Solid torus

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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 interior space surrounded by the torus.

Topological properties[edit]

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}.

See also[edit]

References[edit]

  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 .