Homological instability for moduli spaces of smooth 4-manifolds

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 $\ne 4$. 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.

70 pages, 1 figure, added contents on formally smooth bundles (Section 4)

Cited by in corpus (1)

• The homology of moduli spaces of 4-manifolds may be infinitely generated