# Immersed Lagrangian Floer cohomology via pearly trajectories

## TL;DR

This paper introduces a new approach to Lagrangian Floer cohomology for immersed Lagrangians using pearly trajectories, proving invariance under Hamiltonian deformations and relating it to existing theories.

## Contribution

It defines immersed Lagrangian Floer cohomology over Z_2 using pearly trajectories and establishes its invariance and isomorphism with Hamiltonian-perturbed Floer cohomology.

## Key findings

- Floer cohomology is invariant under Hamiltonian deformations.
- Established an isomorphism with existing Floer cohomology theories.
- Proved a lower bound on the number of non-embedded points in terms of Betti numbers.

## Abstract

We define Lagrangian Floer cohomology over $\mathbb Z_2$-coefficients by counting pearly trajectories for graded, exact Lagrangian immersions that satisfy certain positivity condition on the index of the non-embedded points, and show that it is an invariant of the Lagrangian immersion under Hamiltonian deformations. We also show that it is naturally isomorphic to the Hamiltonian perturbed version of Lagrangian Floer cohomology as defined in [4]. As an application, we prove that the number of non-embedded points of such a Lagrangian in $\mathbb C^n$ is no less than the sum of its Betti numbers.