A Hardy Space Approach to Lagrangian Floer Gluing

TL;DR

This paper introduces a novel Hardy space method for Lagrangian Floer gluing, utilizing intersection theory in Hilbert manifolds to establish convergence and prove key properties of Floer homology.

Contribution

It develops a new Hardy space approach to Lagrangian Floer gluing, providing convergence results and foundational theorems for Floer homology construction.

Findings

Established convergence of moduli spaces in $C^1$ topology.

Proved the boundary map squares to zero in monotone case.

Demonstrated invariance of Lagrangian-Floer homology.

Abstract

We develop a new approach to Lagrangian-Floer gluing. The construction of the gluing map is based on the intersection theory in some Hilbert manifold of paths $\mathcal{P} $. We consider some moduli spaces of perturbed holomorphic curves whose domains are either strips or more general Riemann surfaces with strip-like ends. These moduli spaces can be injectively immersed into the Hilbert manifold $\mathcal{P} $ by taking the restriction to non-Lagrangian boundary. Some subsets $ \mathcal{M}^T(U) $ and $ \mathcal{M}^{\infty} (U) $ of the aforementioned moduli spaces of perturbed holomorphic strips turn out to be embedded submanifolds of the Hilbert manifold $ \mathcal{P} $. The main result is that $ \mathcal{M}^T (U) $ converges in the $C^1 $ topology toward $ \mathcal{M}^{\infty} (U) $. As an application of this convergence property we prove various gluing theorems. We explain the construction of Lagrangian-Floer homology and prove that the square of the boundary map is equal to zero in the monotone case with minimal Maslov number at least three. We also prove the invariance of the homology and include the exposition of the Lagrangian-Floer-Donaldson functor and Seidel homomorphism.