Solid torus

Solid torus

In mathematics, a solid torus is a topological space homeomorphic to S^1 imes 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 imes S^1, the ordinary 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 simplicial homology groups are isomorphic to those of the circle::pi_1(S^1 imes D^2) cong pi_1(S^1) cong mathbb{Z},

:H_k(S^1 imes D^2) cong H_k(S^1) congegin{cases}mathbb{Z} & mbox{ if } k = 0,1 \0 & mbox{ otherwise } end{cases}.


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