SiefertFibration

A Seifert fibration is a way of structuring a three-dimensional manifold such that the manifold is filled with disjoint circles, called Seifert fibers. It provides a "fibered" structure on the manifold, much like the grain in a piece of wood or the threads in a piece of fabric.

When we say that a manifold has a Seifert fibration, we mean that there's a special way to fill the manifold with circles so that around each point in the manifold, the circles look like they're simply twisting around, possibly with different rates of twisting at different points.

More formally, a Seifert fibration of a three-dimensional manifold is a continuous map from the manifold to a two-dimensional orbifold such that the inverse image of each point in the orbifold is a circle. These circles are the Seifert fibers of the fibration. Around each point, there's a neighborhood that looks like the product of a disk in the orbifold and a circle.

The base space of a Seifert fibration is a two-dimensional orbifold, which can be visualized as the space you get when you "collapse" each of the fibers to a single point. The points of this base space correspond to the Seifert fibers in the manifold.

The Seifert fibration also induces an orientation on the Seifert fibers, and this orientation contributes to whether the Seifert manifold is orientable or non-orientable.

An important aspect of Seifert fibrations is the concept of the Euler number or the twisting number. This is a measure of how the Seifert fibers twist as they move through the manifold. For some Seifert manifolds, the twisting is uniform, but for others, there can be differential twisting.

Seifert fibrations are a useful tool in the study of three-dimensional manifolds, as they provide a concrete way to visualize and understand the structure of the manifold.