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Monday, November 12, 2018

**Abstract:** A compact four dimensional completely integrable system $f \colon M \to \mathbb{R}^2$ 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 well studied and have many nice properties; for example, the fibers $f^{-1}(x)$ are connected. Unfortunately, although there are many interesting examples of semitoric systems, the class has some limitations. For example, there are blowups of $S^2 \times S^2$ with Hamiltonian circle actions that cannot be extended to semitoric system. We show that, by allowing certain degenerate singularities, we can expand the class of semitonic systems but still prove that $f^{-1}(x)$ is connected. 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.