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 ${\displaystyle {\mathcal {S}}(\mathbb {T} )}$ .

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.
3. Bello, Hugo J.; Chasco, María Jesús; Domínguez, Xabier (2013). "Extending topological abelian groups by the unit circle". Abstr. Appl. Anal. Article ID 590159. doi:10.1155/2013/590159. Zbl 1295.22009.