A compact four dimensional completely integrable system $f : M \to R^2$ is semitoric if it has only non-degenerate singularities, without hyperbolic blocks, and one of the components of $f$ generates a circle action. Semitoric systems have been extensively studied and have many nice properties: for example, the preimages $f^{-1}(x)$ are all connected. Unfortunately, although there are many interesting examples of semitoric systems, the class has some limitation. For example, there are blowups of $S^2 \times S^2$ with Hamiltonian circle actions which cannot be extended to semitoric systems. We expand the class of semitoric systems by allowing certain degenerate singularities, which we call ephemeral singularities. We prove that the preimage $f^{-1}(x)$ is still connected for this larger class. We hope that this class will be large enough to include not only all compact four manifolds with Hamiltonian circle actions, but more generally all complexity one spaces. Based on joint work with D. Sepe.
