分支复盖圆柱II上拟全纯曲线的粘接
Gluing pseudoholomorphic curves along branched covered cylinders II
摘要：
本文及其前篇（“第一部分”）证明了切触3-流形的辛化中两条指标为1的拟全纯曲线U_+和U_-的通常胶合定理的一个推广。我们假设对于每个嵌入的Reeb轨道γ，U_+在γ覆盖上的负端的重数总和U_-在γ覆盖上的正端的重数总和一致.然而，与通常的胶合故事不同，这里允许个体的多重性不同。在这种情况下，通过在U+和U-之间插入R-不变圆柱的亏格零分支覆盖，可以将U+和U-粘到一条指数为2的曲线上。本文证明了在圆柱分支覆盖的模空间上，这类胶合的符号数等于阻塞丛的某个截面上的零的符号数。第一部分给出了后一个计数的组合公式，并在本文结果的假设下，推导出嵌入接触同调中的微分d满足d^2=0。本文件完成了第一部分所需的所有分析。这里解释的胶合技术原则上适用于更多的胶合问题。我们还证明了一些引理的一般行为的伪全纯曲线的辛，这可能是独立的利益。
英文摘要：
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 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. 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 d in embedded contact homology satisfies d^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.
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