Join (topology)
# Join (topology)

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 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 $A$ with itself is denoted by $A^{\star 2}:=A\star A$. The join is defined in slightly different ways in different contexts

## Geometric sets

If $A$ and $B$ are subsets of the Euclidean space $\mathbb {R} ^{n}$, $then:^{:1}$

   $A\star B\ :=\ \{t\cdot a+(1-t)\cdot b~|~a\in A,b\in B,t\in [0,1]\}$,

that is, the set of all line-segments between a point in $A$ and a point in $B$.

Some $authors^{:5}$ restrict the definition to subsets that are joinable: any two different line-segments, connecting a point of A to a point of B, meet in at most a common endpoint (that is, they do not intersect in their interior). Every two subsets can be made "joinable". For example, if $A$ is in $\mathbb {R} ^{n}$ and $B$ is in $\mathbb {R} ^{m}$, then $A\times \{0^{m}\}\times \{0\}$ and $\{0^{n}\}\times B\times \{1\}$ are joinable in $\mathbb {R} ^{n+m+1}$. The figure above shows an example for m=n=1, where $A$ and $B$ are line-segments.

### Examples

* The join of two simplices is a simplex: the join of an n-dimensional and an m-dimensional simplex is an (m+n+1)-dimensional simplex. Some special cases are:
   * The join of two disjoint points is an interval (m=n=0).
   * The join of a point and an interval is a triangle (m=0, n=1).
   * The join of two line segments is homeomorphic to a solid tetrahedron or disphenoid, illustrated in the figure above right (m=n=1).
   * The join of a point and an (n-1)-dimensional simplex is an n-dimensional simplex.
* The join of a point and a polygon (or any polytope) is a pyramid, like the join of a point and square is a square pyramid. The join of a point and a cube is a cubic pyramid.
* The join of a point and a circle is a cone, and the join of a point and a sphere is a hypercone.

## Topological spaces

If $A$ and $B$ are any topological spaces, then:

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

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:

    ${A\times B\times \{0\}}\xrightarrow {p_{0}} A,$
    ${A\times B\times \{1\}}\xrightarrow {p_{1}} B.$

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 $p_{1},p_{2}$: we form the quotient space

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

where the equivalence relation $\sim$ is 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 $B$.

If $A$ and $B$ are bounded subsets of the Euclidean space $\mathbb {R} ^{n}$, and $A\subseteq U$ and $B\subseteq V$, where $U,V$ are disjoint subspaces of $\mathbb {R} ^{n}$ such that the dimension of their affine hull is $dimU+dimV+1$ (e.g. two non-intersecting non-parallel lines in $\mathbb {R} ^{3}$), then the topological definition reduces to the geometric definition, that is, the "geometric join" is homeomorphic to the "topological $join":^{:75,Prop.4.2.4}$

   ${\big(A\times B\times [0,1])/\sim {\big )}\simeq \{t\cdot a+(1-t)\cdot b~|~a\in A,b\in B,t\in [0,1]\}$

## Abstract simplicial complexes

If $A$ and $B$ are any abstract simplicial complexes, then their join is an abstract simplicial complex defined as $follows:^{:74,Def.4.2.1}$

* The vertex set $V(A\star B)$ is a disjoint union of $V(A)$ and $V(B)$.
* The simplices of $A\star B$ are all disjoint unions of a simplex of $A$ with a simplex of $B$: $A\star B:=\{a\sqcup b:a\in A,b\in B\}$ (in the special case in which $V(A)$ and $V(B)$ are disjoint, the join is simply $\{a\cup b:a\in A,b\in B\}$).

### Examples

* Suppose $A=\{\emptyset ,\{a\}\}$ and $B=\{\emptyset ,\{b\}\}$, that is, two sets with a single point. Then $A\star B=\{\emptyset ,\{a\},\{b\},\{a,b\}\}$, which represents a line-segment. Note that the vertex sets of A and B are disjoint; otherwise, we should have made them disjoint. For example, $A^{\star 2}=A\star A=\{\emptyset ,\{a_{1}\},\{a_{2}\},\{a_{1},a_{2}\}\}$ where $a_1$ and $a_2$ are two copies of the single element in V(A). Topologically, the result is the same as $A\star B$ - a line-segment.
* Suppose $A=\{\emptyset ,\{a\}\}$ and $B=\{\emptyset ,\{b\},\{c\},\{b,c\}\}$. Then $A\star B=P(\{a,b,c\})$, which represents a triangle.
* Suppose $A=\{\emptyset ,\{a\},\{b\}\}$ and $B=\{\emptyset ,\{c\},\{d\}\}$, that is, two sets with two discrete points. then $A\star B$ is a complex with facets $\{a,c\},\{b,c\},\{a,d\},\{b,d\}$, which represents a "square".

The combinatorial definition is equivalent to the topological definition in the following $sense:^{:77,Exercise.3}$ for every two abstract simplicial complexes $A$ and $B$, $||A\star B||$ is homeomorphic to $||A||\star ||B||$, where $||X||$ denotes any geometric realization of the complex $X$.

## Maps

Given two maps $f:A_{1}\to A_{2}$ and $g:B_{1}\to B_{2}$, their join $f\star g:A_{1}\star B_{1}\to A_{2}\star B_{2}$is defined based on the representation of each point in the join $A_{1}\star B_{1}$ as $t\cdot a+(1-t)\cdot b$, for some $a\in A_{1},b\in B_{1}$$:^{:77}$

   $f\star g~(t\cdot a+(1-t)\cdot b)~~=~~t\cdot f(a)+(1-t)\cdot f(b)$

## Special cases

The cone of a topological space $X$, denoted $CX$ , is a join of $X$ with a single point.

The suspension of a topological space $X$, denoted $SX$ , is a join of $X$ with $S^{0}$ (the 0-dimensional sphere, or, the discrete space with two points).

## Properties

### Commutativity

The join of two spaces is commutative up to homeomorphism, i.e. $A\star B\cong B\star A$.

### Associativity

It is not true that the join operation defined above is associative up to homeomorphism for arbitrary topological spaces. However, for locally compact Hausdorff spaces $A,B,C$ we have $(A\star B)\star C\cong A\star (B\star C).$ Therefore, one can define the k-times join of a space with itself, $A^{*k}:=A*\cdots *A$ (k times).

It is possible to define a different join operation $A\;{\hat {\star }}\;B$ which uses the same underlying set as $A\star B$ but a different topology, and this operation is associative for all topological spaces. For locally compact Hausdorff spaces $A$ and $B$, the joins $A\star B$ and $A\;{\hat {\star }}\;B$ coincide.

### Homotopy equivalence

If $A$ and $A'$ are homotopy equivalent, then $A\star B$ and $A'\star B$ are homotopy equivalent $too.^{:77,Exercise.2}$

### Reduced join

Given basepointed CW complexes $(A,a_{0})$ and $(B,b_{0})$, 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

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

This equivalence establishes the isomorphism ${\widetilde {H}}_{n}(A\star B)\cong H_{n-1}(A\wedge B)\ {\bigl=H_{n-1}(A\times B/A\vee B){\bigr )}$.

### Homotopical connectivity

Given two triangulable spaces $A,B$, the homotopical connectivity ($\eta _{\pi }$) of their join is at least the sum of connectivities of its $parts:^{:81,Prop.4.4.3}$

* $\eta _{\pi }(A*B)\geq \eta _{\pi }(A)+\eta _{\pi }(B)$.

As an example, let $A=B=S^{0}$ be a set of two disconnected points. There is a 1-dimensional hole between the points, so $\eta _{\pi }(A)=\eta _{\pi }(B)=1$. The join $A*B$ is a square, which is homeomorphic to a circle that has a 2-dimensional hole, so $\eta _{\pi }(A*B)=2$. The join of this square with a third copy of $S^{0}$ is a octahedron, which is homeomorphic to $S^{2}$ , whose hole is 3-dimensional. In general, the join of n copies of $S^{0}$ is homeomorphic to $S^{n-1}$ and $\eta _{\pi }(S^{n-1})=n$.

## Deleted join

The deleted join of an abstract complex A is an abstract complex containing all disjoint unions of disjoint faces of $A:^{:112}$

   $A_{\Delta }^{*2}:=\{a_{1}\sqcup a_{2}:a_{1},a_{2}\in A,a_{1}\cap a_{2}=\emptyset \}$

### Examples

* Suppose $A=\{\emptyset ,\{a\}\}$ (a single point). Then $A_{\Delta }^{*2}:=\{\emptyset ,\{a_{1}\},\{a_{2}\}\}$, that is, a discrete space with two disjoint points (recall that $A^{\star 2}=\{\emptyset ,\{a_{1}\},\{a_{2}\},\{a_{1},a_{2}\}\}$ = an interval).
* Suppose $A=\{\emptyset ,\{a\},\{b\}\}$ (two points). Then $A_{\Delta }^{*2}$ is a complex with facets $\{a_{1},b_{2}\},\{a_{2},b_{1}\}$ (two disjoint edges).
* Suppose $A=\{\emptyset ,\{a\},\{b\},\{a,b\}\}$ (an edge). Then $A_{\Delta }^{*2}$ is a complex with facets $\{a_{1},b_{1}\},\{a_{1},b_{2}\},\{a_{2},b_{1}\},\{a_{2},b_{2}\}$ (a square). Recall that $A^{\star 2}$ represents a solid tetrahedron.
* Suppose A represents an (n-1)-dimensional simplex (with n vertices). Then the join $A^{\star 2}$ is a (2n-1)-dimensional simplex (with 2n vertices): it is the set of all points $(x_1,...,x_{2n})$ with non-negative coordinates such that $x_1+...+x_{2n}=1.$ The deleted join $A_{\Delta }^{*2}$ can be regarded as a subset of this simplex: it is the set of all points $(x_1,...,x_{2n})$ in that simplex, such that the only nonzero coordinates are some k coordinates in $x_1,..,x_n,$ and the complementary n-k coordinates in $x_{n+1},...,x_{2n}.$

### Properties

The deleted join operation commutes with the join. That is, for every two abstract complexes A and $B:^{:Lem.5.5.2}$

   $(A*B)_{\Delta }^{*2}=(A_{\Delta }^{*2})*(B_{\Delta }^{*2})$

Proof. Each simplex in the left-hand-side complex is of the form $(a_{1}\sqcup b_{1})\sqcup (a_{2}\sqcup b_{2})$, where $a_{1},a_{2}\in A,b_{1},b_{2}\in B$, and $(a_{1}\sqcup b_{1}),(a_{2}\sqcup b_{2})$ are disjoint. Due to the properties of a disjoint union, the latter condition is equivalent to: $a_{1},a_{2}$ are disjoint and $b_{1},b_{2}$ are disjoint.

Each simplex in the right-hand-side complex is of the form $(a_{1}\sqcup a_{2})\sqcup (b_{1}\sqcup b_{2})$, where $a_{1},a_{2}\in A,b_{1},b_{2}\in B$, and $a_{1},a_{2}$ are disjoint and $b_{1},b_{2}$ are disjoint. So the sets of simplices on both sides are exactly the same. □

In particular, the deleted join of the n-dimensional simplex $\Delta ^{n}$ with itself is the n-dimensional crosspolytope, which is homeomorphic to the n-dimensional sphere $S^{n}$$.^{:Cor.5.5.3}$

### Generalization

The n-fold k-wise deleted join of a simplicial complex A is defined as:

   $A_{\Delta (k)}^{*n}:=\{a_{1}\sqcup a_{2}\sqcup \cdots \sqcup a_{n}:a_{1},\cdots ,a_{n}{\text{ are k-wise disjoint faces of }}A\}$,
   where "k-wise disjoint" means that every subset of k have an empty intersection.

In particular, the n-fold n-wise deleted join contains all disjoint unions of n faces whose intersection is empty, and the n-fold 2-wise deleted join is smaller: it contains only the disjoint unions of n faces that are pairwise-disjoint. The 2-fold 2-wise deleted join is just the simple deleted join defined above.

The n-fold 2-wise deleted join of a discrete space with m points is called the (m,n)-chessboard complex.

