Embedded contact homology (ECH) is an invariant of three-manifolds due to Hutchings, Sullivan, and Taubes. It uses a contact structure on a three-manifold to produce an invariant of the underlying topological manifold. The invariant is the homology of a chain complex generated by certain closed orbits of the Reeb vector field (of interest in classical dynamics), whose differential counts certain holomorphic curves in the symplectization of the contact three-manifold. Few nontrivial examples of ECH have been computed. In this talk, I will give some background and context on ECH and then describe the computation of the ECH of circle bundles over Riemann surfaces, in which the relevant holomorphic curves are actually meromorphic sections of complex line bundles.

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Last updated: 24 Mar 2023