Solid torus
3-dimensional object
From Wikipedia, the free encyclopedia
In mathematics, a solid torus is the solid formed by sweeping a disk around a circle.[1] As a topological space, it is homeomorphic to the Cartesian product of the disk and the circle,[2] 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 , the ordinary torus.
Since the disk is contractible, the solid torus has the homotopy type of a circle, .[3] Therefore the fundamental group and homology groups are isomorphic to those of the circle:
