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Extension of a topological group

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In mathematics, more specifically in topological groups, an extension of topological groups, or a topological extension, is a short exact sequence 0 → H → ı X → π G → 0 {\displaystyle 0\to H{\stackrel {\imath }{\to }}X{\stackrel {\pi }{\to }}G\to 0} where H , X {\displaystyle H,X} and G {\displaystyle G} are topological groups and i {\displaystyle i} and π {\displaystyle \pi } are continuous homomorphisms which are also open onto their images. Every extension of topological groups is therefore a group extension.

Classification of extensions of topological groups

We say that the topological extensions

0 → H → i X → π G → 0 {\displaystyle 0\rightarrow H{\stackrel {i}{\rightarrow }}X{\stackrel {\pi }{\rightarrow }}G\rightarrow 0}

and

0 → H → i ′ X ′ → π ′ G → 0 {\displaystyle 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 → X ′ {\displaystyle T:X\to X'} making commutative the diagram of Figure 1.

Figure 1

We say that the topological extension

0 → H → i X → π G → 0 {\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

0 → H → i H H × G → π G G → 0 {\displaystyle 0\rightarrow H{\stackrel {i_{H}}{\rightarrow }}H\times G{\stackrel {\pi _{G}}{\rightarrow }}G\rightarrow 0}

where i H : H → H × G {\displaystyle i_{H}:H\to H\times G} is the natural inclusion over the first factor and π G : H × G → G {\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 0 → H → i X → π G → 0 {\displaystyle 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 → H {\displaystyle R:X\rightarrow H} such that R ∘ i {\displaystyle R\circ i} is the identity map on H {\displaystyle H}

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

Examples

  • Take R {\displaystyle \mathbb {R} } the real numbers and Z {\displaystyle \mathbb {Z} } the integer numbers. Take ı {\displaystyle \imath } the natural inclusion and π {\displaystyle \pi } the natural projection. Then
0 → Z → ı R → π R / Z → 0 {\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 0 → H → ı X → π G → 0 {\displaystyle 0\to H{\stackrel {\imath }{\to }}X{\stackrel {\pi }{\to }}G\to 0} where H , X {\displaystyle H,X} and G {\displaystyle G} are locally compact abelian groups and i {\displaystyle i} and π {\displaystyle \pi } are relatively open continuous homomorphisms.

  • Let be an extension of locally compact abelian groups
0 → H → ı X → π G → 0. {\displaystyle 0\to H{\stackrel {\imath }{\to }}X{\stackrel {\pi }{\to }}G\to 0.}
Take H ∧ , X ∧ {\displaystyle H^{\wedge },X^{\wedge }} and G ∧ {\displaystyle G^{\wedge }} the Pontryagin duals of H , X {\displaystyle H,X} and G {\displaystyle G} and take i ∧ {\displaystyle i^{\wedge }} and π ∧ {\displaystyle \pi ^{\wedge }} the dual maps of i {\displaystyle i} and π {\displaystyle \pi } . Then the sequence
0 → G ∧ → π ∧ X ∧ → ı ∧ H ∧ → 0 {\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 0 → T → i X → π G → 0 {\displaystyle 0\rightarrow \mathbb {T} {\stackrel {i}{\rightarrow }}X{\stackrel {\pi }{\rightarrow }}G\rightarrow 0} where T {\displaystyle \mathbb {T} } is the unit circle and X {\displaystyle X} and G {\displaystyle G} are topological abelian groups.

The class S(T)

A topological abelian group G {\displaystyle G} belongs to the class S ( T ) {\displaystyle {\mathcal {S}}(\mathbb {T} )} if and only if every topological extension of the form 0 → T → i X → π G → 0 {\displaystyle 0\rightarrow \mathbb {T} {\stackrel {i}{\rightarrow }}X{\stackrel {\pi }{\rightarrow }}G\rightarrow 0} splits

  • Every locally compact abelian group belongs to S ( T ) {\displaystyle {\mathcal {S}}(\mathbb {T} )} . In other words every topological extension 0 → T → i X → π G → 0 {\displaystyle 0\rightarrow \mathbb {T} {\stackrel {i}{\rightarrow }}X{\stackrel {\pi }{\rightarrow }}G\rightarrow 0} where G {\displaystyle G} is a locally compact abelian group, splits.
  • Every locally precompact abelian group belongs to S ( T ) {\displaystyle {\mathcal {S}}(\mathbb {T} )} .
  • The Banach space (and in particular topological abelian group) ℓ 1 {\displaystyle \ell ^{1}} does not belong to S ( T ) {\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.
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