m at h . SG ] 1 3 Ju l 2 00 7 Gluing pseudoholomorphic curves along branched covered cylinders I

This paper and its sequel prove a generalization of the usual gluing theorem for two index 1 pseudoholomorphic curves u+ and u− in the symplectization of a contact 3-manifold. We assume that for each embedded Reeb orbit γ, the total multiplicity of the negative ends of u+ at covers of γ agrees with the total multiplicity of the positive ends of u − at covers of γ. However, unlike in the usual gluing story, here the individual multiplicities are allowed to differ. In this situation, one can often glue u+ and u− to an index 2 curve by inserting genus zero branched covers of 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 ∂ in embedded contact homology satisfies ∂ = 0. This paper explains the more algebraic aspects of the story, and proves the above formulas using some analytical results from part II.