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.^[1] Every extension of topological group 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.

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.^[2]

• 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.

References

1. Cabello Sánchez, Félix (2003). "Quasi-homomorphisms". Fundam. Math. 178 (3): 255–270. doi:10.4064/fm178-3-5. Zbl 1051.39032.
2. Fulp, R.O.; Griffith, P.A. (1971). "Extensions of locally compact abelian groups. I, II" (PDF). Trans. Am. Math. Soc. 154: 341–356, 357–363. doi:10.1090/S0002-9947-1971-99931-0. MR 0272870. Zbl 0216.34302.