A Strong Dominability Criterion and an Oka Union Theorem

We prove that an $n$-dimensional complex manifold $Y$ is Oka if and only if it is strongly dominable: for every $y\in Y$, there is an entire map $F:{\mathbb{C}}^n\to Y$ such that $F(0)=y$ and $d{F}_0$ is invertible. The argument converts this pointwise domination into the convex approximation property in every source dimension. We also show that the Oka property extends across proper closed complex analytic subsets: if $A$ is such a subset of a connected complex manifold $X$ and $X\setminus A$ is Oka, then $X$ is Oka. The criterion has broad applications and produces many new examples of Oka manifolds.