From a finite 2-complex X, one can construct a closed, smooth 4-manifold M(X), for example as the boundary of a thickened embedding in 5-dimensional Euclidean space. If X and Y have isomorphic fundmental groups, then J H C Whitehead (1939) proved that X and Y become stably homotopy equivalent after adjoining some number of copies of the 2-sphere. The talk will discuss the analogous 4-dimensional stable and unstable uniqueness question. We define a “quadratic bias invariant” for M(X) based on group cohomology and unitary automorphisms, to distinguish the 4-manifold thickenings of certain families of finite 2-complexes. This leads to the construction of arbitrarily large families of smooth 4-manifolds M(X), by varying over 2-complexes X with a given fundamental group, which are all stably diffeomorphic but pairwise distinct up to homotopy. This is joint work with John Nicholson (University of Glasgow).