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: "~ 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 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 $$dim U + dim V + 1$$ (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": "~ 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: 


 * 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
The combinatorial definition is equivalent to the topological definition in the following sense:  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 $$ X$$.
 * 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 a1 and a2 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".

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 $$ a\in A_1, b\in B_1$$: "$ 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 $$X$$, 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).

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. 

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: 


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

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: "$ A^{*2}_{\Delta} := \{ 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 $$ A^{*2}_{\Delta} := \{ \emptyset, \{a_1\}, \{a_2\} \}$$, 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 $$ A^{*2}_{\Delta} $$ 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 $$ A^{*2}_{\Delta} $$ 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 (x1,...,x2n) with non-negative coordinates such that x1+...+x2n=1. The deleted join $$ A^{*2}_{\Delta} $$ 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:  "$ (A*B)^{*2}_{\Delta} = (A^{*2}_{\Delta}) * (B^{*2}_{\Delta})$"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 $$ S^n$$. 

Generalization
The n-fold k-wise deleted join of a simplicial complex A is defined as: $$ A^{*n}_{\Delta(k)} := \{ a_1\sqcup a_2 \sqcup\cdots \sqcup a_n: a_1,\cdots,a_n \text{ are k-wise disjoint faces of } A \}$$,

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.