Join (topology) explained

In topology, a field of mathematics, the join of two topological spaces

A

and

B

, often denoted by

A\astB

or

A\starB

, 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\starA

. The join is defined in slightly different ways in different contexts

Geometric sets

If

A

and

B

are subsets of the Euclidean space

Rn

, then:[1]

A\starB:= \{ta+(1-t)b~|~a\inA,b\inB,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

Rn

and

B

is in

Rm

, then

A x \{0m\} x \{0\}

and

\{0n\} x B x \{1\}

are joinable in

Rn+m+1

. The figure above shows an example for m=n=1, where

A

and

B

are line-segments.

Examples

Topological spaces

If

A

and

B

are any topological spaces, then:

A\star

B:= A\sqcup
p0

(A x B x

[0,1])\sqcup
p1

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{p0}A,

{A x B x \{1\}}\xrightarrow{p1}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

p1,p2

: we form the quotient space

A\starB:= (A x B x [0,1])/\sim,

where the equivalence relation

\sim

is generated by

(a,b1,0)\sim(a,b2,0)foralla\inAandb1,b2\inB,

(a1,b,1)\sim(a2,b,1)foralla1,a2\inAandb\inB.

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

Rn

, and

A\subseteqU

and

B\subseteqV

, where

U,V

are disjoint subspaces of

Rn

such that the dimension of their affine hull is

dimU+dimV+1

(e.g. two non-intersecting non-parallel lines in

R3

), 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\{ta+(1-t)b~|~a\inA,b\inB,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:

V(A\starB)

is a disjoint union of

V(A)

and

V(B)

.

A\starB

are all disjoint unions of a simplex of

A

with a simplex of

B

:

A\starB:=\{a\sqcupb:a\inA,b\inB\}

(in the special case in which

V(A)

and

V(B)

are disjoint, the join is simply

\{a\cupb:a\inA,b\inB\}

).

Examples

A=\{\emptyset,\{a\}\}

and

B=\{\emptyset,\{b\}\}

, that is, two sets with a single point. Then

A\starB=\{\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\starA=\{\emptyset,\{a1\},\{a2\},\{a1,a2\}\}

where a1 and a2 are two copies of the single element in V(A). Topologically, the result is the same as

A\starB

- a line-segment.

A=\{\emptyset,\{a\}\}

and

B=\{\emptyset,\{b\},\{c\},\{b,c\}\}

. Then

A\starB=P(\{a,b,c\})

, which represents a triangle.

A=\{\emptyset,\{a\},\{b\}\}

and

B=\{\emptyset,\{c\},\{d\}\}

, that is, two sets with two discrete points. then

A\starB

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\starB||

is homeomorphic to

||A||\star||B||

, where

||X||

denotes any geometric realization of the complex

X

.

Maps

Given two maps

f:A1\toA2

and

g:B1\toB2

, their join

f\starg:A1\starB1\toA2\starB2

is defined based on the representation of each point in the join

A1\starB1

as

ta+(1-t)b

, for some

a\inA1,b\inB1

:

f\starg~(ta+(1-t)b)~~=~~tf(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

S0

(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\starB\congB\starA

.

Associativity

A,B,C

we have

(A\starB)\starC\congA\star(B\starC).

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\starB

but a different topology, and this operation is associative for all topological spaces. For locally compact Hausdorff spaces

A

and

B

, the joins

A\starB

and

A\hat{\star}B

coincide.[2]

Homotopy equivalence

If

A

and

A'

are homotopy equivalent, then

A\starB

and

A'\starB

are homotopy equivalent too.

Reduced join

Given basepointed CW complexes

(A,a0)

and

(B,b0)

, the "reduced join"
A\starB
A\star\{b0\

\cup\{a0\}\starB}

is homeomorphic to the reduced suspension

\Sigma(A\wedgeB)

of the smash product. Consequently, since

{A\star\{b0\}\cup\{a0\}\starB}

is contractible, there is a homotopy equivalence

A\starB\simeq\Sigma(A\wedgeB).

This equivalence establishes the isomorphism

\widetilde{H}n(A\starB)\congHn-1(A\wedgeB)l(=Hn-1(A x B/A\veeB)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)\geqη\pi(A)\pi(B)

.

As an example, let

A=B=S0

be a set of two disconnected points. There is a 1-dimensional hole between the points, so

η\pi(A)\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

S0

is a octahedron, which is homeomorphic to

S2

, whose hole is 3-dimensional. In general, the join of n copies of

S0

is homeomorphic to

Sn-1

and

η\pi(Sn-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:

*2
A
\Delta

:=\{a1\sqcupa2:a1,a2\inA,a1\capa2=\emptyset\}

Examples

A=\{\emptyset,\{a\}\}

(a single point). Then
*2
A
\Delta

:=\{\emptyset,\{a1\},\{a2\}\}

, that is, a discrete space with two disjoint points (recall that

A\star=\{\emptyset,\{a1\},\{a2\},\{a1,a2\}\}

= an interval).

A=\{\emptyset,\{a\},\{b\}\}

(two points). Then
*2
A
\Delta

is a complex with facets

\{a1,b2\},\{a2,b1\}

(two disjoint edges).

A=\{\emptyset,\{a\},\{b\},\{a,b\}\}

(an edge). Then
*2
A
\Delta

is a complex with facets

\{a1,b1\},\{a1,b2\},\{a2,b1\},\{a2,b2\}

(a square). Recall that

A\star

represents a solid tetrahedron.

A\star

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
*2
A
\Delta

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

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

*2
(A*B)
\Delta

=

*2
(A
\Delta

)*

*2
(B
\Delta

)

Proof. Each simplex in the left-hand-side complex is of the form

(a1\sqcupb1)\sqcup(a2\sqcupb2)

, where

a1,a2\inA,b1,b2\inB

, and

(a1\sqcupb1),(a2\sqcupb2)

are disjoint. Due to the properties of a disjoint union, the latter condition is equivalent to:

a1,a2

are disjoint and

b1,b2

are disjoint.

Each simplex in the right-hand-side complex is of the form

(a1\sqcupa2)\sqcup(b1\sqcupb2)

, where

a1,a2\inA,b1,b2\inB

, and

a1,a2

are disjoint and

b1,b2

are disjoint. So the sets of simplices on both sides are exactly the same. □

In particular, the deleted join of the n-dimensional simplex

\Deltan

with itself is the n-dimensional crosspolytope, which is homeomorphic to the n-dimensional sphere

Sn

.

Generalization

The n-fold k-wise deleted join of a simplicial complex A is defined as:

*n
A
\Delta(k)

:=\{a1\sqcupa2\sqcup\sqcupan:a1,,anarek-wisedisjointfacesofA\}

,

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

References

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