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 where and are topological groups and and 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

and

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

Figure 1

We say that the topological extension

is a split extension (or splits) if it is equivalent to the trivial extension

where is the natural inclusion over the first factor and is the natural projection over the second factor.

It is easy to prove that the topological extension splits if and only if there is a continuous homomorphism such that is the identity map on

Note that the topological extension splits if and only if the subgroup is a topological direct summand of

Examples

  • Take the real numbers and the integer numbers. Take the natural inclusion and 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 where and are locally compact abelian groups and and are relatively open continuous homomorphisms.[2]

  • Let be an extension of locally compact abelian groups
Take and the Pontryagin duals of and and take and the dual maps of and . Then the sequence
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" (PDF). Trans. Am. Math. Soc. 154: 341–356, 357–363. doi:10.1090/S0002-9947-1971-99931-0. MR 0272870. Zbl 0216.34302.
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