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 $S^1 \times D^2$ 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.

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, $S^{1}$ .^[3] Therefore the fundamental group and homology groups are isomorphic to those of the circle:

{\begin{aligned}\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{aligned}}

References

^ Falconer, Kenneth (2004), Fractal Geometry: Mathematical Foundations and Applications (2nd ed.), John Wiley & Sons, p. 198, ISBN 9780470871355.
^ Matsumoto, Yukio (2002), An Introduction to Morse Theory, Translations of mathematical monographs, 208, American Mathematical Society, p. 188, ISBN 9780821810224.
^ Ravenel, Douglas C. (1992), Nilpotence and Periodicity in Stable Homotopy Theory, Annals of mathematics studies, 128, Princeton University Press, p. 2, ISBN 9780691025728.

[1] Falconer, Kenneth (2004), Fractal Geometry: Mathematical Foundations and Applications (2nd ed.), John Wiley & Sons, p. 198, ISBN 9780470871355.

[2] Matsumoto, Yukio (2002), An Introduction to Morse Theory, Translations of mathematical monographs, 208, American Mathematical Society, p. 188, ISBN 9780821810224.

[3] Ravenel, Douglas C. (1992), Nilpotence and Periodicity in Stable Homotopy Theory, Annals of mathematics studies, 128, Princeton University Press, p. 2, ISBN 9780691025728.

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[2]

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Solid torus

Topological properties

See also

References