On integral rigidity in Seiberg-Witten theory

Francesco Lin; Mike Miller Eismeier

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arXiv · 2409.17884

On integral rigidity in Seiberg-Witten theory

Francesco Lin·Mike Miller Eismeier

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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.

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Francesco Lin, Mike Miller Eismeier. 2024-09-26. On integral rigidity in Seiberg-Witten theory. https://arxiv.org/abs/2409.17884

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