# 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 $0\to H{\stackrel {\imath }{\to }}X{\stackrel {\pi }{\to }}G\to 0$ where $H,X$ and $G$ are topological groups and $i$ and $\pi$ 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
    $0\rightarrow H{\stackrel {i}{\rightarrow }}X{\stackrel {\pi }{\rightarrow }}G\rightarrow 0$
and
    $0\to H{\stackrel {i'}{\rightarrow }}X'{\stackrel {\pi '}{\rightarrow }}G\rightarrow 0$
are equivalent (or congruent) if there exists a topological isomorphism $T:X\to X'$ making commutative the diagram of Figure 1.
We say that the topological extension
    $0\rightarrow H{\stackrel {i}{\rightarrow }}X{\stackrel {\pi }{\rightarrow }}G\rightarrow 0$
is a split extension (or splits) if it is equivalent to the trivial extension
    $0\rightarrow H{\stackrel {i_{H}}{\rightarrow }}H\times G{\stackrel {\pi _{G}}{\rightarrow }}G\rightarrow 0$
where $i_{H}:H\to H\times G$ is the natural inclusion over the first factor and $\pi _{G}:H\times G\to G$ is the natural projection over the second factor.
It is easy to prove that the topological extension $0\rightarrow H{\stackrel {i}{\rightarrow }}X{\stackrel {\pi }{\rightarrow }}G\rightarrow 0$ splits if and only if there is a continuous homomorphism $R:X\rightarrow H$ such that $R\circ i$ is the identity map on $H$
Note that the topological extension $0\rightarrow H{\stackrel {i}{\rightarrow }}X{\stackrel {\pi }{\rightarrow }}G\rightarrow 0$ splits if and only if the subgroup $i(H)$ is a topological direct summand of $X$

## Examples

* Take $\mathbb {R}$ the real numbers and $\mathbb {Z}$ the integer numbers. Take $\imath$ the natural inclusion and $\pi$ the natural projection. Then
       $0\to \mathbb {Z} {\stackrel {\imath }{\to }}\mathbb {R} {\stackrel {\pi }{\to }}\mathbb {R} /\mathbb {Z} \to 0$
    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 $0\to H{\stackrel {\imath }{\to }}X{\stackrel {\pi }{\to }}G\to 0$ where $H,X$ and $G$ are locally compact abelian groups and $i$ and $\pi$ are relatively open continuous homomorphisms.
* Let be an extension of locally compact abelian groups
       $0\to H{\stackrel {\imath }{\to }}X{\stackrel {\pi }{\to }}G\to 0.$
    Take $H^{\wedge },X^{\wedge }$ and $G^{\wedge }$ the Pontryagin duals of $H,X$ and $G$ and take $i^{\wedge }$ and $\pi ^{\wedge }$ the dual maps of $i$ and $\pi$. Then the sequence
       $0\to G^{\wedge }{\stackrel {\pi ^{\wedge }}{\to }}X^{\wedge }{\stackrel {\imath ^{\wedge }}{\to }}H^{\wedge }\to 0$
    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 $0\rightarrow \mathbb {T} {\stackrel {i}{\rightarrow }}X{\stackrel {\pi }{\rightarrow }}G\rightarrow 0$ where $\mathbb {T}$ is the unit circle and $X$ and $G$ are topological abelian groups.

### The class S(T)

A topological abelian group $G$ belongs to the class ${\mathcal {S}}(\mathbb {T} )$ if and only if every topological extension of the form $0\rightarrow \mathbb {T} {\stackrel {i}{\rightarrow }}X{\stackrel {\pi }{\rightarrow }}G\rightarrow 0$ splits
* Every locally compact abelian group belongs to ${\mathcal {S}}(\mathbb {T} )$. In other words every topological extension $0\rightarrow \mathbb {T} {\stackrel {i}{\rightarrow }}X{\stackrel {\pi }{\rightarrow }}G\rightarrow 0$ where $G$ is a locally compact abelian group, splits.
* Every locally precompact abelian group belongs to ${\mathcal {S}}(\mathbb {T} )$.
* The Banach space (and in particular topological abelian group) $\ell ^{1}$ does not belong to ${\mathcal {S}}(\mathbb {T} )$.