A C^0 counterexample to the Arnold conjecture

TL;DR

This paper constructs a counterexample to the Arnold conjecture in dimensions four and higher, showing that Hamiltonian homeomorphisms can have fewer fixed points than predicted, challenging previous assumptions.

Contribution

It provides the first known counterexample to the Arnold conjecture for Hamiltonian homeomorphisms in higher dimensions.

Findings

Existence of Hamiltonian homeomorphisms with a single fixed point in dimensions ≥4

Counterexample disproves the conjecture in higher dimensions

Challenges previous results for symplectic manifolds

Abstract

The Arnold conjecture states that a Hamiltonian diffeomorphism of a closed and connected symplectic manifold must have at least as many fixed points as the minimal number of critical points of a smooth function on the manifold. It is well known that the Arnold conjecture holds for Hamiltonian homeomorphisms of closed symplectic surfaces. The goal of this paper is to provide a counterexample to the Arnold conjecture for Hamiltonian homeomorphisms in dimensions four and higher. More precisely, we prove that every closed and connected symplectic manifold of dimension at least four admits a Hamiltonian homeomorphism with a single fixed point.