Cross sections and pseudo-homomorphisms of topological abelian groups

We say that a mapping ω between two topological abelian groups G and H is a pseudo-homomorphism if the associated map (x,y)∈G×G↦ω(x+y)−ω(x)−ω(y)∈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 π:X→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.