Homological instability for moduli spaces of smooth 4-manifolds

We prove that homological stability fails for the moduli space of any simply connected closed smooth 4-manifold in any degree of homology, unlike what happens in all even dimensions ¤ 4. We detect also the homological discrepancy between various moduli spaces, such as topological and smooth moduli spaces of 4-manifolds, and moduli spaces of 4-manifolds with positive scalar curvature metrics.Some of these discrepancies are special to dimension 4. To prove these results, we use the Seiberg-Witten equations to construct a new characteristic class of families of 4-manifolds, which is unstable and detects the difference between the smooth and topological categories in dimension 4. 1. Introduction 2993 2. The Seiberg-Witten characteristic classes 3004 3. Calculation 3036 4. Moduli space of formally smooth topological bundles 3047 5. Proof of the main results 3054 6. Manifolds that dissolve after one stabilization 3057