Homological instability for moduli spaces of smooth 4-manifolds

TL;DR

This paper demonstrates that homological stability does not hold for moduli spaces of simply-connected closed smooth 4-manifolds, revealing fundamental differences from other dimensions and between smooth and topological categories.

Contribution

It introduces a new unstable characteristic class using Seiberg-Witten equations, highlighting the failure of homological stability and differences between various 4-manifold moduli spaces.

Findings

Homological stability fails in all degrees for 4-manifold moduli spaces.

Detects differences between smooth and topological moduli spaces.

Constructs a new characteristic class using Seiberg-Witten equations.

Abstract

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 dimensions $\neq 4$. 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. 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.