Branched covering representation of non-orientable $4$-manifolds

We show that every closed connected non-orientable PL $4$-manifold $X$ is a simple branched covering of ${\text{RP}}^4$. We also show that $X$ is a simple branched covering of the twisted $S^3$-bundle $S^1\text{simtimes}{S}^3$ if and only if the first Stiefel–Whitney class $w_1(X)$ admits an integral lift. In both cases, the degree of the covering can be any number $d\ge 4$, provided that $d$ has the same parity of the Stiefel–Whitney number $w_1^4[X]$ in the case of ${\text{RP}}^4$. Moreover, the branch set can be assumed to be non-singular if $d\ge 5$ and to have just nodal singularities if $d=4$.