Solid torus

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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.

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

Since the disk D2 is contractible, the solid torus has the homotopy type of S1. Therefore the fundamental group and simplicial 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}.
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