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.

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 ${\displaystyle S^1\times S^1}$, the ordinary torus.

Since the disk ${\displaystyle D^2}$ is contractible, the solid torus has the homotopy type of a circle, ${\displaystyle S^1}$. Therefore the fundamental group and homology groups are isomorphic to those of the circle: $${\displaystyle \begin{array}{cc}{\pi }_1\left(S^1\times D^2\right)& \cong {\pi }_1\left(S^1\right)\cong \mathbb{Z},\\ H_k\left(S^1\times D^2\right)& \cong H_k\left(S^1\right)\cong \{\begin{array}{ll}\mathbb{Z}& \text{if }k=0,1,\\ 0& \text{otherwise}.\end{array}\end{array}}$$