Cross sections and pseudo-homomorphisms of topological abelian groups. (English) Zbl 1372.22002

Summary: We say that a mapping \(\omega\) between two topological abelian groups \(G\) and \(H\) is a pseudo-homomorphism if the associated map \((x, y) \in G \times G \mapsto \omega(x + y) - \omega(x) - \omega(y) \in H\) is continuous. This notion appears naturally in connection with cross sections (continuous right inverses for quotient mappings): given an algebraically splitting, closed subgroup \(H\) of a topological group \(X\) such that the projection \(\pi : X \rightarrow X / H\) admits a cross section, one obtains a pseudo-homomorphism of \(X / H\) to \(H\), and conversely. We show that \(H\) splits as a topological subgroup if and only if the corresponding pseudo-homomorphism can be decomposed as a sum of a homomorphism and a continuous mapping. We also prove that pseudo-homomorphisms between Polish groups satisfy the closed graph theorem. Several examples are given.


22A05 Structure of general topological groups
54H11 Topological groups (topological aspects)
54C65 Selections in general topology
46A30 Open mapping and closed graph theorems; completeness (including \(B\)-, \(B_r\)-completeness)
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