A NOTE ON THURSTON-WINKELNKEMPER'S CONSTRUCTION OF CONTACT FORMS ON 3-MANIFOLDS
A NOTE ON THURSTON-WINKELNKEMPER'S CONSTRUCTION OF CONTACT FORMS ON 3-MANIFOLDS
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关于 THURSTON-WINKELNKEMPER 在 3 歧管上构造接触形式的说明
DOI：
10.18910/5305
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发表时间：
2002
影响因子：
0.4
通讯作者：
A. Mori
中科院分区：
文献类型：
--
作者：
A. Mori
封闭定向3流形3上的接触结构是其上的完全不可积平面场。对于平面场的各同伦类，首先用Martinet[14]证明了3上接触结构的存在性，然后用Lutz[12]、[13]证明了3上接触结构的存在性。以α∧α > 0或< 0的1型α为核，给出了一种共取向接触结构，分别称为正接触形式或负接触形式。Thurston和Winkelnkemper以一种优雅的方式从Alexander关于开卷分解的定理推导出了接触形式的存在。Eliashberg[2]表明，大多数这些接触结构，然而，过于灵活的几何兴趣（见§2）。另一方面，他完整地描述了与多变量复分析密切相关的更具有历史意义的例子，即紧实Stein曲面的严格伪凸边界（[4]）。辛可填充性是其重要的推广之一。最近Loi和Piergallini[11]用Gompf的方法[8]将Eliashberg关于紧凑Stein曲面的描述翻译成Lefschetz纤维化的语言。他们还表明，当且仅当3允许其单行映射是右手dehn -扭转的组合的开卷分解时，3可以作为Stein曲面的边界实现。在本文中，我们改进了Thurston-Winkelnkemper的构造，使我们在给定开卷分解的单映射是沿互不相交曲线的右手dehn -扭转的组合的情况下，得到了一个辛可填充的接触结构（定理2）。我们的构造的一个有趣的副产品是§4中的定理3，它给出了辛可填充接触结构的变形为具有Reeb分量的叶理。请注意，含有Reeb成分的叶理本身是不可辛填充的。我要感谢裁判注意到定理3的重要性。
A contact structure on a closed oriented 3-manifold 3 is a completely nonintegrable plane field on it. The existence of a contact structure on 3 was first proved by Martinet [14] and then Lutz [12], [13] for each homotopy class of plane fields. A co-orientable contact structure is given as the kernel of a 1-form α with α ∧ α > 0 or < 0, which is called a positive or negative contact form respectively. Thurston and Winkelnkemper [17] deduced the existence of a contact form in an elegant way from Alexander’s theorem [1] on open-book decompositions. Eliashberg [2] showed that the most of these contact structures are, however, too flexible for geometrical interest (see §2). On the other hand, he completely characterized more historic examples related closely to the complex analysis in several variables, namely, the strictly pseudo-convex boundary of compact Stein surfaces ([4]). The symplectic fillability is one of its significant generalizations. Recently Loi and Piergallini [11] translated Eliashberg’s characterization of compact Stein surfaces into the language of Lefschetz fibration by using Gompf’s method [8]. They also showed that 3 is realizable as the boundary of a Stein surface if and only if it admits an open-book decomposition whose monodromy map is a composition of right-handed Dehn-twists. In this note, we improve Thurston-Winkelnkemper’s construction so that we obtain a symplectically fillable contact structure in the case where the monodromy map of a given open-book decomposition is a composition of right-handed Dehn-twists along mutually disjoint curves (Theorem 2). An interesting by-product of our construction is Theorem 3 in §4 which gives a deformation of symplectically fillable contact structures into a foliation with a Reeb component. Note that a foliation admitting a Reeb component itself is not symplectically fillable. I would like to thank the referee for noticing the importance of Theorem 3.