Join (topology)

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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.

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[edit]

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[edit]

  • 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[edit]

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[edit]

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[edit]

  • 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[edit]

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[edit]

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[edit]

Commutativity[edit]

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

Associativity[edit]

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[edit]

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

Reduced join[edit]

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[edit]

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[edit]

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

Examples[edit]

  • 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[edit]

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[edit]

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[edit]

References[edit]

  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.