A note on three-fold branched covers of $S^4$

We show that any 4-manifold admitting a $(g;k,0,0)$-trisection is an irregular 3-fold cover of the 4-sphere, branched along an embedded surface with two singularities. A 4-manifold admits such a trisection if and only if it has a handle decomposition with no 1-handles and no 3-handles; it is conjectured that all simply-connected 4-manifolds have this property.