Extension of a topological group explained

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 groups is therefore a group extension.

Classification of extensions of topological groups

We say that the topological extensions

0 → H\stackrel{i}{ → } X\stackrel{\pi}{ → } G → 0 and 0 \to H\stackrel{i'}{ → } X'\stackrel{\pi'}{ → } G → 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 → H\stackrel{i}{ → } X\stackrel{\pi}{ → } G → 0

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

0 → H\stackrel{i _{H}{ → }} H x G\stackrel{\pi _{G}{ → }} G → 0

where

i _{H:} H\to H x G is the natural inclusion over the first factor and \pi _{G:} H x G\to G is the natural projection over the second factor.

It is easy to prove that the topological extension

0 → H\stackrel{i}{ → } X\stackrel{\pi}{ → } G → 0 splits if and only if there is a continuous homomorphism R: X → H such that R\circ i is the identity map on H

Note that the topological extension

0 → H\stackrel{i}{ → } X\stackrel{\pi}{ → } G → 0 splits if and only if the subgroup i(H) is a topological direct summand of X

Examples

• Take

R the real numbers and Z the integer numbers. Take \imath the natural inclusion and \pi the natural projection. Then0 \to Z\stackrel{\imath}{\to} R \stackrel{\pi}{\to} R/ 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 sequence0 \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 → T\stackrel{i}{ → } X\stackrel{\pi}{ → } G → 0 where T is the unit circle and X and G are topological abelian groups.^[3]

The class S(T)

A topological abelian group

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

• Every locally compact abelian group belongs to

l S ( T) . In other words every topological extension 0 → T\stackrel{i}{ → } X\stackrel{\pi}{ → } G → 0 where G is a locally compact abelian group, splits.

• Every locally precompact abelian group belongs to

l S ( T) .

• The Banach space (and in particular topological abelian group)

\ell ^{1} does not belong to l S ( T) .

Notes and References

1. Cabello Sánchez . Félix . Quasi-homomorphisms . 1051.39032 . 178 . 3 . 255–270 . 2003 . Fundam. Math. . 10.4064/fm178-3-5 . free .
2. Fulp . R.O. . Griffith . P.A. . Extensions of locally compact abelian groups. I, II . 0216.34302 . 0272870 . Trans. Am. Math. Soc. . 154 . 341–356, 357–363 . 1971 . 10.1090/S0002-9947-1971-99931-0 . free .
3. Bello . Hugo J. . Chasco . María Jesús . Domínguez . Xabier . Extending topological abelian groups by the unit circle . 1295.22009 . Abstr. Appl. Anal. . 2013 . Article ID 590159 . 10.1155/2013/590159 . free .