Hamiltonian classification of toric fibres and symmetric probes

Hamiltonian classification of toric fibres and symmetric probes JO BRENDELIn a toric symplectic manifold, regular fibres of the moment map are Lagrangian tori which are called toric fibres.We discuss the question of which two toric fibres are equivalent up to a Hamiltonian diffeomorphism of the ambient space.On the construction side of this question, we introduce a new method of constructing equivalences of toric fibres by using a symmetric version of McDuff's probes.On the other hand, we derive some obstructions to such equivalence by using Chekanov's classification of product tori together with a lifting trick from toric geometry.Furthermore, we conjecture that (iterated) symmetric probes yield all possible equivalences and prove this conjecture forThis problem is intimately related to determining the Hamiltonian monodromy group of toric fibres, ie determining which automorphisms of the homology of the toric fibre can be realized by a Hamiltonian diffeomorphism mapping the toric fibre in question to itself.For the above list of examples, we determine the Hamiltonian monodromy group for all toric fibres.