Stable classification of 4-manifolds with 3-manifold fundamental groups

We study closed, oriented 4-manifolds whose fundamental group is that of a closed, oriented, aspherical 3-manifold. We show that two such 4-manifolds are stably diffeomorphic if and only if they have the same w 2 -type and their equivariant intersection forms are stably isometric. We also find explicit algebraic invariants that determine the stable classification for spin manifolds in this class.