# Join (topology) 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 $A\ast B$ or $A\star B$ , 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 line segments joining every point in A to every point in B.

## Examples

• 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 $S^{n+m+1}$ .

## Properties

• 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:
$A\star B\cong C(A)\times B\cup _{A\times B}C(B)\times A.$ • Given basepointed CW complexes (A,a0) and (B,b0), the "reduced join"
${\frac {A\star B}{A\star \{b_{0}\}\cup \{a_{0}\}\star B}}$ is homeomorphic to the reduced suspension

$\Sigma (A\wedge B)$ of the smash product. Consequently, since ${A\star \{b_{0}\}\cup \{a_{0}\}\star B}$ is contractible, there is a homotopy equivalence

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