PD4-complexes with strongly minimal models

Let X be a PD 4 -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 = w 1 ( X ) , the image of [ X ] in H 4 ( π ; Z w ) and the equivariant intersection pairing on π 2 ( X ) . In particular, the homotopy type of an oriented spin 4-manifold with fundamental group a PD 2 -group π is determined by π and this pairing.