In topology, a field of mathematics, the join of two topological spacesA and B, often denoted by ${\displaystyle A\ast B}$ or ${\displaystyle A\star B}$, is defined to be the quotient space

Before you join the meeting, you’ll see a preview screen with options to activate your Mic and Camera. When the session starts, you can access Attendees, Chat, and Mic & Camera settings using the menu bar in the top-right corner. 59 synonyms of join from the Merriam-Webster Thesaurus, plus 80 related words, definitions, and antonyms. Find another word for join. Join: to be adjacent to. Synonyms: abut, adjoin, border (on) Antonyms: demit, break up, dissever Find the right word. GAMES BROWSE THESAURUS WORD OF THE DAY WORDS AT PLAY.

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${\displaystyle (A\times B\times I)/R,}$

where I is the interval [0, 1] and R is the equivalence relation generated by

${\displaystyle (a,b_1,0)\sim (a,b_2,0)\text{for all}a\in A\text{and}{b}_1,b_2\in B,}$

${\displaystyle (a_1,b,1)\sim (a_2,b,1)\text{for all}{a}_1,a_2\in A\text{and}b\in B.}$

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

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

Examples[edit]

• The join of a space X with a one-point space is called the coneCX of X.
• The join of a space X with ${\displaystyle S^0}$ (the 0-dimensional sphere, or, the discrete space with two points) is called the suspension${\displaystyle SX}$ of 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 n+1 pairs of isolated points is an n-dimensional octahedral sphere.
• The join of two abstract simplicial complexes X and 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 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 * Y = { {1,3} , {1,4} , {2,3} , {2,4} } = a 'square' graph.

Properties[edit]

• 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 (A,a₀) and (B,b₀), the 'reduced join'

${\displaystyle \frac{A\star B}{A\star \{b_0\}\cup \{a_0\}\star B}}$

is homeomorphic to the reduced suspension

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

of the smash product. Consequently, since ${\displaystyle A\star \{b_0\}\cup \{a_0\}\star B}$ is contractible, there is a homotopy equivalence

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

See also[edit]

References[edit]

• Hatcher, Allen, Algebraic topology. Cambridge University Press, Cambridge, 2002. xii+544 pp. ISBN0-521-79160-X and ISBN0-521-79540-0
• 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.

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