  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 Loeh 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 $S^2 \times S^1$, 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 $S^3 \times S^1$,
followed by a map whose degree is determined by the torsion of the second
integral homology group of the target.
