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Research on foliations, contact structures and Euler class

叶状结构、接触结构和欧拉级的研究

基本信息

批准号：
18540095
负责人：
MIYOSHI Shigeaki
金额：
$ 1.5万
依托单位：
Chuo University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2006
资助国家：
日本
起止时间：
2006 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-18540095/
关键词：
foliations Thurston norm open book decomposition contact structures confoliation

项目摘要

W. Thurston showed that a foliation on a 3-manifold which has no Reeb component enjoys the property that the Euler class of the tangent bundle satisfies an inequality, Thurston's inequality. On the other hand, the Reeb foliation on the three sphere satisfies Thurston's inequality and a previous research followed by this research showed that there is a class of foliations each of which has Reeb components and satisfi es Thurston's inequality.In the research in 2006, for a class of foliations which are called spinnable foliations, we obtained a sufficient condition for the foliation satisfying Thurston's inequality. Moreover, we revealed an aspect where Thurston's inequality does not hold. They are described by properties of the monodromy diffeomorphisms which determine the spinnable foliationsIn view of the research with respect to the convergence of contact structures to foliations, we studied a finer inequality, the relative version of Thurston's inequality, which deepens the research until 2006. In fact, for spinnable foliations we showed that the relative version implies the absolute version. The same statement for contact structures was known however, it does not hold in general for foliations. Also in 2007, we found the method to construct a foliation which satisfies Thurston's inequality with Euler class of infinite order. Until then, all foliations which satisfies Thurston's inequality have trivial Euler class. Indeed, we can find'a spinnable foliation whose Euler class is of infinite order by the research in 2006. Then we can perform Dehn surgery along the Reeb component and with certain condition on the original spinnable foliation we can conclude that with finitely many exceptions the resultant satisfies Thurston's inequality with Euler class of infinite order by virtue of D. Gabai's sutured manifold theory.

W. Thurston证明了一个在没有Reeb分量的3-流形上的叶状体具有这样的性质：切丛的欧拉类满足一个不等式，即Thurston不等式。另一方面，三球面上的Reeb叶理满足Thurston不等式，并且在2006年的研究中，我们得到了一类具有Reeb分支的叶理满足Thurston不等式的充分条件.此外，我们还揭示了瑟斯顿不等式不成立的一个方面。它们被描述为确定可纺叶理的单值单同态的性质。鉴于接触结构收敛到叶理的研究，我们研究了一个更精细的不等式，Thurston不等式的相对版本，直到2006年，深化了研究。事实上，对于可纺叶理，我们证明了相对版本意味着绝对版本。对于接触构造，同样的陈述是已知的，然而，它一般不适用于叶理。同样在2007年，我们找到了用无限阶Euler类构造满足Thurston不等式的叶层的方法。在此之前，所有满足Thurston不等式的叶理都有平凡的Euler类。实际上，通过2006年的研究，我们发现了一个欧拉类为无穷阶的可旋叶理。然后我们可以沿Reeb分支作沿着Dehn手术，并在原可纺叶理的一定条件下，利用D. Gabai的缝合流形理论