Morse theory and Seiberg-Witten moduli spaces of 3-dimensional cobordisms, I

TL;DR

This paper investigates the properties of Seiberg-Witten moduli spaces on 3-dimensional cobordisms with cylindrical ends, motivated by a variant of the Atiyah-Floer conjecture, focusing on perturbations by specific closed 2-forms.

Contribution

It introduces a new approach to studying Seiberg-Witten moduli spaces on cobordisms with cylindrical ends, considering particular perturbations and Morse functions, as a first step towards understanding related conjectures.

Findings

Analysis of moduli space properties under specific perturbations

Establishment of foundational results for Seiberg-Witten equations on cobordisms

Framework for future exploration of the Atiyah-Floer conjecture variant

Abstract

Motivated by a variant of Atiyah-Floer conjecture proposed in \cite{L2} and its potential generalizations, we study in this article and its sequel as a first step properties of moduli spaces of Seiberg-Witten equations on a 3-dimensional cobordism with cylindrical ends (CCE) \(Y\), perturbed by closed 2-forms of the form \(r*d\ff+w\), where \(r\geq 1\), where \(\ff\) is a harmonic Morse function with certain linear growth at the ends of \(Y\), and \(w\) is a certain closed 2-form.