Solid torus

File:Torus illustration.png

Solid torus

In mathematics, a solid torus is the solid formed by sweeping a disk around a circle. As a topological space, it is homeomorphic to the Cartesian product $S^1\times D^2$ of the disk and the circle, 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, a surface 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 region inside the torus, with a non-zero volume. 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 $S^1\times S^1$, the ordinary torus.

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

See also

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