Strongly minimal PD4-complexes

We consider the homotopy types of PD4-complexes X with fundamental group π such that c.d.π=2 and π has one end. Let β=β2(π;F2) and w=w1(X). Our main result is that (modulo two technical conditions on (π,w)) there are at most β2 orbits of k-invariants determining “strongly minimal” complexes (i.e., those with homotopy intersection pairing λX trivial). The homotopy type of a PD4-complex X with π a PD2-group is determined by π, w, λX and the v2-type of X. Our result also implies that Fox's 2-knot with metabelian group is determined up to homeomorphism by its group.