Lagrangian intersections and a conjecture of Arnol'd

TL;DR

This paper advances the understanding of Lagrangian intersections by proving a degenerate homological Arnol'd conjecture using a novel Lagrangian Ljusternik--Schnirelman theory, providing new bounds and insights.

Contribution

It introduces the concept of Lagrangian fundamental quantum factorizations and applies them to establish lower bounds on intersection counts, extending previous results.

Findings

Established lower bounds for Lagrangian intersections in classical examples.

Proved a degenerate homological Arnol'd conjecture beyond previous cases.

Applied the theory to study intersections of monotone Lagrangians with Hamiltonian images.

Abstract

We prove a degenerate homological Arnol'd conjecture on Lagrangian intersections beyond the case studied by A. Floer and H. Hofer via a new version of Lagrangian Ljusternik--Schnirelman theory. We introduce the notion of (Lagrangian) fundamental quantum factorizations and use them to give some uniform lower bounds of the numbers of Lagrangian intersections for some classical examples including Clifford tori in complex projective spaces. Additionally, we use the Lagrangian Ljusternik-Schnirelman theory to study the size of the intersection of a monotone Lagrangian with its image of a Hamiltonian diffeomorphism.