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

Clasification 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

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]

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". Trans. Am. Math. Soc. 154: 341–356, 357–363. doi:10.1090/S0002-9947-1971-99931-0. MR 0272870. Zbl 0216.34302.