Summary: Let \(X\) be either a \(C^\infty\) paracompact differential manifold or a real analytic manifold. Then for any integer \(n\geq 2\) there is a fundamental system of open neighborhoods \(\{U_{n,\gamma} \}_{\gamma\in\Gamma_n}\) of the small diagonal \(\Delta_{X,n}\) of \(X^n\) in \(X^n\) such that:
(i) each \(U_{n,\gamma}\) is invariant for the action of \(S_n\);
(ii) for each \(n\geq 2\) and each \(\gamma\in \Gamma_n\) there is an \(S_n\)-invariant differentiable submersion (or an \(S_n\)-invariant real analytic submersion) \(u_{n,\gamma}:U_{n, \gamma} \to \Delta_{X,n}\) such that \(u_{n,\gamma}|\Delta_{X,n} = \text{Id}_{\Delta_{X,n}}\).
This is a partially defined solution for the so-called social choice problem.