Solid torus explained

S ^{1} x D ^{2} 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, 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 volume inside the torus. 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} x S ^{1} , the ordinary torus.

Since the disk

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

See also

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

Notes and References

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