> there is a system of generators and transport along each of them reverses the orientation; in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382052.png" /> only one of the generators does not reverse orientation; and in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382053.png" /> only two of the generators do not reverse orientation; the genus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382054.png" /> is at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382055.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382056.png" />, and at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382057.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382058.png" />. The manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382059.png" /> is orientable only for the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382060.png" />. Each Seifert fibration is associated with a system of invariants

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so that, up to fibrewise homeomorphisms, there is exactly one Seifert fibration with a given such system. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382062.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382064.png" /> is the genus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382066.png" /> are the Seifert invariants for the singular fibres; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382067.png" /> is the number of singular fibres; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382068.png" /> in the cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382069.png" />; and, finally, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382070.png" /> is an integer if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382071.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382072.png" /> and a residue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382073.png" /> in the other cases, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382074.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382075.png" /> for at least one fibre. The geometric meaning of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382076.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382077.png" />. In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382078.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382079.png" />, when the orientation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382080.png" /> is reversed, the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382081.png" /> is replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382082.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382083.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382084.png" />.

The second point of interest in the theory of Seifert fibrations is to show that a closed manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382085.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382086.png" />, i.e. their homotopy type is defined by the fundamental group. The [[Fundamental group|fundamental group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382087.png" /> of a manifold equipped with a Seifert fibration is conveniently described in terms of a special system of generators: sections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382088.png" /> on the boundaries of neighbourhoods of singular fibres, elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382089.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382090.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382091.png" /> is non-orientable), whose images in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382092.png" /> are canonical generators, and a non-singular fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382093.png" />. The defining relations for the generators, in the cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382094.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382095.png" />, are

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and in the cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s08382098.png" />,

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s083820101.png" />, depending on whether the fibre orientation is reversed under transport along the corresponding generator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s083820102.png" />. Among the manifolds with small Seifert fibrations are the following: for the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s083820103.png" />, all fibrations with

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for the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s083820108.png" /> — only fibrations with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s083820109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s083820110.png" />; for the types <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s083820111.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s083820112.png" />, fibrations with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s083820113.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s083820114.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s083820115.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s083820116.png" />; for the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s083820117.png" />, fibrations with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s083820118.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s083820119.png" />. All Seifert fibrations of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s083820120.png" /> are large. All small Seifert fibrations have been listed; there are 10 types (see [[#References|[3]]]).

The free actions of finite groups on the three-dimensional sphere commute with the natural action of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s083820121.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s083820122.png" /> with finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s083820123.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s083820124.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s083820125.png" />. Identification of these manifolds makes it possible to construct an explicit resolution of singularities, with the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s083820126.png" /> taken into consideration (and also to present a full description of isolated singularities on surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s083820127.png" /> that admit the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083820/s083820128.png" />). 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|Topology of manifolds]]; [[Three-dimensional manifold|Three-dimensional manifold]]), for example, in order to identify manifolds whose fundamental groups have a centre [[#References|[4]]]. There are also generalizations of the concept to other classes of fibrations with singular fibres.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Seifert, "Topologie driedimensionaler gefaserter Räume" ''Acta Math.'' , '''60''' (1933) pp. 147–238</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> H. Holmann, "Seifertsche Faserräume" ''Math. Ann.'' , '''157''' (1964) pp. 138–166 {{MR|0170349}} {{ZBL|0123.16501}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> P. Orlik, "Seifert Manifolds" , Springer (1972) {{MR|0426001}} {{ZBL|0263.57001}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J. Hempel, "3-manifolds" , Princeton Univ. Press (1976) {{MR|0415619}} {{ZBL|0345.57001}} </TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W.H. Jaco, "Lectures on three-manifold topology" , Amer. Math. Soc. (1980) pp. Chapt. VI {{MR|0565450}} {{ZBL|0433.57001}} </TD></TR></table>