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On integral rigidity in Seiberg-Witten theory

Francesco Lin, Mike Miller Eismeier

TL;DR

This work develops an integral rigidity framework for Seiberg-Witten invariants on closed 4-manifolds containing a non-separating hypersurface by leveraging a self-gluing viewpoint and a detailed chain-level analysis of negative-definite cobordisms. A key contribution is a filtration-compatible description of the induced cobordism maps on monopole Floer homology, with the associated graded map matching cup-homology calculations, enabling explicit trace formulas. The authors introduce RSF-spaces to capture tractable Floer-theoretic conditions and prove a general rigidity theorem: sums of SW invariants tied to a fixed end data on Y are determined by cobordism data $c(W,Y,\mathfrak{s}_Y)$ when $b^+(W)=0$, and vanish otherwise; this yields concrete determinant-type results for Y = $T^3$ and, in particular, for homology 4-tori. The results connect to Meng–Taubes, Donaldson’s TQFT perspective, and Ruberman–Strle-type rigidity, and are illustrated through explicit calculations and examples involving non-separating tori and mapping tori, providing a coherent framework for rigidity phenomena in 4-manifold topology.

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.

On integral rigidity in Seiberg-Witten theory

TL;DR

This work develops an integral rigidity framework for Seiberg-Witten invariants on closed 4-manifolds containing a non-separating hypersurface by leveraging a self-gluing viewpoint and a detailed chain-level analysis of negative-definite cobordisms. A key contribution is a filtration-compatible description of the induced cobordism maps on monopole Floer homology, with the associated graded map matching cup-homology calculations, enabling explicit trace formulas. The authors introduce RSF-spaces to capture tractable Floer-theoretic conditions and prove a general rigidity theorem: sums of SW invariants tied to a fixed end data on Y are determined by cobordism data when , and vanish otherwise; this yields concrete determinant-type results for Y = and, in particular, for homology 4-tori. The results connect to Meng–Taubes, Donaldson’s TQFT perspective, and Ruberman–Strle-type rigidity, and are illustrated through explicit calculations and examples involving non-separating tori and mapping tori, providing a coherent framework for rigidity phenomena in 4-manifold topology.

Abstract

We introduce a framework to prove integral rigidity results for the Seiberg-Witten invariants of a closed -manifold containing a non-separating hypersurface satisfying suitable (chain-level) Floer theoretic conditions. As a concrete application, we show that if has the homology of a four-torus, and it contains a non-separating three-torus, then the sum of all Seiberg-Witten invariants of is determined in purely cohomological terms. Our results can be interpreted as -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 and reducible ones on and its complement. Along the way, we provide a concrete description of the associated graded map (for a suitable filtration) of the map on induced by a negative cobordism between three-manifolds, which might be of independent interest.
Paper Structure (15 sections, 21 theorems, 111 equations)

This paper contains 15 sections, 21 theorems, 111 equations.

Table of Contents

  1. introduction
  2. Gluing formulas in the non-separating case
  3. The map induced by a negative definite cobordism
  4. Review of the three-manifold case
  5. The formula
  6. Proof of Theorem \ref{['thm:constant-term']}
  7. $RSF$-spaces
  8. Definitions.
  9. Examples of $RSF$-spaces
  10. Proof of Theorem \ref{['thm:weak-calculation']}
  11. Examples
  12. Non-separating three-tori
  13. Proof of Theorem \ref{['thm:torus-rigidity']}
  14. Homology 4-tori
  15. Some examples

Key Result

Theorem 1

Consider a homology $4$-torus $X$ which is spin. Then the sum of all (degree zero) Seiberg--Witten invariants $\mathfrak{m}(X)$ of $X$ has the same parity as $\det(X)$.

Theorems & Definitions (58)

  • Theorem : RubermanStrle
  • Theorem 1
  • Remark
  • Remark
  • Theorem 2
  • Remark
  • Remark
  • Theorem 3
  • Remark
  • Proposition 2.1
  • ...and 48 more