Four-manifolds, two-complexes and the quadratic bias invariant

Kreck and Schafer produced the first examples of stably diffeomorphic closed smooth 4-manifolds which are not homotopy equivalent. They were constructed by applying the doubling construction to 2-complexes over certain finite abelian groups of odd order. By extending their methods, we formulate a new homotopy invariant on the class of 4-manifolds arising as doubles of 2-complexes with finite fundamental group. As an application we show that, for any $k\ge 2$, there exist a family of $k$ closed smooth 4-manifolds which are all stably diffeomorphic but are pairwise not homotopy equivalent.