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Gluing pseudoholomorphic curves along branched covered cylinders. I. (English) Zbl 1157.53047

Summary: This paper and its sequel prove a generalization of the usual gluing theorem for two index 1 pseudoholomorphic curves \({\mathbf u}_+\) and \({\mathbf u}_-\) in the symplectization of a contact 3-manifold. We assume that for each embedded Reeb orbit \(\gamma\), the total multiplicity of the negative ends of \({\mathbf u}_+\) at covers of \(\gamma\) agrees with the total multiplicity of the positive ends of \({\mathbf u}_-\) at covers of \(\gamma\). However, unlike in the usual gluing story, here the individual multiplicities are allowed to differ. In this situation, one can often glue \({\mathbf u}_+\) and \({\mathbf u}_-\) to an index 2 curve by inserting genus zero branched covers of \(\mathbb R\)-invariant cylinders between them. We establish a combinatorial formula for the signed count of such gluings. As an application, we deduce that the differential 8 in embedded contact homology satisfies a \(\partial^2 = 0\).
This paper explains the more algebraic aspects of the story, and proves the above formulas using some analytical results from part II [Preprint, arXiv:0705.2074].

MSC:

53D35 Global theory of symplectic and contact manifolds
57R17 Symplectic and contact topology in high or arbitrary dimension
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