Solid torus
{{distinguish|text=its surface which is a regular torus}}
{{short description|3-dimensional object}}
In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle.{{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}}. It is homeomorphic to the Cartesian product of the disk and the circle,{{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}}. 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.
A solid torus is a torus plus the volume inside the torus. Real-world objects that approximate a solid torus include O-rings, non-inflatable lifebuoys, ring doughnuts, and bagels.
Topological properties
The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary. The boundary is homeomorphic to , the ordinary torus.
Since the disk is contractible, the solid torus has the homotopy type of a circle, .{{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}}. Therefore the fundamental group and homology groups are isomorphic to those of the circle:
\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}