1. Introductory survey.
Consider a group extension 1 → A → EG → 1 in some category of groups with
superstructure (topological, analytic, algebraic). Suppose it is split in the category of
abstract groups, i.e., there is a homomorphism σ : G → E of abstract groups such
that π ∘σ is the identity map on G. We are concerned with the question of when it is
possible to conclude that the extension is split as an extension in the given
category.
