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 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.

## Topological properties

The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary. The boundary is homeomorphic to , the ordinary torus.

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

## See also

## References

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