Join (topology)

Geometric join of two line segments. The original spaces are shown in green and blue. The join is a three-dimensional solid 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 $B$.

Definitions

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:

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

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

Some authors 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.

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 $dimU+dimV+1$ (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":

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

Abstract simplicial complexes

If $A$ and $B$ are any abstract simplicial complexes, then their join is an abstract simplicial complex defined as follows:

• The vertex set $V(A\star B)$ is a disjoint union of $V(A)$ and $V(B)$. If $V(A)$ and $V(B)$ are already disjoint, then one can define $V(A\star B):=V(A)\cup V(B)$. Otherwise, one can define, for example, $V(A\star B):=(V(A)\times \{1\})\cup (V(B)\times \{2\})$ (adding 1 and 2 ensures that the elements in the union are disjoint).
• The simplices of $A\star B$ are all unions of a simplex of $A$ with a simplex of $B$. If $V(A)$ and $V(B)$ are disjoint, then $A\star B:=\{a\cup b:a\in A,b\in B\}$.

Examples:

• Suppose $A=\{\emptyset ,\{1\}\}$ and $B=\{\emptyset ,\{2\}\}$, that is, two sets with a single point. Then $A\star B=\{\emptyset ,\{1\},\{2\},\{1,2\}\}$, which represents a line-segment.
• Suppose $A=\{\emptyset ,\{1\}\}$ and $B=\{\emptyset ,\{2\},\{3\},\{2,3\}\}$. Then $A\star B=P(\{1,2,3\})$, which represents a triangle.
• Suppose $A=\{\emptyset ,\{1\},\{2\}\}$ and $B=\{\emptyset ,\{3\},\{4\}\}$, that is, two sets with two discrete points. then $A\star B=\{\{1\},\{2\},\{3\},\{4\},\{1,3\},\{1,4\},\{2,3\},\{2,4\}\}$, which represents a "square".

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 the 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 $a\in A_{1},b\in B_{1}$:

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

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).

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, 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 two spheres is a sphere: the join of $S^{n}$ and $S^{m}$ is the sphere $S^{n+m+1}$. (If $x=(x_{1},\ldots ,x_{n+1})\in S^{n}$ and $y=(y_{1},\ldots ,y_{m+1})\in S^{m}$ are points on the respective unit spheres and the parameter $a\in [0,1]$ describes the location of a point on the line segment joining $x$ to $y$, then $z=(ax_{1},\ldots ,ax_{n+1},{\sqrt {1-a^{2}}}\;y_{1},\ldots ,{\sqrt {1-a^{2}}}\;y_{m+1})\in S^{n+m+1}$.)
• The join of two pairs of isolated points is a square (without interior). The join of a square with a third pair of isolated points is an octahedron (again, without interior). In general, the join of $n+1$ pairs of isolated points is an $n$-dimensional octahedral sphere.

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 $(A\star B)\star C\cong A\star (B\star C).$

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

See also

• Desuspension

References

• 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.