Homological instability for moduli spaces of smooth 4-manifolds Article Swipe

PDF

Related Concepts

Moduli space Mathematics Pure mathematics Manifold (fluid mechanics) Moduli Dimension (graph theory) Homology (biology) Mathematical analysis Topology (electrical circuits) Physics Combinatorics Gene Engineering Quantum mechanics Chemistry Biochemistry Mechanical engineering

Hokuto Konno , Jianfeng Lin ·

YOU? · · 2022 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2211.03043 · OA: W4308613397

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.