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]\displaystyle{ 0\to H\stackrel{\imath}{\to} X \stackrel{\pi}{\to}G\to 0 }[/math] where [math]\displaystyle{ H, X }[/math] and [math]\displaystyle{ G }[/math] are topological groups and [math]\displaystyle{ i }[/math] and [math]\displaystyle{ \pi }[/math] 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

[math]\displaystyle{ 0 \rightarrow H\stackrel{i}{\rightarrow} X\stackrel{\pi}{\rightarrow} G\rightarrow 0 }[/math]

and

[math]\displaystyle{ 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]\displaystyle{ T: X\to X' }[/math] making commutative the diagram of Figure 1.

Figure 1

We say that the topological extension

[math]\displaystyle{ 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]\displaystyle{ 0 \rightarrow H\stackrel{i_H}{\rightarrow} H\times G\stackrel{\pi_G}{\rightarrow} G\rightarrow 0 }[/math]

where [math]\displaystyle{ i_H: H\to H\times G }[/math] is the natural inclusion over the first factor and [math]\displaystyle{ \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]\displaystyle{ 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]\displaystyle{ R: X \rightarrow H }[/math] such that [math]\displaystyle{ R\circ i }[/math] is the identity map on [math]\displaystyle{ H }[/math]

Note that the topological extension [math]\displaystyle{ 0 \rightarrow H\stackrel{i}{\rightarrow} X\stackrel{\pi}{\rightarrow} G\rightarrow 0 }[/math] splits if and only if the subgroup [math]\displaystyle{ i(H) }[/math] is a topological direct summand of [math]\displaystyle{ X }[/math]

Examples

• Take [math]\displaystyle{ \mathbb R }[/math] the real numbers and [math]\displaystyle{ \mathbb Z }[/math] the integer numbers. Take [math]\displaystyle{ \imath }[/math] the natural inclusion and [math]\displaystyle{ \pi }[/math] the natural projection. Then

[math]\displaystyle{ 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]\displaystyle{ 0\to H\stackrel{\imath}{\to} X \stackrel{\pi}{\to}G\to 0 }[/math] where [math]\displaystyle{ H, X }[/math] and [math]\displaystyle{ G }[/math] are locally compact abelian groups and [math]\displaystyle{ i }[/math] and [math]\displaystyle{ \pi }[/math] are relatively open continuous homomorphisms.

• Let be an extension of locally compact abelian groups

[math]\displaystyle{ 0\to H\stackrel{\imath}{\to} X \stackrel{\pi}{\to}G\to 0. }[/math]

Take [math]\displaystyle{ H^\wedge, X^\wedge }[/math] and [math]\displaystyle{ G^\wedge }[/math] the Pontryagin duals of [math]\displaystyle{ H, X }[/math] and [math]\displaystyle{ G }[/math] and take [math]\displaystyle{ i^\wedge }[/math] and [math]\displaystyle{ \pi^\wedge }[/math] the dual maps of [math]\displaystyle{ i }[/math] and [math]\displaystyle{ \pi }[/math]. Then the sequence

[math]\displaystyle{ 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]\displaystyle{ 0 \rightarrow \mathbb T\stackrel{i}{\rightarrow} X\stackrel{\pi}{\rightarrow} G\rightarrow 0 }[/math] where [math]\displaystyle{ \mathbb T }[/math] is the unit circle and [math]\displaystyle{ X }[/math] and [math]\displaystyle{ G }[/math] are topological abelian groups.

The class S(T)

A topological abelian group [math]\displaystyle{ G }[/math] belongs to the class [math]\displaystyle{ \mathcal S (\mathbb T) }[/math] if and only if every topological extension of the form [math]\displaystyle{ 0 \rightarrow \mathbb T\stackrel{i}{\rightarrow} X\stackrel{\pi}{\rightarrow} G\rightarrow 0 }[/math] splits

• Every locally compact abelian group belongs to [math]\displaystyle{ \mathcal S (\mathbb T) }[/math]. In other words every topological extension [math]\displaystyle{ 0 \rightarrow \mathbb T\stackrel{i}{\rightarrow} X\stackrel{\pi}{\rightarrow} G\rightarrow 0 }[/math] where [math]\displaystyle{ G }[/math] is a locally compact abelian group, splits.

• Every locally precompact abelian group belongs to [math]\displaystyle{ \mathcal S (\mathbb T) }[/math].

• The Banach space (and in particular topological abelian group) [math]\displaystyle{ \ell^1 }[/math] does not belong to [math]\displaystyle{ \mathcal S (\mathbb T) }[/math].