Solid torus
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In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle.<ref>Template:Citation.</ref> It is homeomorphic to the Cartesian product <math>S^1 \times D^2</math> of the disk and the circle,<ref>Template:Citation.</ref> 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.
A solid torus is a torus plus the volume inside the torus. Real-world objects that approximate a solid torus include O-rings, non-inflatable lifebuoys, ring doughnuts, and bagels.
Topological propertiesEdit
The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary. The boundary is homeomorphic to <math>S^1 \times S^1</math>, the ordinary torus.
Since the disk <math>D^2</math> is contractible, the solid torus has the homotopy type of a circle, <math>S^1</math>.<ref>Template:Citation.</ref> Therefore the fundamental group and homology groups are isomorphic to those of the circle: <math display=block>\begin{align}
\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{align}</math>