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 , the total multiplicity of the negative ends of u + at covers of agrees with the total multiplicity of the positive ends of u -at covers of . 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 2 = 0.