Join (topology)

In topology, a field of mathematics, the join of two topological spaces "A" and "B", often denoted by $Astar\; B$, is defined to be the quotient space:$A\; imes\; B\; imes\; I\; /\; R,\; ,$where "I" is the interval [0, 1] and "R" is the relation defined by:$(a,\; b\_1,\; 0)\; sim\; (a,\; b\_2,\; 0)\; quadmbox\{for\; all\; \}\; a\; in\; A\; mbox\{\; and\; \}\; b\_1,b\_2\; in\; B,$:$(a\_1,\; b,\; 1)\; sim\; (a\_2,\; b,\; 1)\; quadmbox\{for\; all\; \}\; a\_1,a\_2\; in\; A\; mbox\{\; and\; \}\; b\; in\; B.$In effect, one is collapsing $A\; imes\; B\; imes\; \{0\}$ to $A$ and $A\; imes\; B\; imes\; \{1\}$ to $B$.

Intuitively, $Astar\; B$ 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".

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 $Lambda\; X$ 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 $SX$ of "X".
* The join of the spheres $S^n$ and $S^m$ is the sphere $S^\{n+m+1\}$.

ee also

*Cone (topology)
*Suspension (topology)

References

*Hatcher, Allen, [http://www.math.cornell.edu/~hatcher/AT/ATpage.html "Algebraic topology."] Cambridge University Press, Cambridge, 2002. xii+544 pp. ISBN 0-521-79160-X and ISBN 0-521-79540-0
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