Solid torus

Solid torus

In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle.[1] It is homeomorphic to the Cartesian product $S^1 \times D^2$ of the disk and the circle,[2] endowed with the product topology. A standard way to visualize a solid torus is as a toroid, embedded in 3-space. However, it should be distinguished from a torus, which has the same visual appearance: the torus is the two-dimensional space on the boundary of a toroid, while the solid torus includes also the interior space surrounded by the torus.

Topological properties

The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary. The boundary is homeomorphic to $S^1 \times S^1$, the ordinary torus.

Since the disk $D^2$ is contractible, the solid torus has the homotopy type of a circle, $S^1$.[3] Therefore the fundamental group and homology groups are isomorphic to those of the circle:

$\pi_1(S^1 \times D^2) \cong \pi_1(S^1) \cong \mathbb{Z},$
$H_k(S^1 \times D^2) \cong H_k(S^1) \cong \begin{cases} \mathbb{Z} & \mbox{ if } k = 0,1 \\ 0 & \mbox{ otherwise } \end{cases}.$