Fiber connectivity and bifurcation diagrams of almost toric integrable systems

We describe the bifurcation diagrams of almost toric integrable Hamiltonian systems on a four dimensional symplectic manifold M , not necessarily compact. We prove that, under a weak assumption, the connectivity of the fibers of the induced singular Lagrangian fibration M → R can be detected from the bifurcation diagram alone. In this case, it is possible to give a detailed description of the image of the fibration.