Strongly minimal -complexes

We consider the homotopy types of PD 4 -complexes X with fundamental group π such that c . d . π = 2 and π has one end. Let β = β 2 ( π ; F 2 ) and w = w 1 ( X ) . Our main result is that (modulo two technical conditions on ( π , w ) ) there are at most 2 β orbits of k-invariants determining “strongly minimal” complexes (i.e., those with homotopy intersection pairing λ X trivial). The homotopy type of a PD 4 -complex X with π a PD 2 -group is determined by π, w, λ X and the v 2 -type of X. Our result also implies that Foxʹs 2-knot with metabelian group is determined up to homeomorphism by its group.