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.

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, ${\displaystyle S^{1}}$.^[3] Therefore the fundamental group and homology groups are isomorphic to those of the circle:

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

See also

• Cheerios
• Hyperbolic Dehn surgery
• Reeb foliation
• Whitehead manifold
• Donut

References