Seifert fiber

In three-dimensional topology , a Seifert fiber is understood to be a three-dimensional manifold that is fiberized in a certain way by circles . Such a Seifert-fibered manifold can be imagined as a union of an infinite number of (arbitrarily shaped) circles that either run “parallel” to one another or wrap around discrete “singular” circles. Occasionally Seifert fibers are also referred to as Seifert fiber space in order to distinguish the manifold (the total space) from the fibers.

Seifert fibers play an important role in the geometrization of 3-manifolds , as their geometry and topology are well understood.

Definitions

First, a trivial grain is defined on a full torus , where a circular disk and a circle (a fiber) denote. In the one can imagine the grain in such a way that one takes the disc as a cross section of the full torus, and the circles by rotating a point on the disc around the axis that goes through the “hole” of the torus. ${\displaystyle D^{2}\times S^{1}}$ ${\displaystyle D^{2}}$ ${\displaystyle S^{1}}$ ${\displaystyle \mathbb {R} ^{3}}$

If you cut such a trivially fibrous torus along a disk, twist one of the two cut surfaces by the angle ( and coprime numbers) and glued the two disks together twisted again, you get a -fibre full torus . In the example shown, a Seifert-fibred full torus is obtained by turning the bottom over and gluing it to the top. The numbers indicate which fibers are glued together. ${\displaystyle 360^{\circ }\cdot q/p}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle (p,q)}$ ${\displaystyle (5,2)}$ ${\displaystyle 360^{\circ }\cdot 2/5}$

The central fiber remains unchanged, the remaining fibers are glued to other fibers (in the example with 5) to form a new fiber. This new fiber is wrapped - times along the central fiber (here 5 times) and at the same time - times (here twice) around the central fiber (in the direction of the cross section). ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle q}$

A Seifert fiber is now a 3-manifold that can be broken down into disjoint circles (called fibers ) in such a way that each fiber has an environment that is either isomorphic to the trivially fibered full torus or to a -fibered full torus. In this context, “isomorphic” means that there is a homeomorphism that maps fibers onto fibers. ${\displaystyle M}$ ${\displaystyle (p,q)}$

A fiber is called regular if it has a neighborhood isomorphic to the trivially fibrous full torus, otherwise it is called singular. A fiber is singular if and only if it corresponds to the central fiber of a Seifert-fiber full torus.

properties

A Seifert grain is not a grain in the mathematical sense, but actually a leaf . The term "grain" is of historical origin here. However, a Seifert fiber can also be understood as a singular fiber or Seifert bundle over an orbifold .

Although the topology of a single full torus does not change due to a Seifert fiber, a Seifert fiber of a manifold has topological information about the manifold. This is due to the fact that the Seifert fiber determines how different full torches can be glued along their surfaces. For example, a Seifert fiber is only possible on certain 3-manifolds. The following applies:

The universal superposition of a Seifert-fibered 3-manifold without a border is homeomorphic to the 3-sphere , to the Euclidean space or to the product . The Seifert fiber induces foliation on the overlay as one of the following: ${\displaystyle S^{3}}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle S^{2}\times \mathbb {R} }$

1. a Seifert bundle over with zero, one or two singular fibers ${\displaystyle S^{2}}$
2. a trivial bundle of lines ${\displaystyle \mathbb {R} ^{2}\times \mathbb {R} }$
3. a trivial bundle of lines ${\displaystyle S^{2}\times \mathbb {R} }$

That implies amongst other things, that closed-Seifert fibred 3-manifolds geometrisierbar within the meaning of Thurston are and one of the model geometries , , , , , or wear. On the other hand, there is no Seifert manifold with hyperbolic or sol geometry. ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle S^{3}\,}$ ${\displaystyle S^{2}\times \mathbb {R} }$ ${\displaystyle \mathbb {H} ^{2}\times \mathbb {R} }$ ${\displaystyle {\tilde {\mathrm {SL} }}(2,\mathbb {R} )}$ ${\displaystyle \mathrm {Nil} }$

Since a 3-manifold allows a maximum of one of the model geometries, this results in a characterization of the closed Seifert manifolds in six classes.

Characterization of Seifert fibers

Seifert fiber space conjecture (proved by Casson-Jungreis and Gabai): Let it be an orientable irreducible 3-manifold whose fundamental group is infinite and has a nontrivial normal cyclic subgroup . Then there is a Seifert fiber. ${\displaystyle M}$ ${\displaystyle M}$

history

Seifert fibers were first examined in 1932 by Herbert Seifert (1907-1996). 1979 used William Jaco , Peter Shalen and (regardless of) Klaus Johannson them to define and to prove the JSJ decomposition .

literature

• Herbert Seifert : Topology of 3-dimensional fibered spaces . In: Acta Mathematica . No. 60, 1932, pp. 147-238.
• William H. Jaco, Peter B. Shalen: Seifert fibered spaces in 3 manifolds (= Memoirs of the American Mathematical Society 21, 1 = 220). American Mathematical Soc., Providence RI, 1979, ISBN 0-8218-2220-9 .
• Allen Hatcher: Notes on basic 3-manifold topology (PDF; 385 kB).
• Peter Scott: The Geometries of 3-Manifolds . In: The Bulletin of the London Mathematical Society . No. 15, 1983, ISSN  0024-6093 , pp. 401-487.
• Matthew G. Brin: Seifert fibered spaces (Notes for a one semester cours) . (1993).

Web links

• Mark Jankins, Walter Neumann : Lectures on Seifert manifolds
• J.-P. Préaux: A survey on Seifert fiber space theorem

Individual evidence

1. ^ A. Casson , D. Jungreis: Convergence groups and Seifert fibered 3-manifolds. Invent. Math. 118 (1994) no. 3, 441-456.
2. ^ D. Gabai : Convergence groups are Fuchsian groups. Ann. of Math. (2) 136 (1992) no. 3: 447-510.