# Join

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## Join Meeting

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 spacesA and B, often denoted by ${displaystyle Aast B}$ or ${displaystyle Astar B}$, is defined to be the quotient space

${displaystyle (Atimes Btimes I)/R,}$

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

## Join.nearpod.join

${displaystyle (a,b_{1},0)sim (a,b_{2},0)quad {mbox{for all }}ain A{mbox{ and }}b_{1},b_{2}in B,}$
${displaystyle (a_{1},b,1)sim (a_{2},b,1)quad {mbox{for all }}a_{1},a_{2}in A{mbox{ and }}bin B.}$

At the endpoints, this collapses ${displaystyle Atimes Btimes {0}}$ to ${displaystyle A}$ and ${displaystyle Atimes Btimes {1}}$ to ${displaystyle B}$.

Intuitively, ${displaystyle Astar B}$ 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 coneCX of X.
• The join of a space X with ${displaystyle S^{0}}$ (the 0-dimensional sphere, or, the discrete space with two points) is called the suspension${displaystyle SX}$ of X.
• The join of the spheres ${displaystyle S^{n}}$ and ${displaystyle S^{m}}$ is the sphere ${displaystyle S^{n+m+1}}$.
• The join of two pairs of isolated points is a square (without interior). The join of a square with a third pair of isolated points is an octahedron (again, without interior). In general, the join of n+1 pairs of isolated points is an n-dimensional octahedral sphere.
• The join of two abstract simplicial complexes X and Y on disjoint vertex sets is the abstract simplicial complex ${displaystyle {xcup y xin X,yin Y}}$. I.e., any simplex in the join is the union of a simplex from X and a simplex from Y. For example, if each of X and Y contain two isolated points, X = { {1}, {2} } and Y = { {3}, {4} }, then X * Y = { {1,3} , {1,4} , {2,3} , {2,4} } = a 'square' graph.

## 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:
${displaystyle Astar Bcong C(A)times Bcup _{Atimes B}C(B)times A.}$
• Given basepointed CW complexes (A,a0) and (B,b0), the 'reduced join'
${displaystyle {frac {Astar B}{Astar {b_{0}}cup {a_{0}}star B}}}$

is homeomorphic to the reduced suspension

${displaystyle Sigma (Awedge B)}$

of the smash product. Consequently, since ${displaystyle {Astar {b_{0}}cup {a_{0}}star B}}$ is contractible, there is a homotopy equivalence Mmc's music man.

${displaystyle Astar Bsimeq Sigma (Awedge B).}$