A hierarchy of local symplectic filling obstructions for contact $3$-manifolds
A hierarchy of local symplectic filling obstructions for contact $3$-manifolds
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用于接触 $3$ 流形的局部辛填充障碍物的层次结构
DOI：
10.1215/00127094-2348333
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发表时间：
2010
影响因子：
2.5
通讯作者：
C. Wendl
中科院分区：
文献类型：
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作者：
C. Wendl
我们概括了 3 维接触流形中过度扭曲和吉鲁扭转的熟悉概念，定义了称为平面扭转的局部填充障碍物的无限层次结构，其整数值阶 $k \ge 0$ 可以解释为测量接触流形“紧密度”的等级。我们特别表明，任何具有平面扭转的接触流形不允许接触类型嵌入到任何闭合辛 4-流形中，并且在嵌入接触同调中具有消失的接触不变量，并且我们给出了对于任何 $k \ge 2$ 具有平面 k 扭转但没有 Giroux 扭转的接触流形的示例。我们还表明，支撑打开的书的装订的互补永远不会有平面扭转。背景中的统一思想是根据打开的书籍沿其装订的接触纤维和来分解接触流形。作为这些结果的技术基础，我们建立了放大求和打开书中某些类J全纯曲线的存在性、唯一性和紧性定理；这些也意味着平面性的代数障碍和部分平面域的嵌入。结果进一步应用于弱辛填充（arXiv：1003.3923，与 K. Niederkrueger 联合）、非精确辛配边（arXiv：1008.2456）和辛场论（arXiv：1009.3262，与 J. Latschev 联合）的后续论文中。
We generalize the familiar notions of overtwistedness and Giroux torsion in 3-dimensional contact manifolds, defining an infinite hierarchy of local filling obstructions called planar torsion, whose integer-valued order $k \ge 0$ can be interpreted as measuring a gradation in "degrees of tightness" of contact manifolds. We show in particular that any contact manifold with planar torsion admits no contact type embeddings into any closed symplectic 4-manifold, and has vanishing contact invariant in Embedded Contact Homology, and we give examples of contact manifolds that have planar k-torsion for any $k \ge 2$ but no Giroux torsion. We also show that the complement of the binding of a supporting open book never has planar torsion. The unifying idea in the background is a decomposition of contact manifolds in terms of contact fiber sums of open books along their binding. As the technical basis of these results, we establish existence, uniqueness and compactness theorems for certain classes of J-holomorphic curves in blown up summed open books; these also imply algebraic obstructions to planarity and embeddings of partially planar domains. The results are applied further in followup papers on weak symplectic fillings (arXiv:1003.3923, joint with K. Niederkrueger), non-exact symplectic cobordisms (arXiv:1008.2456) and Symplectic Field Theory (arXiv:1009.3262, joint with J. Latschev).