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 compact interior space enclosed by the torus.

A solid torus is a torus plus the volume inside the torus. Real-world objects that approximate a solid torus include O-rings, non-inflatable lifebuoys, ring doughnuts, and bagels.

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