Loops in the Hamiltonian group: a survey

Details Loops in the Hamiltonian group: a survey Loops in the Hamiltonian group: a survey

2007-11-26

arxiv.org/abs/0711.4086v2

Mathematics

Symplectic Geometry

24 pages, talk given at AMS Summer 2007 Conference, Snowbird, UT; v2 has very minor revisions

Scientific paper

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).

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