Splitting central extensions of \(\mathbb{R}^\ltimes\) by \(\mathbb{R}^\gtrdot\) (Q1806076)

The authors consider the problem of splitting central extensions \(E_{n,m}:\mathbb{R}^n\rightarrowtail E\twoheadrightarrow\mathbb{R}^m\) in the following categories of (a) topological Abelian groups, (b) Abelian groups, (c) topological groups, (d) groups. They prove that (1) \(E_{n,m}\) splits in (a) and (b) for any \(n\) and \(m\). Only \(E_{n,1}\) always splits in (c), whereas, for any \(m,n\geq 1\), there are non-split objects in (d). The case of the category of topological groups is needed for the authors to close a gap found in an economics paper [\textit{A. F. Shorrocks}, Econometrica 52, 1369-1385 (1984; Zbl 0601.90029)] in which, to any continuous function \(f:\mathbb{R}^2\to\mathbb{R}\) such that \(f(x+y,z)+f(x,z) =f(y,z)+ f(x,y+z)\), the authors due to [\textit{J. Erdös}, Periodicum Math.-Phys. Astron. (2) 14, 3-5 (1959; Zbl 0085.32903)] want to find a continuous function \(g:\mathbb{R}\to\mathbb{R}\) such that \(f(x,y)= g(x+y)-g(x)-g(y)\). Indeed, by the Erdős paper mentioned above, we can only demand the existence of \(g\) without assuring the continuity of \(g\). Only the equation \(H^2(\mathbb{R}, \mathbb{R}^n)=0\) in continuous cohomology of \(\mathbb{R}\) with \(\mathbb{R}^n\) as coefficients (i.e., equivalently, \(E_{n,1}\) splits in the category of topological groups) gives the existence of a continuous function \(g\).