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 $p \colon M \darrow{d} N$ if and only if there is an isometric embedding of intersection lattices $d \cdot I_N \hookrightarrow I_M$. Moreover, if such $p$ exists, one can build it in such a way that its branch set $B_p \subset N$ is locally flat PL embedded if $d \geq 5$ and has at most nodal singularities if $d=4$.