Join (topology)

File:Join.svg

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.

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).

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 $S^{n+m+1}$ .