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 join of two line segments. The original spaces are shown in green and blue. The join is a three-dimensional solid, a disphenoid, in gray.

Geometric sets

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

${\displaystyle A\star B\$ ,

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

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 $A$ 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:

{\displaystyle A\star 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

{\displaystyle A\star B\

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":[3]^{: 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:[3]^{: 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 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:[3]^{: 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 $a\in A_{1},b\in B_{1}$ :[3]^: 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.[4]

Homotopy equivalence

If $A$ and $A'$ are homotopy equivalent, then $A\star B$ and $A'\star B$ are homotopy equivalent too.[3]^{: 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:[3]^{: 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:[3]^: 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 (x₁,...,x_2n) with non-negative coordinates such that x₁+...+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 (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:[3]^{: 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 $S^{n}$ .[3]^{: 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.

References

Colin P. Rourke and Brian J. Sanderson (1982). Introduction to Piecewise-Linear Topology. New York: Springer-Verlag. doi:10.1007/978-3-642-81735-9. ISBN 978-3-540-11102-3.
Bryant, John L. (2001-01-01), Daverman, R. J.; Sher, R. B. (eds.), "Chapter 5 - Piecewise Linear Topology", Handbook of Geometric Topology, Amsterdam: North-Holland, pp. 219–259, ISBN 978-0-444-82432-5, retrieved 2022-11-15
Matoušek, Jiří (2007). Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry (2nd ed.). Berlin-Heidelberg: Springer-Verlag. ISBN 978-3-540-00362-5. Written in cooperation with Anders Björner and Günter M. Ziegler , Section 4.3
Fomenko, Anatoly; Fuchs, Dmitry (2016). Homotopical Topology (2nd ed.). Springer. p. 20.

Hatcher, Allen, Algebraic topology. Cambridge University Press, Cambridge, 2002. xii+544 pp. ISBN 0-521-79160-X and ISBN 0-521-79540-0
This article incorporates material from Join on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
Brown, Ronald, Topology and Groupoids Section 5.7 Joins.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] Colin P. Rourke and Brian J. Sanderson (1982). Introduction to Piecewise-Linear Topology. New York: Springer-Verlag. doi:10.1007/978-3-642-81735-9. ISBN 978-3-540-11102-3.

[2] Bryant, John L. (2001-01-01), Daverman, R. J.; Sher, R. B. (eds.), "Chapter 5 - Piecewise Linear Topology", Handbook of Geometric Topology, Amsterdam: North-Holland, pp. 219–259, ISBN 978-0-444-82432-5, retrieved 2022-11-15

[:0-3] Matoušek, Jiří (2007). Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry (2nd ed.). Berlin-Heidelberg: Springer-Verlag. ISBN 978-3-540-00362-5. Written in cooperation with Anders Björner and Günter M. Ziegler , Section 4.3

[4] Fomenko, Anatoly; Fuchs, Dmitry (2016). Homotopical Topology (2nd ed.). Springer. p. 20.

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