Topological rigidity of Hamiltonian loops and quantum homology

Abstract

This paper studies the question of when a loop $ϕ$ in the group Symp$(M,ω)$ of symplectomorphisms of a symplectic manifold $(M,ω)$ is isotopic to a loop that is generated by a time-dependent Hamiltonian function. (Loops with this property are said to be Hamiltonian.) Our main result is that Hamiltonian loops are rigid in the following sense: if $ϕ$ is Hamiltonian with respect to $ω$, and if $ϕ'$ is a small perturbation of $ϕ$ that preserves another symplectic form $ω'$, then $ϕ'$ is Hamiltonian with respect to $ω'$. This allows us to get some new information on the structure of the flux group, i.e. the image of $π_1(Symp(M,ω))$ under the flux homomorphism. We give a complete proof of our result for some manifolds, and sketch the proof in general. The argument uses methods developed by Seidel for studying properties of Hamiltonian loops via the quantum homology of $M$.