- Source: Extension of a topological group
In mathematics, more specifically in topological groups, an extension of topological groups, or a topological extension, is a short exact sequence
0
→
H
→
ı
X
→
π
G
→
0
{\displaystyle 0\to H{\stackrel {\imath }{\to }}X{\stackrel {\pi }{\to }}G\to 0}
where
H
,
X
{\displaystyle H,X}
and
G
{\displaystyle G}
are topological groups and
i
{\displaystyle i}
and
π
{\displaystyle \pi }
are continuous homomorphisms which are also open onto their images. Every extension of topological groups is therefore a group extension.
Classification of extensions of topological groups
We say that the topological extensions
0
→
H
→
i
X
→
π
G
→
0
{\displaystyle 0\rightarrow H{\stackrel {i}{\rightarrow }}X{\stackrel {\pi }{\rightarrow }}G\rightarrow 0}
and
0
→
H
→
i
′
X
′
→
π
′
G
→
0
{\displaystyle 0\to H{\stackrel {i'}{\rightarrow }}X'{\stackrel {\pi '}{\rightarrow }}G\rightarrow 0}
are equivalent (or congruent) if there exists a topological isomorphism
T
:
X
→
X
′
{\displaystyle T:X\to X'}
making commutative the diagram of Figure 1.
We say that the topological extension
0
→
H
→
i
X
→
π
G
→
0
{\displaystyle 0\rightarrow H{\stackrel {i}{\rightarrow }}X{\stackrel {\pi }{\rightarrow }}G\rightarrow 0}
is a split extension (or splits) if it is equivalent to the trivial extension
0
→
H
→
i
H
H
×
G
→
π
G
G
→
0
{\displaystyle 0\rightarrow H{\stackrel {i_{H}}{\rightarrow }}H\times G{\stackrel {\pi _{G}}{\rightarrow }}G\rightarrow 0}
where
i
H
:
H
→
H
×
G
{\displaystyle i_{H}:H\to H\times G}
is the natural inclusion over the first factor and
π
G
:
H
×
G
→
G
{\displaystyle \pi _{G}:H\times G\to G}
is the natural projection over the second factor.
It is easy to prove that the topological extension
0
→
H
→
i
X
→
π
G
→
0
{\displaystyle 0\rightarrow H{\stackrel {i}{\rightarrow }}X{\stackrel {\pi }{\rightarrow }}G\rightarrow 0}
splits if and only if there is a continuous homomorphism
R
:
X
→
H
{\displaystyle R:X\rightarrow H}
such that
R
∘
i
{\displaystyle R\circ i}
is the identity map on
H
{\displaystyle H}
Note that the topological extension
0
→
H
→
i
X
→
π
G
→
0
{\displaystyle 0\rightarrow H{\stackrel {i}{\rightarrow }}X{\stackrel {\pi }{\rightarrow }}G\rightarrow 0}
splits if and only if the subgroup
i
(
H
)
{\displaystyle i(H)}
is a topological direct summand of
X
{\displaystyle X}
Examples
Take
R
{\displaystyle \mathbb {R} }
the real numbers and
Z
{\displaystyle \mathbb {Z} }
the integer numbers. Take
ı
{\displaystyle \imath }
the natural inclusion and
π
{\displaystyle \pi }
the natural projection. Then
0
→
Z
→
ı
R
→
π
R
/
Z
→
0
{\displaystyle 0\to \mathbb {Z} {\stackrel {\imath }{\to }}\mathbb {R} {\stackrel {\pi }{\to }}\mathbb {R} /\mathbb {Z} \to 0}
is an extension of topological abelian groups. Indeed it is an example of a non-splitting extension.
Extensions of locally compact abelian groups (LCA)
An extension of topological abelian groups will be a short exact sequence
0
→
H
→
ı
X
→
π
G
→
0
{\displaystyle 0\to H{\stackrel {\imath }{\to }}X{\stackrel {\pi }{\to }}G\to 0}
where
H
,
X
{\displaystyle H,X}
and
G
{\displaystyle G}
are locally compact abelian groups and
i
{\displaystyle i}
and
π
{\displaystyle \pi }
are relatively open continuous homomorphisms.
Let be an extension of locally compact abelian groups
0
→
H
→
ı
X
→
π
G
→
0.
{\displaystyle 0\to H{\stackrel {\imath }{\to }}X{\stackrel {\pi }{\to }}G\to 0.}
Take
H
∧
,
X
∧
{\displaystyle H^{\wedge },X^{\wedge }}
and
G
∧
{\displaystyle G^{\wedge }}
the Pontryagin duals of
H
,
X
{\displaystyle H,X}
and
G
{\displaystyle G}
and take
i
∧
{\displaystyle i^{\wedge }}
and
π
∧
{\displaystyle \pi ^{\wedge }}
the dual maps of
i
{\displaystyle i}
and
π
{\displaystyle \pi }
. Then the sequence
0
→
G
∧
→
π
∧
X
∧
→
ı
∧
H
∧
→
0
{\displaystyle 0\to G^{\wedge }{\stackrel {\pi ^{\wedge }}{\to }}X^{\wedge }{\stackrel {\imath ^{\wedge }}{\to }}H^{\wedge }\to 0}
is an extension of locally compact abelian groups.
Extensions of topological abelian groups by the unit circle
A very special kind of topological extensions are the ones of the form
0
→
T
→
i
X
→
π
G
→
0
{\displaystyle 0\rightarrow \mathbb {T} {\stackrel {i}{\rightarrow }}X{\stackrel {\pi }{\rightarrow }}G\rightarrow 0}
where
T
{\displaystyle \mathbb {T} }
is the unit circle and
X
{\displaystyle X}
and
G
{\displaystyle G}
are topological abelian groups.
= The class S(T)
=A topological abelian group
G
{\displaystyle G}
belongs to the class
S
(
T
)
{\displaystyle {\mathcal {S}}(\mathbb {T} )}
if and only if every topological extension of the form
0
→
T
→
i
X
→
π
G
→
0
{\displaystyle 0\rightarrow \mathbb {T} {\stackrel {i}{\rightarrow }}X{\stackrel {\pi }{\rightarrow }}G\rightarrow 0}
splits
Every locally compact abelian group belongs to
S
(
T
)
{\displaystyle {\mathcal {S}}(\mathbb {T} )}
. In other words every topological extension
0
→
T
→
i
X
→
π
G
→
0
{\displaystyle 0\rightarrow \mathbb {T} {\stackrel {i}{\rightarrow }}X{\stackrel {\pi }{\rightarrow }}G\rightarrow 0}
where
G
{\displaystyle G}
is a locally compact abelian group, splits.
Every locally precompact abelian group belongs to
S
(
T
)
{\displaystyle {\mathcal {S}}(\mathbb {T} )}
.
The Banach space (and in particular topological abelian group)
ℓ
1
{\displaystyle \ell ^{1}}
does not belong to
S
(
T
)
{\displaystyle {\mathcal {S}}(\mathbb {T} )}
.
References
Kata Kunci Pencarian:
- Ekstensi grup
- Graph database
- Massachusetts Bay Transportation Authority
- Particle swarm optimization
- Daftar masalah matematika yang belum terpecahkan
- Extension of a topological group
- Group extension
- Homeomorphism group
- Alexandroff extension
- Profinite group
- Compact group
- Covering group
- Topological defect
- Direct sum of topological groups
- Galois group
