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

↑ Cabello Sánchez, Félix (2003). "Quasi-homomorphisms". Fundam. Math. 178 (3): 255–270. Zbl 1051.39032. doi:10.4064/fm178-3-5.
↑ 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. MR 0272870. Zbl 0216.34302. doi:10.1090/S0002-9947-1971-99931-0.

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