SiefertManifold - crowlogic/arb4j GitHub Wiki

A Seifert manifold is a type of three-dimensional manifold, which is a space that locally looks like ordinary three-dimensional space. More specifically, a Seifert manifold is one which is equivalently composed of a disjoint union of circles called Seifert fibers, each homeomorphic to the unit circle in $\mathbb{R}^2$.

The definition of Seifert manifolds is quite technical, but here are some of the main points:

• Each Seifert fiber is homeomorphic to a circle, and the entire Seifert fibration has a tubular neighborhood which is a solid torus. This means that around each circle, the manifold looks like a torus.
• The set of these fibers forms a Seifert fibration of the manifold, a kind of composition of the manifold into disjoint circles (the fibers).
• Seifert manifolds can be orientable or non-orientable, meaning that they either do or do not have a consistent 'orientation' or 'direction'.
• Seifert manifolds are classified by several invariants, including the orientability, the Euler number (which is a measure of the 'twisting' of the fibration), and a certain set of integers associated to each component of the base space.
• The base space of a Seifert fibration is a two-dimensional orbifold, which is a generalization of a manifold that allows for certain types of singularities. The base space is the space you get when you "collapse" each of the fibers to a point.

Seifert manifolds are an important class of three-dimensional manifolds and have been extensively studied in the field of algebraic and differential topology. They are named after the German mathematician Herbert Seifert.