Gluing pseudoholomorphic curves along branched covered cylinders II

By Michael Hutchings, Clifford H. Taubes. 2007

155 citations recorded by the source corpus.

This paper and its prequel (“Part I”) 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. This paper shows that the signed count of such gluings equals a signed count of zeroes of a certain section of an obstruction bundle over the moduli space of branched covers of the cylinder. Part I obtained a combinatorial formula for the latter count and, assuming the result of the present paper, deduced that the differential ∂ in embedded contact homology satisfies ∂2 = 0. The present paper completes all of the analysis that was needed in Part I. The gluing technique explained here is in principle applicable to more gluing problems. We also prove some lemmas concerning the generic behavior of pseudoholomorphic curves in symplectizations, which may be of independent interest.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bee89e3643dd776a2bf96389e72a6cda8ff47a65

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Citations in the corpus

Relative asymptotic behavior of pseudoholomorphic half‐cylinders (2007, unknown): “1”
Sutures and contact homology I (2010, methodology): “[T3] C. Taubes,The structure of pseudoholomorphic subvarieties for degenrate almost complex structure and symplectic form onS1 ×B3, Geom.”
The embedding capacity of 4-dimensional symplectic ellipsoids (2009, methodology): “The expected functorial properties of embedded contact homology should imply that cECH(a) ≤ c(a) for all a; see Hutchings–Taubes [11], as well as Remark 2.4.5 (ii).”
Torsion contact forms in three dimensions have two or infinitely many Reeb orbits (2017, unknown): “It is shown in [15, Section 5.3] that @ J is well deﬁned, and in [18, Theorem 7.20] that @ 2J D 0 .”
Embedded contact homology and Seiberg–Witten Floer cohomology II (2008, methodology): “ Hutching uses constructions of Bourgeois and Mohnke [BM] to associate a sign, ±1, to each component of M 1 (Θ - , Θ + ). The full details of this are given in Section 9 and especially Section 9.5 of [HT2]. Let σ(Θ - , Θ + ) denote the sum of these signs when M 1 (Θ - , Θ + ) is non-empty, and 0 oth”