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2014-01-16T04:58:02Z

128.118.101.21:

In [[mathematics]], a '''solid torus''' is a [[topological space]] [[homeomorphic]] to <math>S^1 \times D^2</math>, i.e. the [[cartesian product]] of the [[circle]] with a two dimensional [[ball (mathematics)|disc]] endowed with the [[product topology]]. The solid torus is a [[connected_space|connected]], [[compact_space|compact]], [[Orientation (mathematics)|orientable]] 3-dimensional [[manifold]] with boundary. The boundary is homeomorphic to <math>S^1 \times S^1</math>, the ordinary [[torus]].
[[Image:Torus illustration.png|thumb|right|Solid torus]]
A standard way to picture a solid torus is as a [[toroid_(geometry)|toroid]], embedded in [[3-space]].

Since the disk <math>D^2</math> is [[contractible]], the solid torus has the [[homotopy]] type of <math>S^1</math>. Therefore the [[fundamental group]] and [[Homology_(mathematics)|homology]] groups are [[isomorphism|isomorphic]] to those of the circle:
:<math>\pi_1(S^1 \times D^2) \cong \pi_1(S^1) \cong \mathbb{Z},</math>

:<math>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}.</math>

==See also==
*[[Whitehead manifold]]
*[[Hyperbolic Dehn surgery]]

[[Category:Topology]]

{{topology-stub}}

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