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 w2‐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.