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 ${\displaystyle A}$ and ${\displaystyle B}$, often denoted by ${\displaystyle A\ast B}$ or ${\displaystyle A\star B}$, is a topological space defined as

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

where the cylinder ${\displaystyle A\times B\times [0,1]}$ is attached to the original spaces ${\displaystyle A}$ and ${\displaystyle B}$ along the natural projections of the faces of the cylinder:

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

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

Intuitively, ${\displaystyle A\star B}$ is formed by taking the disjoint union of the two spaces and attaching line segments joining every point in ${\displaystyle A}$ to every point in ${\displaystyle B}$.

Note that usually it is implicitly assumed that ${\displaystyle A}$ and ${\displaystyle B}$ are non-empty, in which case the definition is often phrased a bit differently: instead of attaching the faces of the cylinder ${\displaystyle A\times B\times [0,1]}$ to the spaces ${\displaystyle A}$ and ${\displaystyle B}$, these faces are simply collapsed in a way suggested by the attachment projections ${\displaystyle p_{1},p_{2}}$:

${\displaystyle A\star B\ :=\ (A\times B\times [0,1])/R,}$

where ${\displaystyle R}$ is the equivalence relation generated by

${\displaystyle (a,b_{1},0)\sim (a,b_{2},0)\quad {\mbox{for all }}a\in A{\mbox{ and }}b_{1},b_{2}\in B,}$

${\displaystyle (a_{1},b,1)\sim (a_{2},b,1)\quad {\mbox{for all }}a_{1},a_{2}\in A{\mbox{ and }}b\in B.}$

At the endpoints, this collapses ${\displaystyle A\times B\times \{0\}}$ to ${\displaystyle A}$ and ${\displaystyle A\times B\times \{1\}}$ to ${\displaystyle B}$.

Examples[]

• The join of a space ${\displaystyle X}$ with a one-point space is called the cone ${\displaystyle CX}$ of ${\displaystyle X}$.
• The join of a space ${\displaystyle X}$ with ${\displaystyle S^{0}}$ (the 0-dimensional sphere, or, the discrete space with two points) is called the suspension ${\displaystyle SX}$ of ${\displaystyle X}$.
• The join of the spheres ${\displaystyle S^{n}}$ and ${\displaystyle S^{m}}$ is the sphere ${\displaystyle S^{n+m+1}}$.
• 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 ${\displaystyle n+1}$ pairs of isolated points is an ${\displaystyle n}$-dimensional octahedral sphere.
• The join of two abstract simplicial complexes ${\displaystyle X}$ and ${\displaystyle Y}$ on disjoint vertex sets is the abstract simplicial complex ${\displaystyle \{x\cup y|x\in X,y\in Y\}}$. I.e., any simplex in the join is the union of a simplex from ${\displaystyle X}$ and a simplex from ${\displaystyle Y}$. For example, if each of ${\displaystyle X}$ and ${\displaystyle Y}$ contain two isolated points, ${\displaystyle X=\{\{1\},\{2\}\}}$ and ${\displaystyle Y=\{\{3\},\{4\}\}}$, then ${\displaystyle X\star Y=\{\{1,3\},\{1,4\},\{2,3\},\{2,4\}\}}$, a "square" graph.

Properties[]

• The join of two spaces is homeomorphic to a sum of cartesian products of cones over the spaces and the spaces themselves, where the sum is taken over the cartesian product of the spaces:

${\displaystyle A\star B\cong C(A)\times B\cup _{A\times B}C(B)\times A.}$

• Given basepointed CW complexes ${\displaystyle (A,a_{0})}$ and ${\displaystyle (B,b_{0})}$, the "reduced join"

${\displaystyle {\frac {A\star B}{A\star \{b_{0}\}\cup \{a_{0}\}\star B}}}$

is homeomorphic to the reduced suspension

${\displaystyle \Sigma (A\wedge B)}$

of the smash product. Consequently, since ${\displaystyle {A\star \{b_{0}\}\cup \{a_{0}\}\star B}}$ is contractible, there is a homotopy equivalence

${\displaystyle A\star B\simeq \Sigma (A\wedge B).}$

This equivalence establishes the isomorphism ${\displaystyle {\widetilde {H}}_{n}(A\star B)\cong H_{n-1}(A\wedge B)\ {\bigl (}=H_{n-1}(A\times B/A\vee B){\bigr )}}$.

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.