Branched coverings of simply connected manifolds

We construct branched double coverings by certain direct products of manifolds for connected sums of copies of sphere bundles over the 2-sphere. As an application we answer a question of Kotschick and Löh up to dimension five. More precisely, we show that(1)every simply connected, closed four-manifold admits a branched double covering by a product of the circle with a connected sum of copies of S2×S1S2×S1, followed by a collapsing map;(2)every simply connected, closed five-manifold admits a branched double covering by a product of the circle with a connected sum of copies of S3×S1S3×S1, followed by a map whose degree is determined by the torsion of the second integral homology group of the target.