This is an announcement for the paper “Smooth surjections and surjective restrictions” by Richard M. Aron<http://arxiv.org/find/math/1/au:+Aron_R/0/1/0/all/0/1>, Jesús A. Jaramillo<http://arxiv.org/find/math/1/au:+Jaramillo_J/0/1/0/all/0/1>, Enrico Le Donne<http://arxiv.org/find/math/1/au:+Donne_E/0/1/0/all/0/1>. Abstract: Given a surjective mapping $f: E\rightarrow F$ between Banach spaces, we investigate the existence of a subspace $G$ of $E$, with the same density character as $F$, such that the restriction of $f$ to $G$ remains surjective. We obtain a positive answer whenever $f$ is continuous and uniformly open. In the smooth case, we deduce a positive answer when $f$ is a $C_1$ -smooth surjection whose set of critical values is countable. Finally we show that, when $f$ takes values in the Euclidean space $R_n$, in order to obtain this result it is not sufficient to assume that the set of critical values of $f$ has zero-measure. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1607.01725