Join (topology)
, is a topological space formed by taking the disjoint union of the two spaces, and attaching line segments joining every point in
,that is, the set of all line-segments between a point in
Some authors[2]: 5 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).
along the natural projections of the faces of the cylinder: Usually it is implicitly assumed that
are non-empty, in which case the definition is often phrased a bit differently: instead of attaching the faces of the cylinder
, these faces are simply collapsed in a way suggested by the attachment projections
: we form the quotient space where the equivalence relation
are bounded subsets of the Euclidean space
), then the topological definition reduces to the geometric definition, that is, the "geometric join" is homeomorphic to the "topological join":[3]: 75, Prop.4.2.4
are any abstract simplicial complexes, then their join is an abstract simplicial complex defined as follows:[3]: 74, Def.4.2.1 The combinatorial definition is equivalent to the topological definition in the following sense:[3]: 77, Exercise.3 for every two abstract simplicial complexes
denotes any geometric realization of the complex
is defined based on the representation of each point in the join
(the 0-dimensional sphere, or, the discrete space with two points).
The join of two spaces is commutative up to homeomorphism, i.e.
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
Therefore, one can define the k-times join of a space with itself,
It is possible to define a different join operation
For locally compact Hausdorff spaces
) of their join is at least the sum of connectivities of its parts:[3]: 81, Prop.4.4.3 As an example, let
be a set of two disconnected points.
is a square, which is homeomorphic to a circle that has a 2-dimensional hole, so
The deleted join of an abstract complex A is an abstract complex containing all disjoint unions of disjoint faces of A:[3]: 112
Each simplex in the left-hand-side complex is of the form
Due to the properties of a disjoint union, the latter condition is equivalent to:
Each simplex in the right-hand-side complex is of the form
□ In particular, the deleted join of the n-dimensional simplex
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 n-fold 2-wise deleted join of a discrete space with m points is called the (m,n)-chessboard complex.
