UCLA

In this thesis, we give a proposed construction of cobordism maps in embedded contact homology via a count of $J$-holomorphic curves. We prove a new index formula in the $L$-supersimple setting of Bao-Honda and use it to classify degenerations of $1$-dimensional moduli spaces of curves in exact symplectic cobordisms. We correct these degenerations by using obstruction bundle techniques of Hutchings-Taubes to continue the moduli spaces by gluing in branched covers of trivial cylinders. We then use a new evaluation map to cut out a $1$-dimensional family in the resulting moduli space, modulo two conjectures on the behavior of this map. Finally, we analyze the new endpoints of the continuations and use them to complete our proposed definition of the cobordism maps.