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.

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{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} ==See also== Cheerios Hyperbolic Dehn surgery Reeb foliation Whitehead manifold Doughnut ==References==

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