Join (topology)

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
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 or , is defined to be the quotient space

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

At the endpoints, this collapses to and to .

Intuitively, 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[edit]

  • The join of two spaces is homeomorphic to a sum of cartesian products of cones over the spaces and the spaces themselves, where the sum is taken over the cartesian product of the spaces:
  • Given basepointed CW complexes (A,a0) and (B,b0), the "reduced join"

is homeomorphic to the reduced suspension

of the smash product. Consequently, since is contractible, there is a homotopy equivalence

Examples[edit]

  • 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 (the 0-dimensional sphere, or, the discrete space with two points) is called the suspension of X.
  • The join of the spheres and is the sphere .

See also[edit]

References[edit]