We seek algebraic characterizations of closed connected 4-manifolds M with universal covering space S² × R² or S³ × R. We show that if and π = π ₁(M) virtually has infinite Abelianization then M is double covered by an S²-bundle over an aspherical surface and π is the group of a manifold with geometry S² × E² or S² × H². If χ(M) = 0 and π is virtually Z² then there are just nine possible groups; if moreover π / π′ is infinite then M is simple homotopy equivalent to an S² × E²-manifold, and if the torsion in π is central there are only finitely many homeomorphism types of such manifolds. If χ(M) = 0 and in π is virtually Z then its maximal finite normal subgroup is either the group of an S³-manifold or is one of the groups Q(8a, b, c) × Z / dZ. One such manifolds has π Q(24, 13, 1) × Z and so is not homotopy equivalent to an S³ × E¹-manifold.