Seifert fibered space

A Seifert fibered space is a three-manifold decomposed into circles so that every fiber has a standard fibered solid-torus neighborhood. The quotient is a two-dimensional orbifold, whose cone points correspond to exceptional fibers.

A vertical surface is a union of Seifert fibers over an arc or a closed curve in the base orbifold. Typical connected examples are an annulus and a torus.

A horizontal surface is a properly embedded topological surface transverse to every Seifert fiber. Its projection to the base orbifold is an orbifold covering. The covering degree is its intersection number with a regular Seifert fiber.

A Seifert fiber is one of the circles of a Seifert fibered space. A regular fiber has a product neighborhood; an exceptional fiber has a fibered solid torus neighborhood with multiplicity greater than one.

The Hopf circle action on $S^3$ descends through the antipodal quotient to a free circle action on ${\mathbb{RP}}^3$, giving a Seifert fibration over $S^2$ with no exceptional fibers. The weighted actionalso descends and, after dividing by its generic order-two kernel, gives a fibration over $S^2$ with one exceptional fiber of multiplicity three.