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

Geometric sets

If A and B are subsets of the Euclidean space, then: ","that is, the set of all line-segments between a point in A and a point in B. 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 A is in and B is in, then and are joinable in. The figure above shows an example for m=n=1, where A and B are line-segments.

Examples

Topological spaces

If A and B are any topological spaces, then: where the cylinder is attached to the original spaces A and B along the natural projections of the faces of the cylinder: Usually it is implicitly assumed that A and B 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 A and B, these faces are simply collapsed in a way suggested by the attachment projections p_1,p_2: we form the quotient space where the equivalence relation \sim is generated by At the endpoints, this collapses to A and to B. If A and B are bounded subsets of the Euclidean space, and and , where U, V 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 A and B are any abstract simplicial complexes, then their join is an abstract simplicial complex defined as follows:

Examples

The combinatorial definition is equivalent to the topological definition in the following sense: for every two abstract simplicial complexes A and B, is homeomorphic to, where ||X|| denotes any geometric realization of the complex X.

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 X, denoted CX, is a join of X with a single point. The suspension of a topological space X, denoted SX, is a join of X with S^0 (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 A, B, C 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 A\star B but a different topology, and this operation is associative for all topological spaces. For locally compact Hausdorff spaces A and B, the joins A\star B and coincide.

Homotopy equivalence

If A and A' are homotopy equivalent, then A\star B and A'\star B are homotopy equivalent too.

Reduced join

Given basepointed CW complexes (A, a_0) and (B, b_0), the "reduced join" is homeomorphic to the reduced suspension""of the smash product. Consequently, since is contractible, there is a homotopy equivalence This equivalence establishes the isomorphism.

Homotopical connectivity

Given two triangulable spaces A, B, the homotopical connectivity (\eta_{\pi}) of their join is at least the sum of connectivities of its parts: As an example, let A = B = S^0 be a set of two disconnected points. There is a 1-dimensional hole between the points, so. The join A * B 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 S^0 is a octahedron, which is homeomorphic to **S^2 **, whose hole is 3-dimensional. In general, the join of n copies of S^0 is homeomorphic to **S^{n-1} ** 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

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: a_1,a_2 are disjoint and b_1,b_2 are disjoint. Each simplex in the right-hand-side complex is of the form, where , and a_1,a_2 are disjoint and b_1,b_2 are disjoint. So the sets of simplices on both sides are exactly the same. □ In particular, the deleted join of the n-dimensional simplex \Delta^n with itself is the n-dimensional crosspolytope, which is homeomorphic to the n-dimensional sphere S^n.

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.