Homological instability for moduli spaces of smooth 4-manifolds

Konno, Hokuto; Lin, Jianfeng

doi:10.2140/gt.2026.30.2993

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Homological instability for moduli spaces of smooth 4-manifolds

Hokuto Konno and Jianfeng Lin

Geometry & Topology 30 (2026) 2993–3068

DOI: 10.2140/gt.2026.30.2993

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 even 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. 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.

Keywords

4-manifold, Seiberg-Witten theory, diffeomorphism group

Mathematical Subject Classification

Primary: 57K41, 57S05

References

Publication

Received: 10 June 2024

Revised: 8 February 2026

Accepted: 4 May 2026

Published: 30 August 2026

Proposed: András I Stipsicz

Seconded: Simon Donaldson, Nathalie Wahl

Authors

Hokuto Konno
Graduate School of Mathematical Sciences The University of Tokyo Tokyo Japan	RIKEN iTHEMS Saitama Japan

Jianfeng Lin
Yau Mathematical Sciences Center Tsinghua University Beijing China

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Volume 30, issue 8 (2026)