Abstract
A twisted sum in the category of topological Abelian groups is a short exact sequence0→Y→X→Z→0where all maps are assumed to be continuous and open onto their images. The twisted sum splits if it is equivalent to0→Y→Y×Z→Z→0. We study the classSTG𝕋of topological groupsGfor which every twisted sum0→𝕋→X→G→0splits. We prove that this class contains Hausdorff locally precompact groups, sequential direct limits of locally compact groups, and topological groups withℒ∞topologies. We also prove that it is closed by taking open and dense subgroups, quotients by dually embedded subgroups, and coproducts. As means to find further subclasses ofSTG𝕋, we use the connection between extensions of the form0→𝕋→X→G→0and quasi-characters onG, as well as three-space problems for topological groups. The subject is inspired on some concepts known in the framework of topological vector spaces such as the notion of𝒦-space, which were interpreted for topological groups by Cabello.
Highlights
Introduction and PreliminariesIn the theory of topological vector spaces a property P is said to be a 3-space property if whenever a closed subspace Y of a space X and the corresponding quotient X/Y both have property P, X has property P.A short exact sequence of topological vector spaces 0 → Y →ı X →π Z → 0 will be called a twisted sum, and the space X will be called an extension of Z by Y when both ı and π are continuous and open onto their images
At this point the following definition, originally introduced in [4], comes across as natural: a F-space X is said to be a K-space if, whenever Y is an F-space and L is a subspace of Y with dimension one such that Y/L ≅ X, the corresponding twisted sum splits
Observe that a necessary condition for the splitting of the twisted sum of topological Abelian groups 0 → H →ı X →π G → 0 is that i(H) be a dually embedded subgroup of
Summary
A twisted sum in the category of topological Abelian groups is a short exact sequence 0 → Y → X → Z → 0 where all maps are assumed to be continuous and open onto their images. The twisted sum splits if it is equivalent to 0 → Y → Y × Z → Z → 0. We study the class STG(T) of topological groups G for which every twisted sum 0 → T → X → G → 0 splits. We prove that this class contains Hausdorff locally precompact groups, sequential direct limits of locally compact groups, and topological groups with L∞ topologies. The subject is inspired on some concepts known in the framework of topological vector spaces such as the notion of K-space, which were interpreted for topological groups by Cabello
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