Loops in the Hamiltonian group: a survey

TL;DR

This survey reviews recent advances in understanding the homotopy properties of Hamiltonian loops across various symplectic manifolds, highlighting conditions for non-contractibility, examples of non-preserving loops, and infinite Hofer diameter.

Contribution

It compiles recent results on Hamiltonian loop homotopy properties, introduces new conditions for non-contractibility, and discusses the use of classical and quantum methods in symplectic topology.

Findings

Certain circle actions do not contract in the Hamiltonian group.

Existence of loops that do not preserve any symplectic form.

Hamiltonian group can have infinite Hofer diameter under new conditions.

Abstract

This note describes some recent results about the homotopy properties of Hamiltonian loops in various manifolds, including toric manifolds and one point blow ups. We describe conditions under which a circle action does not contract in the Hamiltonian group, and construct an example of a loop $\ga$ of diffeomorphisms of a symplectic manifold M with the property that none of the loops smoothly isotopic to $\ga$ preserve any symplectic form on M. We also discuss some new conditions under which the Hamiltonian group has infinite Hofer diameter. Some of the methods used are classical (Weinstein's action homomorphism and volume calculations), while others use quantum methods (the Seidel representation and spectral invariants).