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 ${\displaystyle 0\to H{\stackrel {\imath }{\to }}X{\stackrel {\pi }{\to }}G\to 0}$ where ${\displaystyle H,X}$ and ${\displaystyle G}$ are topological groups and ${\displaystyle i}$ and ${\displaystyle \pi }$ 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

${\displaystyle 0\rightarrow H{\stackrel {i}{\rightarrow }}X{\stackrel {\pi }{\rightarrow }}G\rightarrow 0}$

and

${\displaystyle 0\to H{\stackrel {i'}{\rightarrow }}X'{\stackrel {\pi '}{\rightarrow }}G\rightarrow 0}$

are equivalent (or congruent) if there exists a topological isomorphism ${\displaystyle T:X\to X'}$ making commutative the diagram of Figure 1.

Figure 1

We say that the topological extension

${\displaystyle 0\rightarrow H{\stackrel {i}{\rightarrow }}X{\stackrel {\pi }{\rightarrow }}G\rightarrow 0}$

is a split extension (or splits) if it is equivalent to the trivial extension

${\displaystyle 0\rightarrow H{\stackrel {i_{H}}{\rightarrow }}H\times G{\stackrel {\pi _{G}}{\rightarrow }}G\rightarrow 0}$

where ${\displaystyle i_{H}:H\to H\times G}$ is the natural inclusion over the first factor and ${\displaystyle \pi _{G}:H\times G\to G}$ is the natural projection over the second factor.

It is easy to prove that the topological extension ${\displaystyle 0\rightarrow H{\stackrel {i}{\rightarrow }}X{\stackrel {\pi }{\rightarrow }}G\rightarrow 0}$ splits if and only if there is a continuous homomorphism ${\displaystyle R:X\rightarrow H}$ such that ${\displaystyle R\circ i}$ is the identity map on ${\displaystyle H}$

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

Examples

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

${\displaystyle 0\to \mathbb {Z} {\stackrel {\imath }{\to }}\mathbb {R} {\stackrel {\pi }{\to }}\mathbb {R} /\mathbb {Z} \to 0}$

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 ${\displaystyle 0\to H{\stackrel {\imath }{\to }}X{\stackrel {\pi }{\to }}G\to 0}$ where ${\displaystyle H,X}$ and ${\displaystyle G}$ are locally compact abelian groups and ${\displaystyle i}$ and ${\displaystyle \pi }$ are relatively open continuous homomorphisms.^[2]

• Let be an extension of locally compact abelian groups

${\displaystyle 0\to H{\stackrel {\imath }{\to }}X{\stackrel {\pi }{\to }}G\to 0.}$

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

${\displaystyle 0\to G^{\wedge }{\stackrel {\pi ^{\wedge }}{\to }}X^{\wedge }{\stackrel {\imath ^{\wedge }}{\to }}H^{\wedge }\to 0}$

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 ${\displaystyle 0\rightarrow \mathbb {T} {\stackrel {i}{\rightarrow }}X{\stackrel {\pi }{\rightarrow }}G\rightarrow 0}$ where ${\displaystyle \mathbb {T} }$ is the unit circle and ${\displaystyle X}$ and ${\displaystyle G}$ are topological abelian groups.^[3]

The class S(T)

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

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

• Every locally precompact abelian group belongs to ${\displaystyle {\mathcal {S}}(\mathbb {T} )}$.

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