Join (topology)

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

Geometric sets

If $A$ and $B$ are subsets of the Euclidean space $\mathbb^n$, then:

$A\star B\ :=\ \$,

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 $\mathbb^n$ and $B$ is in $\mathbb^m$, then $A\times\\times\$ and $\\times B\times\$ are joinable in $\mathbb^$. 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_(A\times B \times ,1\sqcup_B,
where the cylinder A\times B \times ,1/math> is attached to the original spaces $A$ and $B$ along the natural projections of the faces of the cylinder:
:$\xrightarrow A,$
:$\xrightarrow B.$

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\times B \times ,1/math> 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 space
: A\star B\ :=\ (A\times B \times ,1)/ \sim,
where the equivalence relation $\sim$ is generated by
:$(a, b_1, 0) \sim (a, b_2, 0) \quad\mbox a \in A \mbox b_1,b_2 \in B,$

:$(a_1, b, 1) \sim (a_2, b, 1) \quad\mbox a_1,a_2 \in A \mbox b \in B.$
At the endpoints, this collapses $A\times B\times \$ to $A$ and $A\times B\times \$ to $B$.

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

\big((A\times B \times ,1)/ \sim\big) \simeq \

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 := \$ (in the special case in which $V(A)$ and $V( B)$ are disjoint, the join is simply $\$).

Examples

* Suppose $A = \$ and $B = \$, that is, two sets with a single point. Then $A \star 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^ = A \star A = \$ 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 = \$ and $B = \$. Then $A \star B = P(\)$, which represents a triangle.
* Suppose $A = \$ and $B = \$, that is, two sets with two discrete points. then $A\star B$ is a complex with facets $\, \, \, \$, 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 $t\cdot a +(1-t)\cdot b$, for some $a\in A_1, b\in B_1$:''''

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

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

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 $A, B, C$ 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, $A^ := A * \cdots * A$ (''k'' times).

It is possible to define a different join operation $A\; \hat\;B$ 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 $A \;\hat\;B$ coincide.

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"
::$\frac$
is homeomorphic to the reduced suspension

$\Sigma(A\wedge B)$

of the smash product. Consequently, since $$ is contractible, there is a homotopy equivalence
:$A\star B\simeq \Sigma(A\wedge B).$
This equivalence establishes the isomorphism $\widetilde_n(A\star B)\cong H_(A\wedge B)\ \bigl( =H_(A\times B / A\vee B)\bigr)$.

Homotopical connectivity

Given two triangulable spaces $A, B$, the homotopical connectivity ($\eta_$) of their join is at least the sum of connectivities of its parts:'', Section 4.3''

* $\eta_(A*B) \geq \eta_(A)+\eta_(B)$.

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 $\eta_(A)=\eta_(B)=1$. The join $A * B$ is a square, which is homeomorphic to a circle that has a 2-dimensional hole, so $\eta_(A * B)=2$. 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 $S^$ and $\eta_(S^)=n$.

Deleted join

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

$A^_ := \$

Examples

* Suppose $A = \$ (a single point). Then $A^_ := \$, that is, a discrete space with two disjoint points (recall that $A^ =\$ = an interval).
* Suppose $A = \$ (two points). Then $A^_$ is a complex with facets $\, \$ (two disjoint edges).
* Suppose $A = \$ (an edge). Then $A^_$ is a complex with facets $\, \, \, \$ (a square). Recall that $A^$ represents a solid tetrahedron.
* Suppose ''A'' represents an (''n''-1)-dimensional simplex (with ''n'' vertices). Then the join $A^$ is a (''2n-''1)-dimensional simplex (with 2''n'' vertices): it is the set of all points (x₁,...,x_2n) with non-negative coordinates such that x₁+...+x_2n=1. The deleted join $A^_$ 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'':''''

$(A*B)^_ = (A^_) * (B^_)$

''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^_ := \$,

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
*{{PlanetMath attribution, id=3985, title=Join
* Brown, Ronald
''Topology and Groupoids''
Section 5.7 Joins.

Algebraic topology
Operations on structures