From Seiberg-Witten to Gromov: MCE and Singular Symplectic Forms

TL;DR

This paper extends Taubes's convergence theorem to non-compact 4-manifolds with cylindrical ends, providing new technical tools that could unify various Floer homology theories and their TQFT applications.

Contribution

It proves variants of Taubes's convergence theorem for non-compact manifolds with cylindrical ends, simplifying previous technical approaches and enabling broader applications in Floer theory.

Findings

Extended Taubes's convergence theorem to non-compact manifolds

Simplified key technical ingredients from prior work

Potential applications to TQFTs and link invariants

Abstract

Motivated by various possible generalizations of Taubes's \(SW=Gr\) theorem [T] to Floer-theoretic setting, we prove certain variants of Taubes's convergence theorem in \cite{T} (the first part of his proof of \(SW=Gr\)). In place of the closed symplectic 4-manifold considered in [T], this article considers non-compact manifolds with cylindrical ends, equipped with a self-dual harmonic 2-form with non-degenerate zeroes. This extends and simplifies some central technical ingredients of the author's prior work in [LT] and [KLT5]. Other expected applications include: extending the \(HM=PFH\) theorem in [T] and the \(HM=HF\) theorem in [KLT1]-[KLT5] to TQFTs on both sides [L1]; definitions of large-perturbation Seiberg-Witten analogs of Heegaard Floer theory's link Floer homologies and link cobordism invariants.