On integral rigidity in Seiberg-Witten theory

TL;DR

This paper develops a framework for proving integral rigidity of Seiberg-Witten invariants in 4-manifolds with specific hypersurfaces, revealing new cohomological determination results and connections to TQFT.

Contribution

It introduces a novel approach to integral rigidity in Seiberg-Witten theory, linking solutions on 4-manifolds and 3-manifolds, and describes the associated graded map of cobordism-induced maps.

Findings

Sum of Seiberg-Witten invariants determined by cohomology for certain 4-manifolds.

Concrete description of the graded map induced by negative cobordisms.

Framework generalizes Donaldson's TQFT approach to Seiberg-Witten invariants.

Abstract

We introduce a framework to prove integral rigidity results for the Seiberg-Witten invariants of a closed $4$-manifold $X$ containing a non-separating hypersurface $Y$ satisfying suitable (chain-level) Floer theoretic conditions. As a concrete application, we show that if $X$ has the homology of a four-torus, and it contains a non-separating three-torus, then the sum of all Seiberg-Witten invariants of $X$ is determined in purely cohomological terms. Our results can be interpreted as $(3+1)$-dimensional versions of Donaldson's TQFT approach to the formula of Meng-Taubes, and build upon a subtle interplay between irreducible solutions to the Seiberg-Witten equations on $X$ and reducible ones on $Y$ and its complement. Along the way, we provide a concrete description of the associated graded map (for a suitable filtration) of the map on $\overline{HM}_*$ induced by a negative cobordism between three-manifolds, which might be of independent interest.