In mathematics, a solid torus is the solid formed by sweeping a disk around a circle. As a topological space, it is homeomorphic to the Cartesian product S 1 × D 2 {\displaystyle S^{1}\times D^{2}} of the disk and the circle, 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, a surface 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 region inside the torus, with a non-zero volume. 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 S 1 × S 1 {\displaystyle S^{1}\times S^{1}} , the ordinary torus. Since the disk D 2 {\displaystyle D^{2}} is contractible, the solid torus has the homotopy type of a circle, S 1 {\displaystyle S^{1}} . Therefore the fundamental group and homology groups are isomorphic to those of the circle:
π 1 ( S 1 × D 2 ) ≅ π 1 ( S 1 ) ≅ Z , H k ( S 1 × D 2 ) ≅ H k ( S 1 ) ≅ { Z if k = 0 , 1 , 0 otherwise . {\displaystyle {\begin{aligned}\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{aligned}}}
See also Cheerios Hyperbolic Dehn surgery Reeb foliation Whitehead manifold Donut
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