Join (topology)

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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.


  • 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)


  • 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}.

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