Branched coverings of simply connected $4$-manifolds

We show that, given $d\ge 4$ and two closed connected oriented PL $4$-manifolds $M$ and $N$ such that $N$ has a handle decomposition with no $1$- and $3$-handles, there exists a $d$-fold (simple) branched covering $p\text{\hspace{0.17em}⁣}:M\to N$ if and only if there is an isometric embedding of lattices $d\cdot I_N\hookrightarrow I_M$. Here $I_N$ and $I_M$ respectively denote the intersection lattices of $N$ and $M$. In particular, we characterize the manifolds which are branched covers of the K3 surface.