Solid torus

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In mathematics, a solid torus is a topological space homeomorphic to $S^{1}\times D^{2}$ , i.e. the cartesian product of the circle with a two dimensional disc endowed with the product topology. The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary. The boundary is homeomorphic to $S^{1}\times S^{1}$ , the ordinary torus.

Solid torus

A standard way to picture a solid torus is as a toroid, embedded in 3-space.

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

$\pi _{1}(S^{1}\times D^{2})\cong \pi _{1}(S^{1})\cong {\mathbb {Z}},$

$H_{k}(S^{1}\times D^{2})\cong H_{k}(S^{1})\cong {\begin{cases}{\mathbb {Z}}&{\mbox{ if }}k=0,1\\0&{\mbox{ otherwise }}\end{cases}}.$

Solid torus

See also