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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.

MSC:
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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