Solid torus

The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary. The boundary is homeomorphic to ${\displaystyle S^{1}\times S^{1}}$, the ordinary torus.

Since the disk ${\displaystyle D^{2}}$ is contractible, the solid torus has the homotopy type of a circle, ${\displaystyle S^{1}}$.[3] Therefore the fundamental group and homology groups are isomorphic to those of the circle:

${\displaystyle {\begin{aligned}\pi _{1}\left(S^{1}\times D^{2}\right)&\cong \pi _{1}\left(S^{1}\right)\cong \mathbb {Z} ,\\H_{k}\left(S^{1}\times D^{2}\right)&\cong H_{k}\left(S^{1}\right)\cong {\begin{cases}\mathbb {Z} &{\text{if }}k=0,1,\\0&{\text{otherwise}}.\end{cases}}\end{aligned}}}$

• Cheerios
• Hyperbolic Dehn surgery
• Reeb foliation
• Whitehead manifold