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
If and are subsets of the Euclidean space , then: [1]: 1
,
that is, the set of all line-segments between a point in and 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). 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.
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": [3]: 75, Prop.4.2.4
If and 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 and , is homeomorphic to , where denotes any geometric realization of the complex .
Given two maps and , their join is defined based on the representation of each point in the join as , for some : [3]: 77
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).
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 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. [4]
If and are homotopy equivalent, then and are homotopy equivalent too. [3]: 77, Exercise.2
Given basepointed CW complexes and , 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 .
Given two triangulable spaces , the homotopical connectivity () 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. 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 .
The deleted join of an abstract complex A is an abstract complex containing all disjoint unions of disjoint faces of A: [3]: 112
The deleted join operation commutes with the join. That is, for every two abstract complexes A and B: [3]: Lem.5.5.2
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 . [3]: Cor.5.5.3
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.
Written in cooperation with Anders Björner and Günter M. Ziegler, Section 4.3
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
If and are subsets of the Euclidean space , then: [1]: 1
,
that is, the set of all line-segments between a point in and 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). 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.
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": [3]: 75, Prop.4.2.4
If and 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 and , is homeomorphic to , where denotes any geometric realization of the complex .
Given two maps and , their join is defined based on the representation of each point in the join as , for some : [3]: 77
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).
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 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. [4]
If and are homotopy equivalent, then and are homotopy equivalent too. [3]: 77, Exercise.2
Given basepointed CW complexes and , 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 .
Given two triangulable spaces , the homotopical connectivity () 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. 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 .
The deleted join of an abstract complex A is an abstract complex containing all disjoint unions of disjoint faces of A: [3]: 112
The deleted join operation commutes with the join. That is, for every two abstract complexes A and B: [3]: Lem.5.5.2
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 . [3]: Cor.5.5.3
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.
Written in cooperation with Anders Björner and Günter M. Ziegler, Section 4.3