The Arnold conjecture for singular symplectic manifolds

TL;DR

This paper proves the Arnold conjecture for a broad class of singular symplectic manifolds, introducing new techniques to relate singular and smooth structures, and explores Floer homology in this context.

Contribution

It introduces novel methods to associate smooth symplectic forms to singular structures, proving the Arnold conjecture for $b^m$-symplectic manifolds and extending Floer homology to singular settings.

Findings

Lower bounds on 1-periodic Hamiltonian orbits depending on topology

Validation of Arnold conjecture for $b^{2m}$-symplectic manifolds

Enhanced lower bounds for $b^m$-symplectic surfaces

Abstract

In this article, we study the Hamiltonian dynamics on singular symplectic manifolds and prove the Arnold conjecture for a large class of $b^m$-symplectic manifolds. Novel techniques are introduced to associate smooth symplectic forms to the original singular symplectic structure, under some mild conditions. These techniques yield the validity of the Arnold conjecture for singular symplectic manifolds across multiple scenarios. More precisely, we prove a lower bound on the number 1-periodic Hamiltonian orbits for $b^{2m}$-symplectic manifolds depending only on the topology of the manifold. Moreover, for $b^m$-symplectic surfaces, we improve the lower bound depending on the topology of the pair $(M,Z)$. We then venture into the study of Floer homology to this singular realm and we conclude with a list of open questions.