Orbifold Floer spectral invariants, symmetric product links and Weyl laws

2025/11/29 by Cheuk Yu Mak, Mak, Cheuk Yu, Sobhan Seyfaddini +3
Mathematics · #Geometric and Algebraic Topology #Advanced Combinatorial Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2512.00454

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.

Citations

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Related

• Orbifold Hamiltonian Floer theory for global quotients
• Kodaira-Spencer map, Lagrangian Floer theory and orbifold Jacobian algebras
• Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds
• Seidel Representation for Symplectic Orbifolds
• Chain level Floer theory and Hofer's geometry of the Hamiltonian diffeomorphism group