Join (topology)

From Wikipedia, the free encyclopedia
Geometric join of two line segments. The original spaces are shown in green and blue. The join is a three-dimensional solid in gray.

In topology, a field of mathematics, the join of two topological spaces A and B, often denoted by A\star B, is defined to be the quotient space

(A\times B\times I)/R,\,

where I is the interval [0, 1] and R is the equivalence relation generated by

(a,b_{1},0)\sim (a,b_{2},0)\quad {\mbox{for all }}a\in A{\mbox{ and }}b_{1},b_{2}\in B,
(a_{1},b,1)\sim (a_{2},b,1)\quad {\mbox{for all }}a_{1},a_{2}\in A{\mbox{ and }}b\in B.

At the endpoints, this collapses A\times B\times \{0\} to A and A\times B\times \{1\} to B.

Intuitively, A\star B is formed by taking the disjoint union of the two spaces and attaching a line segment joining every point in A to every point in B.

Properties

  • The join is homeomorphic to sum of cartesian products of cones over spaces and spaces itself, where sum is taken over cartesian product of spaces:
A\star B\cong C(A)\times B\cup _{{A\times B}}C(B)\times A

and is homotopy equivalent to suspension of smash product of spaces:

A\star B\simeq \Sigma (A\wedge B)

Examples

  • The join of subsets of n-dimensional Euclidean space A and B is homotopy equivalent to the space of paths in n-dimensional Euclidean space, beginning in A and ending in B.
  • The join of a space X with a one-point space is called the cone CX of X.
  • The join of a space X with S^{0} (the 0-dimensional sphere, or, the discrete space with two points) is called the suspension SX of X.
  • The join of the spheres S^{n} and S^{m} is the sphere S^{{n+m+1}}.

See also

References

This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.