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Title:
Embedded contact homology as a field theory
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Abstract:
We show that embedded contact homology has a "completion", which is a functor from the category whose objects are contact 3-manifolds, and whose morphisms are four-dimensional strong symplectic cobordisms, to the category of modules over a Novikov ring. The construction uses Seiberg-Witten theory. However, unlike in Seiberg-Witten theory, the 3-manifolds may be disconnected, and the functor recovers the Seiberg-Witten invariant of a closed symplectic four-manifold without needing to use a "mixed" invariant. This structure has applications to functoriality of the ECH contact invariant and some obstructions to symplectic embeddings into closed four-manifolds.
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