SiefertFiber

A Seifert fiber in a Seifert manifold is essentially a one-dimensional submanifold, which is a simple loop or circle. It's important to note that the concept of a Seifert fiber is directly related to the Seifert fibration of the manifold, and it doesn't make sense to talk about Seifert fibers without considering the overall fibration.

In the context of a Seifert fibration, the Seifert fibers are all the disjoint circles that the fibration is composed of.

The Seifert fibers can be thought of as the "building blocks" of the Seifert manifold, and they play a crucial role in the topology and structure of the manifold. Each Seifert fiber has a tubular neighborhood, which means that if you were to 'zoom in' on the fiber in the manifold, the local structure around the fiber would look like a solid torus (a doughnut shape).

The Seifert fibers of a Seifert fibration are often visualized as a bundle of circles that together form the manifold. Each fiber can be given a direction, forming an oriented circle. However, in non-orientable Seifert manifolds, there might be fibers that cannot be consistently oriented.

One interesting aspect of Seifert fibers is that they may not all be topologically equivalent. That is, there might be more than one type of fiber in a Seifert manifold, classified according to how they are embedded in the manifold.

Overall, Seifert fibers are a key part of the structure of Seifert manifolds and they provide a way to understand and study these manifolds.