Join (topology)
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In topology, a field of mathematics, the join of two topological spaces and , often denoted by or , is a topological space defined as
where the cylinder is attached to the original spaces and along the natural projections of the faces of the cylinder:
Intuitively, is formed by taking the disjoint union of the two spaces and attaching line segments joining every point in to every point in .
Note that usually it is implicitly assumed that and are non-empty, in which case the definition is often phrased a bit differently: instead of attaching the faces of the cylinder to the spaces and , these faces are simply collapsed in a way suggested by the attachment projections :
where is the equivalence relation generated by
At the endpoints, this collapses to and to .
Examples
- 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 (the 0-dimensional sphere, or, the discrete space with two points) is called the suspension of X.
- The join of the spheres and is the sphere .
- 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 . 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
- 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:
- Given basepointed CW complexes (A,a0) and (B,b0), the "reduced join"
is homeomorphic to the reduced suspension
of the smash product. Consequently, since is contractible, there is a homotopy equivalence
See also
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.