join (topology)

Image:Join.svgs. The original spaces are shown in green and blue. The join is a three-dimensional solid, a disphenoid, 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 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:{{Cite book |last=Colin P. Rourke and Brian J. Sanderson |url=https://link.springer.com/book/10.1007/978-3-642-81735-9 |title=Introduction to Piecewise-Linear Topology |publisher=Springer-Verlag |year=1982 |location=New York |language=en |doi=10.1007/978-3-642-81735-9|isbn=978-3-540-11102-3 }}{{Rp|page=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{{Citation |last=Bryant |first=John L. |title=Chapter 5 - Piecewise Linear Topology |date=2001-01-01 |url=https://www.sciencedirect.com/science/article/pii/B9780444824325500068 |work=Handbook of Geometric Topology |pages=219–259 |editor-last=Daverman |editor-first=R. J. |place=Amsterdam |publisher=North-Holland |language=en |isbn=978-0-444-82432-5 |access-date=2022-11-15 |editor2-last=Sher |editor2-first=R. B.}}{{Rp|page=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 $\dim U + \dim V + 1$ (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":{{Rp|page=75|location=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:{{Rp|page=74|location=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 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".

The combinatorial definition is equivalent to the topological definition in the following sense:{{Rp|page=77|location=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 $a\in A_1, b\in B_1$ :{{Rp|page=77|location=}}

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

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).

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.{{Cite book |last1=Fomenko |first1=Anatoly |title=Homotopical Topology |last2=Fuchs |first2=Dmitry |publisher=Springer |year=2016 |edition=2nd |pages=20}}

= Homotopy equivalence =

If $A$ and $A'$ are homotopy equivalent, then $A\star B$ and $A'\star B$ are homotopy equivalent too.{{Rp|page=77|location=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:{{Cite Matousek 2007}}, Section 4.3{{Rp|page=81|location=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:{{Rp|location=|page=112}}

$A^{*2}_{\Delta} := \{ a_1\sqcup a_2: a_1,a_2\in A, a_1\cap a_2 = \emptyset \}$

= 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,...,x_2n) with non-negative coordinates such that x₁+...+x_2n=1. The deleted join $A^{*2}_{\Delta}$ 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.

= Properties =

The deleted join operation commutes with the join. That is, for every two abstract complexes A and B:{{Rp|location=Lem.5.5.2|page=}}

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

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 $S^n$ .{{Rp|location=Cor.5.5.3|page=}}

= 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.

References

Hatcher, Allen, [http://pi.math.cornell.edu/~hatcher/AT/ATpage.html Algebraic topology.] Cambridge University Press, Cambridge, 2002. xii+544 pp. {{ISBN|0-521-79160-X}} and {{ISBN|0-521-79540-0}}
{{PlanetMath attribution|id=3985|title=Join}}
Brown, Ronald, [http://pages.bangor.ac.uk/~mas010/topgpds.html Topology and Groupoids] Section 5.7 Joins.

Category:Algebraic topology

Category:Operations on structures

join (topology)

Geometric sets

= Examples =

Topological spaces

Abstract simplicial complexes

= Examples =

Maps

Special cases

Properties

= Commutativity =

= Associativity =

= Homotopy equivalence =

= Reduced join =

= Homotopical connectivity =

Deleted join

= Examples =

= Properties =

= Generalization =

See also

References