Solid torus

Not to be confused with its surface which is a regular torus.

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

1. Falconer, Kenneth (2004), Fractal Geometry: Mathematical Foundations and Applications (2nd ed.), John Wiley & Sons, p. 198, ISBN 9780470871355.
2. Matsumoto, Yukio (2002), An Introduction to Morse Theory, Translations of mathematical monographs, vol. 208, American Mathematical Society, p. 188, ISBN 9780821810224.
3. Ravenel, Douglas C. (1992), Nilpotence and Periodicity in Stable Homotopy Theory, Annals of mathematics studies, vol. 128, Princeton University Press, p. 2, ISBN 9780691025728.