Английская Википедия: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 <math> 0\to H\stackrel{\imath}{\to} X \stackrel{\pi}{\to}G\to 0 </math> where <math>H, X</math> and <math>G</math> are topological groups and <math>i</math> and <math>\pi</math> are continuous homomorphisms which are also open onto their images.[1] Every extension of topological groups is therefore a group extension.
Classification of extensions of topological groups
We say that the topological extensions
- <math>0 \rightarrow H\stackrel{i}{\rightarrow} X\stackrel{\pi}{\rightarrow} G\rightarrow 0</math>
and
- <math>0\to H\stackrel{i'}{\rightarrow} X'\stackrel{\pi'}{\rightarrow} G\rightarrow 0</math>
are equivalent (or congruent) if there exists a topological isomorphism <math>T: X\to X'</math> making commutative the diagram of Figure 1.
We say that the topological extension
- <math>0 \rightarrow H\stackrel{i}{\rightarrow} X\stackrel{\pi}{\rightarrow} G\rightarrow 0</math>
is a split extension (or splits) if it is equivalent to the trivial extension
- <math>0 \rightarrow H\stackrel{i_H}{\rightarrow} H\times G\stackrel{\pi_G}{\rightarrow} G\rightarrow 0</math>
where <math>i_H: H\to H\times G</math> is the natural inclusion over the first factor and <math>\pi_G: H\times G\to G</math> is the natural projection over the second factor.
It is easy to prove that the topological extension <math>0 \rightarrow H\stackrel{i}{\rightarrow} X\stackrel{\pi}{\rightarrow} G\rightarrow 0</math> splits if and only if there is a continuous homomorphism <math>R: X \rightarrow H</math> such that <math>R\circ i</math> is the identity map on <math>H</math>
Note that the topological extension <math>0 \rightarrow H\stackrel{i}{\rightarrow} X\stackrel{\pi}{\rightarrow} G\rightarrow 0</math> splits if and only if the subgroup <math>i(H)</math> is a topological direct summand of <math>X</math>
Examples
- Take <math>\mathbb R </math> the real numbers and <math>\mathbb Z </math> the integer numbers. Take <math>\imath </math> the natural inclusion and <math>\pi </math> the natural projection. Then
- <math> 0\to \mathbb Z\stackrel{\imath}{\to} \mathbb R \stackrel{\pi}{\to}\mathbb R/\mathbb Z\to 0 </math>
- 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 <math> 0\to H\stackrel{\imath}{\to} X \stackrel{\pi}{\to}G\to 0 </math> where <math>H, X</math> and <math>G</math> are locally compact abelian groups and <math>i</math> and <math>\pi</math> are relatively open continuous homomorphisms.[2]
- Let be an extension of locally compact abelian groups
- <math> 0\to H\stackrel{\imath}{\to} X \stackrel{\pi}{\to}G\to 0. </math>
- Take <math>H^\wedge, X^\wedge</math> and <math>G^\wedge</math> the Pontryagin duals of <math>H, X</math> and <math>G</math> and take <math>i^\wedge</math> and <math>\pi^\wedge</math> the dual maps of <math>i</math> and <math>\pi</math>. Then the sequence
- <math> 0\to G^\wedge\stackrel{\pi^\wedge}{\to} X^\wedge \stackrel{\imath^\wedge}{\to}H^\wedge\to 0 </math>
- 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 <math>0 \rightarrow \mathbb T\stackrel{i}{\rightarrow} X\stackrel{\pi}{\rightarrow} G\rightarrow 0</math> where <math>\mathbb T</math> is the unit circle and <math>X</math> and <math>G</math> are topological abelian groups.[3]
The class S(T)
A topological abelian group <math>G</math> belongs to the class <math>\mathcal S (\mathbb T)</math> if and only if every topological extension of the form <math>0 \rightarrow \mathbb T\stackrel{i}{\rightarrow} X\stackrel{\pi}{\rightarrow} G\rightarrow 0</math> splits
- Every locally compact abelian group belongs to <math>\mathcal S (\mathbb T)</math>. In other words every topological extension <math>0 \rightarrow \mathbb T\stackrel{i}{\rightarrow} X\stackrel{\pi}{\rightarrow} G\rightarrow 0</math> where <math>G</math> is a locally compact abelian group, splits.
- Every locally precompact abelian group belongs to <math>\mathcal S (\mathbb T)</math>.
- The Banach space (and in particular topological abelian group) <math>\ell^1</math> does not belong to <math>\mathcal S (\mathbb T)</math>.
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