Seifert fiber space

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

Definitions

First, we define a trivial fibration on a Volltorus, with a circular disk and a circle denotes ( a fiber ). In one can imagine the fibration so that you take the disc as a cross -section of the Volltorus, and the circles by rotation of a point on the disc about the axis passing through the "hole" of the torus.

If you cut such a trivial fibered torus along a disc on, twisted one of the two cut surfaces by angle 360 ° · q / p ( p and q are prime integers ) and glued the two discs so twisted back together, we obtain a (p, q) - fibered Volltorus. In the illustrated example, to obtain a (5,2) - Seifert fibrillated Volltorus, by rotating the bottom around 360 ° × 2 /5 and glued to the top. The numbers indicate which fibers are glued together there.

The central fiber remains unchanged, the remaining fibers are bonded to each other fibers (in example 5 ) to a new fiber. This new fiber wraps times along the central fiber (here 5 times) and there time (here 2 times) around the central fiber ( in the direction of the cross section ) around.

A Seifert fibration is now a 3-manifold which can be decomposed as in disjoint cycles ( called fibers ) that each fiber has a neighborhood that is either isomorphic to the trivial fibered Volltorus or to a (p, q) - fibered Volltorus. " Isomorph " means in this context that there is a homeomorphism, the fibers maps to fibers.

A fiber is called regular if it has a neighborhood isomorphic to the trivial fibered Volltorus, otherwise it is called singular. A fiber is singular if and only if it corresponds to the central fiber of a Seifert fibered - Volltorus.

Properties

A Seifert fibration is not a fibration in the mathematical sense, but actually a foliation. The term " fibration " has historic origins here. However, can a Seifert fibration as a singular fibration or Seifert bundle conceive a Orbifaltigkeit.

Although the topology of a single Volltorus not changed by a Seifertfaserung, has a Seifert fibration of a manifold topological information about the manifold. The reason is that the Seifert fibration defines how different Volltori can be glued along their surfaces. For example, a Seifert fibration possible only on certain 3-manifolds. The following applies:

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

History

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