On the homotopy types of closed 4-manifolds covered by S2 × R2

We show that if M is a closed 4-manifold such that π2(M) ≅ Z and χ(M) ⩽ 0 then its homotopy type is determined up to a finite ambiguity by π = π1(M). We also show that (for given π) the action of π on π2(M) and the orientation character ω1(M) determine each other, and find strong constraints on the possible k-invariants when χ(M) = 0. Finally we show that Wh(π) = 0.