Join (topology)
In topology, a field of mathematics, the join of two topological spaces  and , often denoted by  or , is a topological space formed by taking the disjoint union of the two spaces, and attaching line segments joining every point in  to every point in . The join of a space  with itself is denoted by . The join is defined in slightly different ways in different contexts

Geometric sets 
If  and  are subsets of the Euclidean space , then:,that is, the set of all line-segments between a point in  and a point in . Some authors restrict the definition to subsets that are joinable: any two different line-segments, connecting a point of A to a point of B, meet in at most a common endpoint (that is, they do not intersect in their interior). Every two subsets can be made "joinable". For example, if  is in  and  is in , then  and  are joinable in . The figure above shows an example for m=n=1, where   and  are line-segments. Examples 
 The join of two simplices is a simplex: the join of an n-dimensional and an m-dimensional simplex is an (m+n+1)-dimensional simplex. Some special cases are:
 The join of two disjoint points is an interval (m=n=0). The join of a point and an interval is a triangle (m=0, n=1). The join of two line segments is homeomorphic to a solid tetrahedron or disphenoid, illustrated in the figure above right (m=n=1). The join of a point and an (n-1)-dimensional simplex is an n-dimensional simplex. The join of a point and a polygon (or any polytope) is a pyramid, like the join of a point and square is a square pyramid. The join of a point and a cube is a cubic pyramid. The join of a point and a circle is a cone, and the join of a point and a sphere is a hypercone. Topological spaces 
If  and  are any topological spaces, then:

where the cylinder  is attached to the original spaces  and  along the natural projections of the faces of the cylinder:

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 : we form the quotient space

where the equivalence relation  is generated by

At the endpoints, this collapses  to  and  to . If  and  are bounded subsets of the Euclidean space , and  and , where  are disjoint subspaces of  such that the dimension of their affine hull is  (e.g. two non-intersecting non-parallel lines in ), then the topological definition reduces to the geometric definition, that is, the "geometric join" is homeomorphic to the "topological join":

Abstract simplicial complexes 
If  and  are any abstract simplicial complexes, then their join is an abstract simplicial complex defined as follows:

 The vertex set  is a disjoint union of  and . The simplices of  are all disjoint unions of a simplex of  with a simplex of :  (in the special case in which   and  are disjoint, the join is simply ). Examples 
 Suppose  and , that is, two sets with a single point. Then , which represents a line-segment. Note that the vertex sets of A and B are disjoint; otherwise, we should have made them disjoint. For example,  where a1 and a2 are two copies of the single element in V(A). Topologically, the result is the same as  - a line-segment. Suppose  and . Then , which represents a triangle. Suppose  and , that is, two sets with two discrete points. then  is a complex with facets , which represents a "square". The combinatorial definition is equivalent to the topological definition in the following sense: for every two abstract simplicial complexes  and ,  is homeomorphic to , where  denotes any geometric realization of the complex . Maps 
Given two maps  and , their join is defined based on the representation of each point in the join  as , for some :

Special cases 
The cone of a topological space , denoted  , is a join of  with a single point. The suspension of a topological space , denoted  , is a join of  with  (the 0-dimensional sphere, or, the discrete space with two points). Properties

Commutativity 
The join of two spaces is commutative up to homeomorphism, i.e. . Associativity 
It is not true that the join operation defined above is associative up to homeomorphism for arbitrary topological spaces. However, for locally compact Hausdorff spaces  we have  Therefore, one can define the k-times join of a space with itself,  (k times). It is possible to define a different join operation  which uses the same underlying set as  but a different topology, and this operation is associative for all topological spaces. For locally compact Hausdorff spaces  and , the joins  and  coincide. Homotopy equivalence 
If  and  are homotopy equivalent, then  and  are homotopy equivalent too. Reduced join 
Given basepointed CW complexes  and , the "reduced join"

is homeomorphic to the reduced suspensionof the smash product. Consequently, since  is contractible, there is a homotopy equivalence

This equivalence establishes the isomorphism . Homotopical connectivity 
Given two triangulable spaces , the homotopical connectivity () of their join is at least the sum of connectivities of its parts:

 . As an example, let  be a set of two disconnected points. There is a 1-dimensional hole between the points, so . The join  is a square, which is homeomorphic to a circle that has a 2-dimensional hole, so . The join of this square with a third copy of  is a octahedron, which is homeomorphic to  , whose hole is 3-dimensional. In general, the join of n copies of  is homeomorphic to   and . Deleted join 
The deleted join of an abstract complex A is an abstract complex containing all disjoint unions of disjoint faces of A:

Examples 

 Suppose  (a single point). Then , that is, a discrete space with two disjoint points (recall that  = an interval). Suppose  (two points). Then  is a complex with facets  (two disjoint edges). Suppose  (an edge). Then  is a complex with facets  (a square). Recall that  represents a solid tetrahedron. Suppose A represents an (n-1)-dimensional simplex (with n vertices). Then the join   is a (2n-1)-dimensional simplex (with 2n vertices): it is the set of all points (x1,...,x2n) with non-negative coordinates such that x1+...+x2n=1. The deleted join  can be regarded as a subset of this simplex: it is the set of all points (x1,...,x2n) in that simplex, such that the only nonzero coordinates are some k coordinates in x1,..,xn, and the complementary n-k coordinates in xn+1,...,x2n. Properties 
The deleted join operation commutes with the join. That is, for every two abstract complexes A and B: Proof. Each simplex in the left-hand-side complex is of the form  , where , and  are disjoint. Due to the properties of a disjoint union, the latter condition is equivalent to:  are disjoint and  are disjoint. Each simplex in the right-hand-side complex is of the form  , where , and   are disjoint and  are disjoint. So the sets of simplices on both sides are exactly the same. □

In particular, the deleted join of the n-dimensional simplex  with itself is the n-dimensional crosspolytope, which is homeomorphic to the n-dimensional sphere . Generalization 
The n-fold k-wise deleted join of a simplicial complex A is defined as:,

where "k-wise disjoint" means that every subset of k have an empty intersection.In particular, the n-fold n-wise deleted join contains all disjoint unions of n faces whose intersection is empty, and the n-fold 2-wise deleted join is smaller: it contains only the disjoint unions of n faces that are pairwise-disjoint. The 2-fold 2-wise deleted join is just the simple deleted join defined above. The n-fold 2-wise deleted join of a discrete space with m points is called the (m,n)-chessboard complex. See also

Desuspension

References

Hatcher, Allen, Algebraic topology. Cambridge University Press, Cambridge, 2002. xii+544 pp. and 

 Brown, Ronald, Topology and Groupoids Section 5.7 Joins. Algebraic topology
Operations on structures