Solid torus

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:

\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}}.

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 .

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Solid torus

Topological properties

See also

References