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Homological instability for moduli spaces of smooth 4-manifolds
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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.
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Families of diffeomorphisms, embeddings, and positive scalar curvature metrics via Seiberg-Witten theory
A new gluing theorem for parameterized Seiberg-Witten invariants gives infinite rank Z^∞ summands in higher homotopy and homology of diffeomorphism groups of 4-manifolds that are topologically trivial.
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