Solid torus
In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle.[1] It is homeomorphic to the Cartesian product [math]\displaystyle{ S^1 \times D^2 }[/math] of the disk and the circle,[2] 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 properties
The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary. The boundary is homeomorphic to [math]\displaystyle{ S^1 \times S^1 }[/math], the ordinary torus.
Since the disk [math]\displaystyle{ D^2 }[/math] is contractible, the solid torus has the homotopy type of a circle, [math]\displaystyle{ S^1 }[/math].[3] Therefore the fundamental group and homology groups are isomorphic to those of the circle: [math]\displaystyle{ \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]
See also
- Cheerios
- Hyperbolic Dehn surgery
- Reeb foliation
- Whitehead manifold
- Donut
References
- ↑ Falconer, Kenneth (2004), Fractal Geometry: Mathematical Foundations and Applications (2nd ed.), John Wiley & Sons, p. 198, ISBN 9780470871355, https://books.google.com/books?id=JXnGzv7X6wcC&pg=PA198.
- ↑ Matsumoto, Yukio (2002), An Introduction to Morse Theory, Translations of mathematical monographs, 208, American Mathematical Society, p. 188, ISBN 9780821810224, https://books.google.com/books?id=TtKyqozvgIwC&pg=PA188.
- ↑ Ravenel, Douglas C. (1992), Nilpotence and Periodicity in Stable Homotopy Theory, Annals of mathematics studies, 128, Princeton University Press, p. 2, ISBN 9780691025728, https://books.google.com/books?id=RA18_pxdPK4C&pg=PA2.
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