# 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 ${\displaystyle S^{1}\times D^{2}}$ of the disk and the circle,[2] endowed with the product topology.

Solid torus

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 compact interior space enclosed by the torus.

## Topological properties

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

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

{\displaystyle {\begin{aligned}\pi _{1}\left(S^{1}\times D^{2}\right)&\cong \pi _{1}\left(S^{1}\right)\cong \mathbb {Z} ,\\H_{k}\left(S^{1}\times D^{2}\right)&\cong H_{k}\left(S^{1}\right)\cong {\begin{cases}\mathbb {Z} &{\text{if }}k=0,1,\\0&{\text{otherwise}}.\end{cases}}\end{aligned}}}