Construction of holomorphic quilts in Cartesian product of closed surfaces

TL;DR

This paper modifies existing proofs to construct immersed holomorphic quilts in closed surfaces, aiming to compare Lagrangian Floer theories and support conjectures about their isomorphism after certain twists.

Contribution

It introduces a new construction of immersed holomorphic quilts in closed surfaces and applies this to compare different Lagrangian Floer theories, providing evidence for a conjectured isomorphism.

Findings

Constructed specific immersed holomorphic quilts in closed surfaces.

Provided a potential example supporting the conjecture on Floer homology isomorphism.

Linked Lagrangian Floer theory with quilted Floer theory through this construction.

Abstract

In this article, we modify the proof of holomorphic quilts from Wehrheim and Woodward in \cite{wehrheim2009floer} to construct a specific type of immersed holomorphic quilt, where the symplectic manifolds are closed surfaces. The application is to compare Lagrangian Floer theory with quilted Lagrangian Floer theory, as they relate through Lagrangian correspondence. A potential example is provided to support Bottman and Wehrheim's conjecture \cite{bottman2018gromov} regarding the isomorphism between Lagrangian Floer homology and quilted Lagrangian Floer homology after twisting by bounding cochains.