Surjective holomorphic maps onto Oka manifolds

Let $X$ be a connected Oka manifold, and let $S$ be a Stein manifold with $\mathrm{d}\mathrm{i}\mathrm{m}S\ge \mathrm{d}\mathrm{i}\mathrm{m}X$. We show that every continuous map $S\to X$ishomotopictoasurjectivestronglydominatingholomorphicmap$S\to X$. We also find strongly dominating algebraic morphisms from the affine $n$-space onto any compact $n$-dimensional algebraically subelliptic manifold. Motivated by these results, we propose a new holomorphic flexibility property of complex manifolds, the basic Oka property with surjectivity, which could potentially provide another characterization of the class of Oka manifolds.