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.

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.

• 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.

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 ${\displaystyle \dim U+\dim V+1}$ (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}

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

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 ${\displaystyle \{a\cup b:a\in A,b\in B\}}$).

• 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 ${\displaystyle X}$.

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

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

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).

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

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

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}

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

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 ${\displaystyle \eta _{\pi }(S^{n-1})=n}$.

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

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

• 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,...,x_2n) with non-negative coordinates such that x₁+...+x_2n=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,...,x_2n) in that simplex, such that the only nonzero coordinates are some k coordinates in x₁,..,x_n, and the complementary n-k coordinates in x_n+1,...,x_2n.

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}

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.