Flag word

In mathematics, a solid torus is a topological space homeomorphic to $S^1\times D^2$, i.e. the cartesian product of the circle with a two dimensional disc endowed with the product topology. 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.

Solid torus

A standard way to picture a solid torus is as a toroid, embedded in 3-space.

Since the disk $D^2$ is contractible, the solid torus has the homotopy type of $S^1$. Therefore the fundamental group and homology groups are isomorphic to those of the circle:

${\pi }_1(S^1\times D^2)\cong {\pi }_1(S^1)\cong \mathbb{\mathbb{Z}},$

$H_k(S^1\times D^2)\cong H_k(S^1)\cong \{\begin{array}{ll}\mathbb{\mathbb{Z}}& \text{ if }k=0,1\\ 0& \text{ otherwise }\end{array}.$

See also

• Whitehead manifold
• Hyperbolic Dehn surgery