“Solid torus”的意思、由来-开放百科全书

In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle.^[1] 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, 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.

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:

See also

References

1. ^{{citation|title=Fractal Geometry: Mathematical Foundations and Applications|first=Kenneth|last=Falconer|edition=2nd|publisher=John Wiley & Sons|year=2004|isbn=9780470871355|page=198|url=https://books.google.com/books?id=JXnGzv7X6wcC&pg=PA198}}.
2. ^{{citation|title=An Introduction to Morse Theory|volume=208|series=Translations of mathematical monographs|first=Yukio|last=Matsumoto|publisher=American Mathematical Society|year=2002|isbn= 9780821810224|page=188|url=https://books.google.com/books?id=TtKyqozvgIwC&pg=PA188}}.
3. ^{{citation|title=Nilpotence and Periodicity in Stable Homotopy Theory|volume= 128 |series= Annals of mathematics studies|first=Douglas C.|last=Ravenel|publisher=Princeton University Press|year=1992|isbn= 9780691025728 |page=2|url=https://books.google.com/books?id=RA18_pxdPK4C&pg=PA2}}.