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 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 line segments joining every point in A to every point 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 .
  • 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 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

See also[edit]


  • Hatcher, Allen, Algebraic topology. Cambridge University Press, Cambridge, 2002. xii+544 pp. ISBN 0-521-79160-X and ISBN 0-521-79540-0
  • This article incorporates material from Join on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
  • Brown, Ronald, Topology and Groupoids Section 5.7 Joins.