View source for Solid torus

{{distinguish|text=its surface which is a regular [[torus]]}}
{{short description|3-dimensional object}}
[[Image:Torus illustration.png|thumb|Solid torus]]
In [[mathematics]], a '''solid torus''' is the [[topological space]] formed by sweeping a [[Disk (mathematics)|disk]] around a [[circle]].<ref>{{citation|title=Fractal Geometry: Mathematical Foundations and Applications|first=Kenneth|last=Falconer|edition=2nd|publisher=[[John Wiley & Sons]]|year=2004|isbn=9780470871355|page=198|url=https://books.google.com/books?id=JXnGzv7X6wcC&pg=PA198}}.</ref> It is [[homeomorphic]] to the [[Cartesian product]] <math>S^1 \times D^2</math> of the disk and the circle,<ref>{{citation|title=An Introduction to Morse Theory|volume=208|series=Translations of mathematical monographs|first=Yukio|last=Matsumoto|publisher=[[American Mathematical Society]]|year=2002|isbn= 9780821810224|page=188|url=https://books.google.com/books?id=TtKyqozvgIwC&pg=PA188}}.</ref> endowed with the [[product topology]].

A standard way to visualize a solid torus is as a [[toroid (geometry)|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-ring]]s, non-inflatable [[lifebuoy]]s, ring [[doughnut]]s, and [[bagel]]s.

==Topological properties==

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]].

Since the disk <math>D^2</math> is [[contractible]], the solid torus has the [[homotopy]] type of a circle, <math>S^1</math>.<ref>{{citation|title=Nilpotence and Periodicity in Stable Homotopy Theory|volume= 128 |series= Annals of mathematics studies|first=Douglas C.|last=Ravenel|publisher=[[Princeton University Press]]|year=1992|isbn= 9780691025728 |page=2|url=https://books.google.com/books?id=RA18_pxdPK4C&pg=PA2}}.</ref> Therefore the [[fundamental group]] and [[Homology (mathematics)|homology]] groups are [[isomorphism|isomorphic]] to those of the circle:
<math display=block>\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]]
*[[Doughnut|Donut]]

==References==
{{reflist}}

{{Manifolds}}

[[Category:3-manifolds]]

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