Mackey-Moore cohomology and topological extensions of Polish groups

Abstract

Topological group extensions with Abelian kernels are analyzed using factor sets and following the pattern of the work of Eilenberg and MacLane on extensions of groups without topology. In this analysis, the Eilenberg-MacLane cohomology is replaced by the Mackey-Moore one, whose cochains are Borel mappings and which is especially suitable in the case of Polish groups (Hausdorff second countable complete groups). The connection between cohomology groups of degree 2 and equivalence classes of topological group extensions with Abelian kernels is established. A fundamental sequence of cohomology groups and group homomorphisms is proven to be exact, and it is shown that, in some interesting cases, the low degree cohomology groups of topological semidirect products are determined by the corresponding cohomology groups of the factors.