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 $0\to H\stackrel{ı}{\to }X\stackrel{\pi }{\to }G\to 0$ 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

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

and

$0\to H\stackrel{i^{\prime }}{\to }{X}^{\prime }\stackrel{{\pi }^{\prime }}{\to }G\to 0$

are equivalent (or congruent) if there exists a topological isomorphism $T:X\to X^{\prime }$ making commutative the diagram of Figure 1.

File:Extensions groups2.png

Figure 1

We say that the topological extension

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

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

$0\to H\stackrel{i_H}{\to }H\times G\stackrel{{\pi }_G}{\to }G\to 0$

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 $0\to H\stackrel{i}{\to }X\stackrel{\pi }{\to }G\to 0$ splits if and only if there is a continuous homomorphism $R:X\to H$ such that $R\circ i$ is the identity map on $H$ Note that the topological extension $0\to H\stackrel{i}{\to }X\stackrel{\pi }{\to }G\to 0$ splits if and only if the subgroup $i(H)$ is a topological direct summand of $X$

Examples

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

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

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 $0\to H\stackrel{ı}{\to }X\stackrel{\pi }{\to }G\to 0$ 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

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

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

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

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 $0\to \mathbb{𝕋}\stackrel{i}{\to }X\stackrel{\pi }{\to }G\to 0$ 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 ${𝒮}(\mathbb{𝕋})$ if and only if every topological extension of the form $0\to \mathbb{𝕋}\stackrel{i}{\to }X\stackrel{\pi }{\to }G\to 0$ splits

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