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

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.

Real-world objects that approximate a solid torus include O-rings, non-inflatable lifebuoys, ring doughnuts, and bagels.

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

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:

{\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}}}

This topology-related article is a stub.

You can help Wikipedia by expanding it.