The embedded contact homology index revisited
The embedded contact homology index revisited
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重新审视嵌入式接触同源索引
DOI：
10.1090/crmp/049/10
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发表时间：
2008
期刊：
影响因子：
--
通讯作者：
M. Hutchings
中科院分区：
文献类型：
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作者：
M. Hutchings
设Y是具有接触形式的封闭定向3-流形，使得所有Reeb轨道都是非退化的。嵌入接触同调（ECH）指数关联一个整数到每个相对的2维同调类的表面，其边界是两个联盟的Reeb轨道之间的差异。这个整数决定了ECH上的相对分次; ECH微分计数Y的辛化中相对同调类的ECH指数为1的全纯曲线。一个已知的指数不等式意味着这样的曲线（大部分）是嵌入的，并满足一些额外的约束。本文证明了关于ECH指标的四个新结果。首先，我们将ECH上的相对分次细化为绝对分次，它将Y上的定向2-平面场的同伦类与Reeb轨道的每个并集相关联。其次，我们将ECH指标不等式推广到具有Hamilton结构的三流形之间的辛配边，并简化了证明。第三，我们建立了全纯曲线的并和重覆盖的ECH指标的一般不等式。最后，我们定义了一个新的相对滤子，或切触3-流形的任何其他类型的切触同调，它类似于ECH指标，并与全纯曲线的Euler特征有关。这并不给新的拓扑不变量，除非可能在特殊情况下，但它是一个有用的计算工具。
Let Y be a closed oriented 3-manifold with a contact form such that all Reeb orbits are nondegenerate. The embedded contact homology (ECH) index associates an integer to each relative 2-dimensional homology class of surfaces whose boundary is the difference between two unions of Reeb orbits. This integer determines the relative grading on ECH; the ECH differential counts holomorphic curves in the symplectization of Y whose relative homology classes have ECH index 1. A known index inequality implies that such curves are (mostly) embedded and satisfy some additional constraints. In this paper we prove four new results about the ECH index. First, we refine the relative grading on ECH to an absolute grading, which associates to each union of Reeb orbits a homotopy class of oriented 2-plane fields on Y. Second, we extend the ECH index inequality to symplectic cobordisms between three-manifolds with Hamiltonian structures, and simplify the proof. Third, we establish general inequalities on the ECH index of unions and multiple covers of holomorphic curves in cobordisms. Finally, we define a new relative filtration on ECH, or any other kind of contact homology of a contact 3-manifold, which is similar to the ECH index and related to the Euler characteristic of holomorphic curves. This does not give new topological invariants except possibly in special situations, but it is a useful computational tool.