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 a topological space defined as

A\star B\ :=\ A\sqcup _{p_{0}}(A\times B\times [0,1])\sqcup _{p_{1}}B,

where the cylinder $A\times B\times [0,1]$ is attached to the original spaces $A$ and $B$ along the natural projections of the faces of the cylinder:

{A\times B\times \{0\}}\xrightarrow {p_{0}} A,

{A\times B\times \{1\}}\xrightarrow {p_{1}} 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 $B$ .

Note that usually it is implicitly assumed that $A$ and $B$ are non-empty, in which case the definition is often phrased a bit differently: instead of attaching the faces of the cylinder $A\times B\times [0,1]$ to the spaces $A$ and $B$ , these faces are simply collapsed in a way suggested by the attachment projections $p_{1},p_{2}$ :

A\star B\ :=\ (A\times B\times [0,1])/R,

where $R$ 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 $B$ .

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 $S^{n+m+1}$ .
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 $\{x\cup y|x\in X,y\in Y\}$ . 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: