Join (topology) explained

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 A ^{\star} : = A\star A . The join is defined in slightly different ways in different contexts

Geometric sets

If

A and B are subsets of the Euclidean space R ^{n} , then:^[1]

A\star B : = \{ t ⋅ a + ( 1- t) ⋅ b ~|~ a\in A, b\in B, t\in [ 0 , 1 ]\} ,

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 R ^{n} and B is in R ^{m} , then A x \{ 0 ^{m} \} x \{ 0 \} and \{ 0 ^{n} \} x B x \{ 1 \} are joinable in R ^{n+m+1} . The figure above shows an example for m=n=1, where A and B are line-segments.

Examples

• 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

If

A and B are any topological spaces, then:A\star B : = A\sqcup _{p _{0}} (A x B x [ 0 , 1 ])\sqcup _{p _{1}} B, where the cylinder A x B x [ 0 , 1 ] is attached to the original spaces A and B along the natural projections of the faces of the cylinder:{A x B x \{ 0 \}} \xrightarrow{p _{0}} A, {A x B x \{ 1 \}} \xrightarrow{p _{1}} B.

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 A x B x [ 0 , 1 ] 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 spaceA\star B : = (A x B x [ 0 , 1 ] )/ \sim, where the equivalence relation \sim is generated by(a, b _{1,} 0 ) \sim (a, b _{2,} 0 ) forall a \in A and b _{1,b} _{2} \in B, (a _{1,} b, 1 ) \sim (a _{2,} b, 1 ) forall a _{1,a} _{2} \in A and b \in B. At the endpoints, this collapses A x B x \{ 0 \} to A and A x B x \{ 1 \} to B .

If

A and B are bounded subsets of the Euclidean space R ^{n} , and A\subseteq U and B \subseteq V , where U, V are disjoint subspaces of R ^{n} such that the dimension of their affine hull is dim U + dim V + 1 (e.g. two non-intersecting non-parallel lines in R ^{3} ), then the topological definition reduces to the geometric definition, that is, the "geometric join" is homeomorphic to the "topological join":

( (A x B x [ 0 , 1 ] )/ \sim ) \simeq \{ t ⋅ a + ( 1- t) ⋅ b ~|~ a\in A, b\in B, t\in [ 0 , 1 ]\}

Abstract simplicial complexes

If

A and B are any abstract simplicial complexes, then their join is an abstract simplicial complex defined as follows:

• The vertex set

V(A\star B) is a disjoint union of V(A) and V( B) .

• The simplices of

A\star B are all disjoint unions of a simplex of A with a simplex of B : A\star B : = \{ a\sqcup b: a\in A, b\in B \} (in the special case in which V(A) and V( B) are disjoint, the join is simply \{ a\cup b: a\in A, b\in B \} ).

Examples

• Suppose

A = \{ \emptyset, \{a\} \} and B = \{\emptyset, \{b\} \} , that is, two sets with a single point. Then A \star B = \{ \emptyset, \{a\}, \{b\}, \{a,b\} \} , 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, A ^{\star} = A \star A = \{ \emptyset, \{a _{1\},} \{a _{2\},} \{a _{1,a} _{2\}} \} where a₁ and a₂ are two copies of the single element in V(A). Topologically, the result is the same as A \star B - a line-segment.

• Suppose

A = \{ \emptyset, \{a\} \} and B = \{\emptyset, \{b\}, \{c\}, \{b,c\} \} . Then A \star B = P(\{a,b,c\}) , which represents a triangle.

• Suppose

A = \{ \emptyset, \{a\}, \{b\} \} and B = \{\emptyset, \{c\}, \{d\} \} , that is, two sets with two discrete points. then A\star B is a complex with facets \{a,c\}, \{b,c\}, \{a,d\}, \{b,d\} , which represents a "square".The combinatorial definition is equivalent to the topological definition in the following sense: for every two abstract simplicial complexes A and B , ||A\star B|| is homeomorphic to ||A||\star ||B|| , where ||X|| denotes any geometric realization of the complex X .

Maps

Given two maps

f:A _{1\to} A _{2} and g:B _{1\to} B _{2} , their join f\star g:A _{1\star} B _{1} \to A _{2\star} B _{2} is defined based on the representation of each point in the join A _{1\star} B _{1} as t ⋅ a + ( 1- t) ⋅ b , for some a\in A _{1,} b\in B _{1} :

f\star g ~(t ⋅ a + ( 1- t) ⋅ b) ~~ = ~~ t ⋅ f(a) + ( 1- t) ⋅ g(b)

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.

A\star B\cong B\star A .

Associativity

A, B, C we have (A\star B)\star C \cong A\star(B\star C). Therefore, one can define the k-times join of a space with itself, A ^{*k} : = A * … * A (k times).

It is possible to define a different join operation

A \hat{\star} B 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 A \hat{\star} B coincide.^[2]

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"(A\star B)/(A \star \{b _{0\}) \cup \{a _{0\}} \star B} is homeomorphic to the reduced suspension

\Sigma(A\wedge B)

of the smash product. Consequently, since {A \star \{b _{0\}} \cup \{a _{0\}} \star B} is contractible, there is a homotopy equivalenceA\star B\simeq \Sigma(A\wedge B). This equivalence establishes the isomorphism \widetilde{H} _{n(A\star} B)\cong H _{n-1} (A\wedge B) l ( = H _{n-1} (A x B / A\vee B) r ) .

Homotopical connectivity

Given two triangulable spaces

A, B , the homotopical connectivity (η _{\pi} ) of their join is at least the sum of connectivities of its parts:^[3]

η _{\pi} (A*B) \geq η _{\pi} (A) +η _{\pi} (B) .

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 η _{\pi} (A) =η _{\pi} (B) =1 . The join A * B is a square, which is homeomorphic to a circle that has a 2-dimensional hole, so η _{\pi} (A * B) =2 . 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 η _{\pi} (S ^{n-1} ) = n .

Deleted join

The deleted join of an abstract complex A is an abstract complex containing all disjoint unions of disjoint faces of A:

A ^{* 2} _{\Delta} : = \{ a _{1\sqcup} a _{2:} a _{1,a} _{2\in} A, a _{1\cap} a _{2} = \emptyset \}

Examples

• Suppose

A = \{ \emptyset, \{a\} \} (a single point). Then A ^{* 2} _{\Delta} : = \{ \emptyset, \{a _{1\},} \{a _{2\}} \} , that is, a discrete space with two disjoint points (recall that A ^{\star} = \{ \emptyset, \{a _{1\},} \{a _{2\},} \{a _{1,a} _{2\}} \} = an interval).

• Suppose

A = \{ \emptyset, \{a\} ,\{b\}\} (two points). Then A ^{* 2} _{\Delta} is a complex with facets \{a _{1,} b _{2\},} \{a _{2,} b _{1\}} (two disjoint edges).

• Suppose

A = \{ \emptyset, \{a\} ,\{b\}, \{a,b\}\} (an edge). Then A ^{* 2} _{\Delta} is a complex with facets \{a _{1,b} _{1\},} \{a _{1,} b _{2\},} \{a _{2,} b _{1\},} \{a _{2,b} _{2\}} (a square). Recall that A ^{\star} represents a solid tetrahedron.

• Suppose A represents an (n-1)-dimensional simplex (with n vertices). Then the join

A ^{\star} is a (2n-1)-dimensional simplex (with 2n vertices): it is the set of all points (x₁,...,x_2n) with non-negative coordinates such that x₁+...+x_2n=1. The deleted join A ^{* 2} _{\Delta} can be regarded as a subset of this simplex: it is the set of all points (x₁,...,x_2n) in that simplex, such that the only nonzero coordinates are some k coordinates in x₁,..,x_n, and the complementary n-k coordinates in x_n+1,...,x_2n.

Properties

The deleted join operation commutes with the join. That is, for every two abstract complexes A and B:

(A*B) ^{* 2} _{\Delta} = (A ^{* 2} _{\Delta} ) * (B ^{* 2} _{\Delta} )

Proof. Each simplex in the left-hand-side complex is of the form (a _{1} \sqcup b _{1)} \sqcup (a _{2\sqcup} b _{2)} , where a _{1,a} _{2\in} A, b _{1,b} _{2\in} B , and (a _{1} \sqcup b _{1),} (a _{2\sqcup} b _{2)} 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

(a _{1} \sqcup a _{2)} \sqcup (b _{1\sqcup} b _{2)} , where a _{1,a} _{2\in} A, b _{1,b} _{2\in} B , 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:

A ^{*n} _{\Delta(k)} : = \{ a _{1\sqcup} a _{2} \sqcup … \sqcup a _{n:} a _{1, … ,a} _{n} arek-wisedisjointfacesof A \} ,

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

• Desuspension

References

• Hatcher, Allen, Algebraic topology. Cambridge University Press, Cambridge, 2002. xii+544 pp. and
  • Brown, Ronald, Topology and Groupoids Section 5.7 Joins.

Notes and References

1. Book: Colin P. Rourke and Brian J. Sanderson . Introduction to Piecewise-Linear Topology . Springer-Verlag . 1982 . New York . en . 10.1007/978-3-642-81735-9. 978-3-540-11102-3 .
2. Book: Fomenko . Anatoly . Homotopical Topology . Fuchs . Dmitry . Springer . 2016 . 2nd . 20.
3. , Section 4.3