- Source: Join (topology)
In topology, a field of mathematics, the join of two topological spaces
A
{\displaystyle A}
and
B
{\displaystyle B}
, often denoted by
A
∗
B
{\displaystyle A\ast B}
or
A
⋆
B
{\displaystyle 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
{\displaystyle A}
to every point in
B
{\displaystyle B}
. The join of a space
A
{\displaystyle A}
with itself is denoted by
A
⋆
2
:=
A
⋆
A
{\displaystyle A^{\star 2}:=A\star A}
. The join is defined in slightly different ways in different contexts
Geometric sets
If
A
{\displaystyle A}
and
B
{\displaystyle B}
are subsets of the Euclidean space
R
n
{\displaystyle \mathbb {R} ^{n}}
, then:: 1
A
⋆
B
:=
{
t
⋅
a
+
(
1
−
t
)
⋅
b
|
a
∈
A
,
b
∈
B
,
t
∈
[
0
,
1
]
}
{\displaystyle A\star B\ :=\ \{t\cdot a+(1-t)\cdot b~|~a\in A,b\in B,t\in [0,1]\}}
,that is, the set of all line-segments between a point in
A
{\displaystyle A}
and a point in
B
{\displaystyle B}
.
Some authors: 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
A
{\displaystyle A}
is in
R
n
{\displaystyle \mathbb {R} ^{n}}
and
B
{\displaystyle B}
is in
R
m
{\displaystyle \mathbb {R} ^{m}}
, then
A
×
{
0
m
}
×
{
0
}
{\displaystyle A\times \{0^{m}\}\times \{0\}}
and
{
0
n
}
×
B
×
{
1
}
{\displaystyle \{0^{n}\}\times B\times \{1\}}
are joinable in
R
n
+
m
+
1
{\displaystyle \mathbb {R} ^{n+m+1}}
. The figure above shows an example for m=n=1, where
A
{\displaystyle A}
and
B
{\displaystyle 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
{\displaystyle A}
and
B
{\displaystyle B}
are any topological spaces, then:
A
⋆
B
:=
A
⊔
p
0
(
A
×
B
×
[
0
,
1
]
)
⊔
p
1
B
,
{\displaystyle A\star B\ :=\ A\sqcup _{p_{0}}(A\times B\times [0,1])\sqcup _{p_{1}}B,}
where the cylinder
A
×
B
×
[
0
,
1
]
{\displaystyle A\times B\times [0,1]}
is attached to the original spaces
A
{\displaystyle A}
and
B
{\displaystyle B}
along the natural projections of the faces of the cylinder:
A
×
B
×
{
0
}
→
p
0
A
,
{\displaystyle {A\times B\times \{0\}}\xrightarrow {p_{0}} A,}
A
×
B
×
{
1
}
→
p
1
B
.
{\displaystyle {A\times B\times \{1\}}\xrightarrow {p_{1}} B.}
Usually it is implicitly assumed that
A
{\displaystyle A}
and
B
{\displaystyle B}
are non-empty, in which case the definition is often phrased a bit differently: instead of attaching the faces of the cylinder
A
×
B
×
[
0
,
1
]
{\displaystyle A\times B\times [0,1]}
to the spaces
A
{\displaystyle A}
and
B
{\displaystyle B}
, these faces are simply collapsed in a way suggested by the attachment projections
p
1
,
p
2
{\displaystyle p_{1},p_{2}}
: we form the quotient space
A
⋆
B
:=
(
A
×
B
×
[
0
,
1
]
)
/
∼
,
{\displaystyle A\star B\ :=\ (A\times B\times [0,1])/\sim ,}
where the equivalence relation
∼
{\displaystyle \sim }
is generated by
(
a
,
b
1
,
0
)
∼
(
a
,
b
2
,
0
)
for all
a
∈
A
and
b
1
,
b
2
∈
B
,
{\displaystyle (a,b_{1},0)\sim (a,b_{2},0)\quad {\mbox{for all }}a\in A{\mbox{ and }}b_{1},b_{2}\in B,}
(
a
1
,
b
,
1
)
∼
(
a
2
,
b
,
1
)
for all
a
1
,
a
2
∈
A
and
b
∈
B
.
{\displaystyle (a_{1},b,1)\sim (a_{2},b,1)\quad {\mbox{for all }}a_{1},a_{2}\in A{\mbox{ and }}b\in B.}
At the endpoints, this collapses
A
×
B
×
{
0
}
{\displaystyle A\times B\times \{0\}}
to
A
{\displaystyle A}
and
A
×
B
×
{
1
}
{\displaystyle A\times B\times \{1\}}
to
B
{\displaystyle B}
.
If
A
{\displaystyle A}
and
B
{\displaystyle B}
are bounded subsets of the Euclidean space
R
n
{\displaystyle \mathbb {R} ^{n}}
, and
A
⊆
U
{\displaystyle A\subseteq U}
and
B
⊆
V
{\displaystyle B\subseteq V}
, where
U
,
V
{\displaystyle U,V}
are disjoint subspaces of
R
n
{\displaystyle \mathbb {R} ^{n}}
such that the dimension of their affine hull is
d
i
m
U
+
d
i
m
V
+
1
{\displaystyle dimU+dimV+1}
(e.g. two non-intersecting non-parallel lines in
R
3
{\displaystyle \mathbb {R} ^{3}}
), then the topological definition reduces to the geometric definition, that is, the "geometric join" is homeomorphic to the "topological join":: 75, Prop.4.2.4
(
(
A
×
B
×
[
0
,
1
]
)
/
∼
)
≃
{
t
⋅
a
+
(
1
−
t
)
⋅
b
|
a
∈
A
,
b
∈
B
,
t
∈
[
0
,
1
]
}
{\displaystyle {\big (}(A\times B\times [0,1])/\sim {\big )}\simeq \{t\cdot a+(1-t)\cdot b~|~a\in A,b\in B,t\in [0,1]\}}
Abstract simplicial complexes
If
A
{\displaystyle A}
and
B
{\displaystyle B}
are any abstract simplicial complexes, then their join is an abstract simplicial complex defined as follows:: 74, Def.4.2.1
The vertex set
V
(
A
⋆
B
)
{\displaystyle V(A\star B)}
is a disjoint union of
V
(
A
)
{\displaystyle V(A)}
and
V
(
B
)
{\displaystyle V(B)}
.
The simplices of
A
⋆
B
{\displaystyle A\star B}
are all disjoint unions of a simplex of
A
{\displaystyle A}
with a simplex of
B
{\displaystyle B}
:
A
⋆
B
:=
{
a
⊔
b
:
a
∈
A
,
b
∈
B
}
{\displaystyle A\star B:=\{a\sqcup b:a\in A,b\in B\}}
(in the special case in which
V
(
A
)
{\displaystyle V(A)}
and
V
(
B
)
{\displaystyle V(B)}
are disjoint, the join is simply
{
a
∪
b
:
a
∈
A
,
b
∈
B
}
{\displaystyle \{a\cup b:a\in A,b\in B\}}
).
= Examples
=Suppose
A
=
{
∅
,
{
a
}
}
{\displaystyle A=\{\emptyset ,\{a\}\}}
and
B
=
{
∅
,
{
b
}
}
{\displaystyle B=\{\emptyset ,\{b\}\}}
, that is, two sets with a single point. Then
A
⋆
B
=
{
∅
,
{
a
}
,
{
b
}
,
{
a
,
b
}
}
{\displaystyle 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
⋆
2
=
A
⋆
A
=
{
∅
,
{
a
1
}
,
{
a
2
}
,
{
a
1
,
a
2
}
}
{\displaystyle A^{\star 2}=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
⋆
B
{\displaystyle A\star B}
- a line-segment.
Suppose
A
=
{
∅
,
{
a
}
}
{\displaystyle A=\{\emptyset ,\{a\}\}}
and
B
=
{
∅
,
{
b
}
,
{
c
}
,
{
b
,
c
}
}
{\displaystyle B=\{\emptyset ,\{b\},\{c\},\{b,c\}\}}
. Then
A
⋆
B
=
P
(
{
a
,
b
,
c
}
)
{\displaystyle A\star B=P(\{a,b,c\})}
, which represents a triangle.
Suppose
A
=
{
∅
,
{
a
}
,
{
b
}
}
{\displaystyle A=\{\emptyset ,\{a\},\{b\}\}}
and
B
=
{
∅
,
{
c
}
,
{
d
}
}
{\displaystyle B=\{\emptyset ,\{c\},\{d\}\}}
, that is, two sets with two discrete points. then
A
⋆
B
{\displaystyle A\star B}
is a complex with facets
{
a
,
c
}
,
{
b
,
c
}
,
{
a
,
d
}
,
{
b
,
d
}
{\displaystyle \{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:: 77, Exercise.3 for every two abstract simplicial complexes
A
{\displaystyle A}
and
B
{\displaystyle B}
,
|
|
A
⋆
B
|
|
{\displaystyle ||A\star B||}
is homeomorphic to
|
|
A
|
|
⋆
|
|
B
|
|
{\displaystyle ||A||\star ||B||}
, where
|
|
X
|
|
{\displaystyle ||X||}
denotes any geometric realization of the complex
X
{\displaystyle X}
.
Maps
Given two maps
f
:
A
1
→
A
2
{\displaystyle f:A_{1}\to A_{2}}
and
g
:
B
1
→
B
2
{\displaystyle g:B_{1}\to B_{2}}
, their join
f
⋆
g
:
A
1
⋆
B
1
→
A
2
⋆
B
2
{\displaystyle 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
⋆
B
1
{\displaystyle A_{1}\star B_{1}}
as
t
⋅
a
+
(
1
−
t
)
⋅
b
{\displaystyle t\cdot a+(1-t)\cdot b}
, for some
a
∈
A
1
,
b
∈
B
1
{\displaystyle a\in A_{1},b\in B_{1}}
:: 77
f
⋆
g
(
t
⋅
a
+
(
1
−
t
)
⋅
b
)
=
t
⋅
f
(
a
)
+
(
1
−
t
)
⋅
g
(
b
)
{\displaystyle 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
{\displaystyle X}
, denoted
C
X
{\displaystyle CX}
, is a join of
X
{\displaystyle X}
with a single point.
The suspension of a topological space
X
{\displaystyle X}
, denoted
S
X
{\displaystyle SX}
, is a join of
X
{\displaystyle X}
with
S
0
{\displaystyle 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
⋆
B
≅
B
⋆
A
{\displaystyle 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
{\displaystyle A,B,C}
we have
(
A
⋆
B
)
⋆
C
≅
A
⋆
(
B
⋆
C
)
.
{\displaystyle (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
{\displaystyle A^{*k}:=A*\cdots *A}
(k times).
It is possible to define a different join operation
A
⋆
^
B
{\displaystyle A\;{\hat {\star }}\;B}
which uses the same underlying set as
A
⋆
B
{\displaystyle A\star B}
but a different topology, and this operation is associative for all topological spaces. For locally compact Hausdorff spaces
A
{\displaystyle A}
and
B
{\displaystyle B}
, the joins
A
⋆
B
{\displaystyle A\star B}
and
A
⋆
^
B
{\displaystyle A\;{\hat {\star }}\;B}
coincide.
= Homotopy equivalence
=If
A
{\displaystyle A}
and
A
′
{\displaystyle A'}
are homotopy equivalent, then
A
⋆
B
{\displaystyle A\star B}
and
A
′
⋆
B
{\displaystyle A'\star B}
are homotopy equivalent too.: 77, Exercise.2
= Reduced join
=Given basepointed CW complexes
(
A
,
a
0
)
{\displaystyle (A,a_{0})}
and
(
B
,
b
0
)
{\displaystyle (B,b_{0})}
, the "reduced join"
A
⋆
B
A
⋆
{
b
0
}
∪
{
a
0
}
⋆
B
{\displaystyle {\frac {A\star B}{A\star \{b_{0}\}\cup \{a_{0}\}\star B}}}
is homeomorphic to the reduced suspension
Σ
(
A
∧
B
)
{\displaystyle \Sigma (A\wedge B)}
of the smash product. Consequently, since
A
⋆
{
b
0
}
∪
{
a
0
}
⋆
B
{\displaystyle {A\star \{b_{0}\}\cup \{a_{0}\}\star B}}
is contractible, there is a homotopy equivalence
A
⋆
B
≃
Σ
(
A
∧
B
)
.
{\displaystyle A\star B\simeq \Sigma (A\wedge B).}
This equivalence establishes the isomorphism
H
~
n
(
A
⋆
B
)
≅
H
n
−
1
(
A
∧
B
)
(
=
H
n
−
1
(
A
×
B
/
A
∨
B
)
)
{\displaystyle {\widetilde {H}}_{n}(A\star B)\cong H_{n-1}(A\wedge B)\ {\bigl (}=H_{n-1}(A\times B/A\vee B){\bigr )}}
.
= Homotopical connectivity
=Given two triangulable spaces
A
,
B
{\displaystyle A,B}
, the homotopical connectivity (
η
π
{\displaystyle \eta _{\pi }}
) of their join is at least the sum of connectivities of its parts:: 81, Prop.4.4.3
η
π
(
A
∗
B
)
≥
η
π
(
A
)
+
η
π
(
B
)
{\displaystyle \eta _{\pi }(A*B)\geq \eta _{\pi }(A)+\eta _{\pi }(B)}
.
As an example, let
A
=
B
=
S
0
{\displaystyle A=B=S^{0}}
be a set of two disconnected points. There is a 1-dimensional hole between the points, so
η
π
(
A
)
=
η
π
(
B
)
=
1
{\displaystyle \eta _{\pi }(A)=\eta _{\pi }(B)=1}
. The join
A
∗
B
{\displaystyle A*B}
is a square, which is homeomorphic to a circle that has a 2-dimensional hole, so
η
π
(
A
∗
B
)
=
2
{\displaystyle \eta _{\pi }(A*B)=2}
. The join of this square with a third copy of
S
0
{\displaystyle S^{0}}
is a octahedron, which is homeomorphic to
S
2
{\displaystyle S^{2}}
, whose hole is 3-dimensional. In general, the join of n copies of
S
0
{\displaystyle S^{0}}
is homeomorphic to
S
n
−
1
{\displaystyle S^{n-1}}
and
η
π
(
S
n
−
1
)
=
n
{\displaystyle \eta _{\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:: 112
A
Δ
∗
2
:=
{
a
1
⊔
a
2
:
a
1
,
a
2
∈
A
,
a
1
∩
a
2
=
∅
}
{\displaystyle A_{\Delta }^{*2}:=\{a_{1}\sqcup a_{2}:a_{1},a_{2}\in A,a_{1}\cap a_{2}=\emptyset \}}
= Examples
=Suppose
A
=
{
∅
,
{
a
}
}
{\displaystyle A=\{\emptyset ,\{a\}\}}
(a single point). Then
A
Δ
∗
2
:=
{
∅
,
{
a
1
}
,
{
a
2
}
}
{\displaystyle A_{\Delta }^{*2}:=\{\emptyset ,\{a_{1}\},\{a_{2}\}\}}
, that is, a discrete space with two disjoint points (recall that
A
⋆
2
=
{
∅
,
{
a
1
}
,
{
a
2
}
,
{
a
1
,
a
2
}
}
{\displaystyle A^{\star 2}=\{\emptyset ,\{a_{1}\},\{a_{2}\},\{a_{1},a_{2}\}\}}
= an interval).
Suppose
A
=
{
∅
,
{
a
}
,
{
b
}
}
{\displaystyle A=\{\emptyset ,\{a\},\{b\}\}}
(two points). Then
A
Δ
∗
2
{\displaystyle A_{\Delta }^{*2}}
is a complex with facets
{
a
1
,
b
2
}
,
{
a
2
,
b
1
}
{\displaystyle \{a_{1},b_{2}\},\{a_{2},b_{1}\}}
(two disjoint edges).
Suppose
A
=
{
∅
,
{
a
}
,
{
b
}
,
{
a
,
b
}
}
{\displaystyle A=\{\emptyset ,\{a\},\{b\},\{a,b\}\}}
(an edge). Then
A
Δ
∗
2
{\displaystyle A_{\Delta }^{*2}}
is a complex with facets
{
a
1
,
b
1
}
,
{
a
1
,
b
2
}
,
{
a
2
,
b
1
}
,
{
a
2
,
b
2
}
{\displaystyle \{a_{1},b_{1}\},\{a_{1},b_{2}\},\{a_{2},b_{1}\},\{a_{2},b_{2}\}}
(a square). Recall that
A
⋆
2
{\displaystyle A^{\star 2}}
represents a solid tetrahedron.
Suppose A represents an (n-1)-dimensional simplex (with n vertices). Then the join
A
⋆
2
{\displaystyle A^{\star 2}}
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
A
Δ
∗
2
{\displaystyle A_{\Delta }^{*2}}
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:: Lem.5.5.2
(
A
∗
B
)
Δ
∗
2
=
(
A
Δ
∗
2
)
∗
(
B
Δ
∗
2
)
{\displaystyle (A*B)_{\Delta }^{*2}=(A_{\Delta }^{*2})*(B_{\Delta }^{*2})}
Proof. Each simplex in the left-hand-side complex is of the form
(
a
1
⊔
b
1
)
⊔
(
a
2
⊔
b
2
)
{\displaystyle (a_{1}\sqcup b_{1})\sqcup (a_{2}\sqcup b_{2})}
, where
a
1
,
a
2
∈
A
,
b
1
,
b
2
∈
B
{\displaystyle a_{1},a_{2}\in A,b_{1},b_{2}\in B}
, and
(
a
1
⊔
b
1
)
,
(
a
2
⊔
b
2
)
{\displaystyle (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
{\displaystyle a_{1},a_{2}}
are disjoint and
b
1
,
b
2
{\displaystyle b_{1},b_{2}}
are disjoint.
Each simplex in the right-hand-side complex is of the form
(
a
1
⊔
a
2
)
⊔
(
b
1
⊔
b
2
)
{\displaystyle (a_{1}\sqcup a_{2})\sqcup (b_{1}\sqcup b_{2})}
, where
a
1
,
a
2
∈
A
,
b
1
,
b
2
∈
B
{\displaystyle a_{1},a_{2}\in A,b_{1},b_{2}\in B}
, and
a
1
,
a
2
{\displaystyle a_{1},a_{2}}
are disjoint and
b
1
,
b
2
{\displaystyle 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
Δ
n
{\displaystyle \Delta ^{n}}
with itself is the n-dimensional crosspolytope, which is homeomorphic to the n-dimensional sphere
S
n
{\displaystyle S^{n}}
.: Cor.5.5.3
= Generalization
=The n-fold k-wise deleted join of a simplicial complex A is defined as:
A
Δ
(
k
)
∗
n
:=
{
a
1
⊔
a
2
⊔
⋯
⊔
a
n
:
a
1
,
⋯
,
a
n
are k-wise disjoint faces of
A
}
{\displaystyle A_{\Delta (k)}^{*n}:=\{a_{1}\sqcup a_{2}\sqcup \cdots \sqcup a_{n}:a_{1},\cdots ,a_{n}{\text{ are k-wise disjoint faces of }}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. ISBN 0-521-79160-X and ISBN 0-521-79540-0
This article incorporates material from Join on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
Brown, Ronald, Topology and Groupoids Section 5.7 Joins.
Kata Kunci Pencarian:
- Subgrup normal
- Join (topology)
- Join
- Comparison of topologies
- Suspension (topology)
- Topology
- Cone (topology)
- Pointless topology
- Network topology
- Order topology
- Algebraic topology
