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 join of two line segments. The original spaces are shown in green and blue. The join is a three-dimensional solid, a disphenoid, in gray.

Geometric sets

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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.

Examples

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  • 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

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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 

 

Abstract simplicial complexes

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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 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

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  • 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:[3]: 77, Exercise.3  for every two abstract simplicial complexes   and  ,   is homeomorphic to  , where   denotes any geometric realization of the complex  .

Maps

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Given two maps   and  , their join   is defined based on the representation of each point in the join   as  , for some  :[3]: 77 

 

Special cases

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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

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Commutativity

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The join of two spaces is commutative up to homeomorphism, i.e.  .

Associativity

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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]

Homotopy equivalence

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If   and   are homotopy equivalent, then   and   are homotopy equivalent too.[3]: 77, Exercise.2 

Reduced join

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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  .

Homotopical connectivity

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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  .

Deleted join

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The deleted join of an abstract complex A is an abstract complex containing all disjoint unions of disjoint faces of A:[3]: 112 

 

Examples

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  • 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

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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 

Generalization

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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

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References

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  1. ^ Colin P. Rourke and Brian J. Sanderson (1982). Introduction to Piecewise-Linear Topology. New York: Springer-Verlag. doi:10.1007/978-3-642-81735-9. ISBN 978-3-540-11102-3.
  2. ^ Bryant, John L. (2001-01-01), Daverman, R. J.; Sher, R. B. (eds.), "Chapter 5 - Piecewise Linear Topology", Handbook of Geometric Topology, Amsterdam: North-Holland, pp. 219–259, ISBN 978-0-444-82432-5, retrieved 2022-11-15
  3. ^ a b c d e f g h i Matoušek, Jiří (2007). Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry (2nd ed.). Berlin-Heidelberg: Springer-Verlag. ISBN 978-3-540-00362-5. Written in cooperation with Anders Björner and Günter M. Ziegler , Section 4.3
  4. ^ Fomenko, Anatoly; Fuchs, Dmitry (2016). Homotopical Topology (2nd ed.). Springer. p. 20.