Topology of integrable systems on 4-manifolds

We will give a survey on the topology of integrable Hamiltonian systems on 4-manifolds. Open questions and problems will be also discussed. Recall that, from a topological point of view, an integrable Hamiltonian system can be treated as a singular Lagrangian fibration on a smooth symplectic 2n-manifold whose generic fibres are n-dimensional tori. By a singularity, we mean either a singular point or a singular fibre of the fibration. The topological structure of such singularities is very important for understanding the dynamics of integrable systems both globally and locally. Our goal is to describe topological invariants of such singularities and obtain their classification up to fibrewise homeomorphism (for time being we forget about symplectic structure). The next step is to combine these singularities together to study the global structure of the fibration. For many integrable systems, this structure is completely determined by topological properties of singularities.