# Join (topology)

In topology, a field of mathematics, the join of two topological spaces ${\displaystyle A}$ and ${\displaystyle B}$, often denoted by ${\displaystyle A\ast B}$ or ${\displaystyle A\star B}$, is a topological space formed by taking the disjoint union of the two spaces, and attaching line segments joining every point in ${\displaystyle A}$ to every point in ${\displaystyle B}$. The join of a space ${\displaystyle A}$ with itself is denoted by ${\displaystyle A^{\star 2}:=A\star A}$. The join is defined in slightly different ways in different contexts

## Geometric sets

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

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

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

Some authors[2]: 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 ${\displaystyle A}$  is in ${\displaystyle \mathbb {R} ^{n}}$  and ${\displaystyle B}$  is in ${\displaystyle \mathbb {R} ^{m}}$ , then ${\displaystyle A\times \{0^{m}\}\times \{0\}}$  and ${\displaystyle \{0^{n}\}\times B\times \{1\}}$  are joinable in ${\displaystyle \mathbb {R} ^{n+m+1}}$ . The figure above shows an example for m=n=1, where ${\displaystyle A}$  and ${\displaystyle B}$  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 ${\displaystyle A}$  and ${\displaystyle B}$  are any topological spaces, then:

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

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

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

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

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

where the equivalence relation ${\displaystyle \sim }$  is generated by

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

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

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

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

## Abstract simplicial complexes

If ${\displaystyle A}$  and ${\displaystyle B}$  are any abstract simplicial complexes, then their join is an abstract simplicial complex defined as follows:[3]: 74, Def.4.2.1

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

### Examples

• Suppose ${\displaystyle A=\{\emptyset ,\{a\}\}}$  and ${\displaystyle B=\{\emptyset ,\{b\}\}}$ , that is, two sets with a single point. Then ${\displaystyle A\star B=\{\emptyset ,\{a\},\{b\},\{a,b\}\}}$ , 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, ${\displaystyle A^{\star 2}=A\star A=\{\emptyset ,\{a_{1}\},\{a_{2}\},\{a_{1},a_{2}\}\}}$  where a1 and a2 are two copies of the single element in V(A). Topologically, the result is the same as ${\displaystyle A\star B}$  - a line-segment.
• Suppose ${\displaystyle A=\{\emptyset ,\{a\}\}}$  and ${\displaystyle B=\{\emptyset ,\{b\},\{c\},\{b,c\}\}}$ . Then ${\displaystyle A\star B=P(\{a,b,c\})}$ , which represents a triangle.
• Suppose ${\displaystyle A=\{\emptyset ,\{a\},\{b\}\}}$  and ${\displaystyle B=\{\emptyset ,\{c\},\{d\}\}}$ , that is, two sets with two discrete points. then ${\displaystyle A\star B}$  is a complex with facets ${\displaystyle \{a,c\},\{b,c\},\{a,d\},\{b,d\}}$ , which represents a "square".

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

## Maps

Given two maps ${\displaystyle f:A_{1}\to A_{2}}$  and ${\displaystyle g:B_{1}\to B_{2}}$ , their join ${\displaystyle f\star g:A_{1}\star B_{1}\to A_{2}\star B_{2}}$ is defined based on the representation of each point in the join ${\displaystyle A_{1}\star B_{1}}$  as ${\displaystyle t\cdot a+(1-t)\cdot b}$ , for some ${\displaystyle a\in A_{1},b\in B_{1}}$ :[3]: 77

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

## Special cases

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

The suspension of a topological space ${\displaystyle X}$ , denoted ${\displaystyle SX}$  , is a join of ${\displaystyle X}$  with ${\displaystyle S^{0}}$  (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. ${\displaystyle A\star B\cong B\star A}$ .

### 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 ${\displaystyle A,B,C}$  we have ${\displaystyle (A\star B)\star C\cong A\star (B\star C).}$  Therefore, one can define the k-times join of a space with itself, ${\displaystyle A^{*k}:=A*\cdots *A}$  (k times).

It is possible to define a different join operation ${\displaystyle A\;{\hat {\star }}\;B}$  which uses the same underlying set as ${\displaystyle A\star B}$  but a different topology, and this operation is associative for all topological spaces. For locally compact Hausdorff spaces ${\displaystyle A}$  and ${\displaystyle B}$ , the joins ${\displaystyle A\star B}$  and ${\displaystyle A\;{\hat {\star }}\;B}$  coincide.[4]

### Homotopy equivalence

If ${\displaystyle A}$  and ${\displaystyle A'}$  are homotopy equivalent, then ${\displaystyle A\star B}$  and ${\displaystyle A'\star B}$  are homotopy equivalent too.[3]: 77, Exercise.2

### Reduced join

Given basepointed CW complexes ${\displaystyle (A,a_{0})}$  and ${\displaystyle (B,b_{0})}$ , the "reduced join"

${\displaystyle {\frac {A\star B}{A\star \{b_{0}\}\cup \{a_{0}\}\star B}}}$

is homeomorphic to the reduced suspension

${\displaystyle \Sigma (A\wedge B)}$

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

${\displaystyle A\star B\simeq \Sigma (A\wedge B).}$

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

### Homotopical connectivity

Given two triangulable spaces ${\displaystyle A,B}$ , the homotopical connectivity (${\displaystyle \eta _{\pi }}$ ) of their join is at least the sum of connectivities of its parts:[3]: 81, Prop.4.4.3

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

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

## Deleted join

The deleted join of an abstract complex A is an abstract complex containing all disjoint unions of disjoint faces of A:[3]: 112

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

### Examples

• Suppose ${\displaystyle A=\{\emptyset ,\{a\}\}}$  (a single point). Then ${\displaystyle A_{\Delta }^{*2}:=\{\emptyset ,\{a_{1}\},\{a_{2}\}\}}$ , that is, a discrete space with two disjoint points (recall that ${\displaystyle A^{\star 2}=\{\emptyset ,\{a_{1}\},\{a_{2}\},\{a_{1},a_{2}\}\}}$  = an interval).
• Suppose ${\displaystyle A=\{\emptyset ,\{a\},\{b\}\}}$  (two points). Then ${\displaystyle A_{\Delta }^{*2}}$  is a complex with facets ${\displaystyle \{a_{1},b_{2}\},\{a_{2},b_{1}\}}$  (two disjoint edges).
• Suppose ${\displaystyle A=\{\emptyset ,\{a\},\{b\},\{a,b\}\}}$  (an edge). Then ${\displaystyle A_{\Delta }^{*2}}$  is a complex with facets ${\displaystyle \{a_{1},b_{1}\},\{a_{1},b_{2}\},\{a_{2},b_{1}\},\{a_{2},b_{2}\}}$  (a square). Recall that ${\displaystyle A^{\star 2}}$  represents a solid tetrahedron.
• Suppose A represents an (n-1)-dimensional simplex (with n vertices). Then the join ${\displaystyle A^{\star 2}}$  is a (2n-1)-dimensional simplex (with 2n vertices): it is the set of all points (x1,...,x2n) with non-negative coordinates such that x1+...+x2n=1. The deleted join ${\displaystyle A_{\Delta }^{*2}}$  can be regarded as a subset of this simplex: it is the set of all points (x1,...,x2n) in that simplex, such that the only nonzero coordinates are some k coordinates in x1,..,xn, and the complementary n-k coordinates in xn+1,...,x2n.

### Properties

The deleted join operation commutes with the join. That is, for every two abstract complexes A and B:[3]: Lem.5.5.2

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

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

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

In particular, the deleted join of the n-dimensional simplex ${\displaystyle \Delta ^{n}}$  with itself is the n-dimensional crosspolytope, which is homeomorphic to the n-dimensional sphere ${\displaystyle S^{n}}$ .[3]: Cor.5.5.3

### Generalization

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

${\displaystyle A_{\Delta (k)}^{*n}:=\{a_{1}\sqcup a_{2}\sqcup \cdots \sqcup a_{n}:a_{1},\cdots ,a_{n}{\text{ are k-wise disjoint faces of }}A\}}$ , 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.