Branched coverings of simply connected $4$-manifolds

We show that, given $d \geq 4$ 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 $p \colon M \rightarrow N$ 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.