Gluing pseudoholomorphic curves along branched covered cylinders I

Details Gluing pseudoholomorphic curves along branched covered cylinders I Gluing pseudoholomorphic curves along branched covered cylinders I

2007-01-10

arxiv.org/abs/math/0701300v3

Mathematics

Symplectic Geometry

100 pages; slightly revised with new pictures following referee's suggestions; to appear in Journal of Symplectic Geometry

Scientific paper

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 gamma, the total multiplicity of the negative ends of u_+ at covers of gamma agrees with the total multiplicity of the positive ends of 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 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 d in embedded contact homology satisfies d^2=0. This paper explains the more algebraic aspects of the story, and proves the above formulas using some analytical results from part II.

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