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

In Euclidean spaces

The simplest definition is: 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 $B$.

In general topological spaces

Formally, the join is defined as

$A\star B\ :=\ A\sqcup _{p_{0}}(A\times B\times [0,1])\sqcup _{p_{1}}B,$

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:

${A\times B\times \{0\}}\xrightarrow {p_{0}} A,$

${A\times B\times \{1\}}\xrightarrow {p_{1}} B.$

Note that 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 $B$.

Examples

• The join of two disjoint points is an interval.
• The join of a point and an interval is a triangle. In general, the join of a point and an (n-1)-dimensional simplex is an n-dimensional simplex.
• The join of a space $X$ with a one-point space is called the cone $CX$ of $X$.
• The join of a space $X$ with $S^{0}$ (the 0-dimensional sphere, or, the discrete space with two points) is called the suspension $SX$ of $X$.
• The join of two line segments is homeomorphic to a solid tetrahedron, illustrated in the figure above right.
• The join of the spheres $S^{n}$ 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.

Join of abstract simplicial complexes

The join of two abstract simplicial complexes $X$ and $Y$ on disjoint vertex sets is the abstract simplicial complex $\{x\cup y|x\in X,y\in Y\}$. That is, any simplex in the join is the union of a simplex from $X$ and a simplex from $Y$.

For example, if each of $X$ and $Y$ contain two isolated points, $X=\{\{1\},\{2\}\}$ and $Y=\{\{3\},\{4\}\}$, then $X\star Y=\{\{1,3\},\{1,4\},\{2,3\},\{2,4\}\}$, a "square" graph.

Properties

Commutativity and associativity

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

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 $A,B,C$ 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.

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

The homotopical connectivity ($\eta _{\pi }$) of a 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.