Extension of a topological group

Classification of extensions of topological groups

We say that the topological extensions

0 → H → i X → π G → 0

and

0 → H → i ′ X ′ → π ′ G → 0

are equivalent (or congruent) if there exists a topological isomorphism T : X → X ′ making commutative the diagram of Figure 1.

We say that the topological extension

0 → H → i X → π G → 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

where i H : H → H × G is the natural inclusion over the first factor and π G : H × G → 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 splits if and only if there is a continuous homomorphism R : X → H such that R ∘ i is the identity map on H

Note that the topological extension 0 → H → i X → π 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 ı the natural inclusion and π the natural projection. Then

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 where H , X and G are locally compact abelian groups and i and π are relatively open continuous homomorphisms.

Let be an extension of locally compact abelian groups

Take H ∧ , X ∧ and G ∧ the Pontryagin duals of H , X and G and take i ∧ and π ∧ the dual maps of i and π . Then the sequence 0 → G ∧ → π ∧ X ∧ → ı ∧ H ∧ → 0 is an extension of locally compact abelian groups.