Join (topology)

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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 $A\star B$ , is defined to be the quotient space

$A \times B \times I / R, \,$

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

$(a, b_1, 0) \sim (a, b_2, 0) \quad\mbox{for all } a \in A \mbox{ and } b_1,b_2 \in B,$

$(a_1, b, 1) \sim (a_2, b, 1) \quad\mbox{for all } a_1,a_2 \in A \mbox{ and } b \in B.$

In effect, one is collapsing $A\times B\times \{0\}$ to $A$ and $A\times B\times \{1\}$ to $B$ .

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

[edit] Examples

The join of A and B, regarded as subsets of n-dimensional Euclidean space is homotopy equivalent to the space of paths in n-dimensional Euclidean space, beginning in A and ending in B.
The join of a space X with a one-point space is called the cone $Λ X$ 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 $S X$ of X.
The join of the spheres $S n$ and $S m$ is the sphere $S n + m + 1$ .

[edit] See also

[edit] 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 GFDL.

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Categories: PlanetMath sourced articles | Algebraic topology | Binary operations