Seifert fiber space

Definition

Let $a \ge 1$ and $b$ be coprime integers, let $D^2$ be the closed unit disk and let $\rho \colon D^2 \to D^2$ be the rotation by the angle $2\pi b / a$. The fibered solid torus $T(a, b)$ is the quotient of $D^2 \times [0, 1]$ by $(x, 0) \sim (\rho(x), 1)$, decomposed into the circles that are the images of the segments $\set{x} \times [0, 1]$. Its central circle, the image of $\set{0} \times [0, 1]$, is called regular if $a = 1$ and exceptional if $a > 1$.

A Seifert fiber space is a closed 3-manifold $M$ with a decomposition of $M$ into disjoint circles, the fibers, such that the following holds. Each fiber $C$ has a neighborhood $U$ that is a union of fibers. There is a homeomorphism from $U$ to some $T(a, b)$ that maps fibers to fibers and $C$ to the central circle.