# Solid torus

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.
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:
$\pi_1(S^1 \times D^2) \cong \pi_1(S^1) \cong \mathbb{Z},$
$H_k(S^1 \times D^2) \cong H_k(S^1) \cong \begin{cases} \mathbb{Z} & \mbox{ if } k = 0,1 \\ 0 & \mbox{ otherwise } \end{cases}.$