Orbifold Floer spectral invariants, symmetric product links and Weyl laws

TL;DR

This paper introduces a spectral invariant-based strategy using orbifold Floer cohomology on symmetric products to prove the smooth closing lemma for Hamiltonian diffeomorphisms, with applications to area-preserving maps on the 2-sphere.

Contribution

It develops a novel approach linking orbifold quantum cohomology idempotents to Lagrangian links, providing a new proof of the smooth closing lemma in certain symplectic settings.

Findings

Provides a new proof of the smooth closing lemma for area-preserving diffeomorphisms of the 2-sphere.

Establishes a framework connecting orbifold Floer cohomology to the existence of Lagrangian links.

Highlights open problems in constructing Lagrangian links in higher dimensions.

Abstract

We explain a strategy, based on spectral invariants on symmetric product orbifolds, for proving the smooth closing lemma for Hamiltonian diffeomorphisms of a symplectic manifold when the orbifold quantum cohomologies of its symmetric products possess suitable idempotents. We relate the existence of such idempotents to the manifold containing a sequence of Lagrangian links, whose number of components tends to infinity, satisfying a number of properties. Orbifold Floer cohomology for global quotient orbifolds is used axiomatically, and is constructed in a companion paper. We illustrate this strategy by giving a new proof of the smooth closing lemma for area-preserving diffeomorphisms of the 2-sphere. The construction of suitable Lagrangian links in higher dimensions remains an intriguing open problem.