PD4-complexes with strongly minimal models

Let X be a PD4-complex with fundamental group π. We give conditions on the algebraic 2-type of X under which the homotopy type of X is determined by π, w=w1(X), the image of [X] in H4(π;Zw) and the equivariant intersection pairing on π2(X). In particular, the homotopy type of an oriented spin 4-manifold with fundamental group a PD2-group π is determined by π and this pairing.