Solid torus

Solid torus

In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle. It is homeomorphic to the Cartesian product $S^{1}\times D^{2}$ of the disk and the circle, 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, $S^{1}$. Therefore the fundamental group and homology groups are isomorphic to those of the circle:

$\pi _{1}(S^{1}\times D^{2})\cong \pi _{1}(S^{1})\cong \mathbb {Z} ,$

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

See also

• Hyperbolic Dehn surgery
• Reeb foliation
• Whitehead manifold