Research on foliations, contact structures and Euler class

叶状结构、接触结构和欧拉级的研究

• 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

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 表明，没有 Reeb 分量的 3 流形上的叶状结构具有切丛的欧拉级满足不等式（Thurston 不等式）的性质。另一方面，三个球体上的 Reeb 叶状结构满足 Thurston 不等式，并且本研究之前的一项研究表明，存在一类叶状结构，每个叶状结构都具有 Reeb 分量并满足 Thurston 不等式。在 2006 年的研究中，对于一类称为可旋转叶状结构的叶状结构，我们得到了满足瑟斯顿不等式的叶状结构的充分条件。此外，我们还揭示了瑟斯顿不等式不成立的一个方面。它们是通过决定可旋转叶状结构的单向微分同胚的性质来描述的。考虑到接触结构与叶状结构收敛性的研究，我们研究了一种更精细的不等式，即瑟斯顿不等式的相对版本，该研究一直深入到2006年。事实上，对于可旋转的叶状结构，我们表明相对版本意味着绝对版本。对于接触结构的相同说法是已知的，但是，它通常不适用于叶状结构。同样在2007年，我们找到了构造满足Thurston不等式与无限阶欧拉类的叶状结构的方法。直到那时，所有满足瑟斯顿不等式的叶状结构都具有平凡的欧拉类。事实上，通过2006年的研究，我们可以找到欧拉级无限阶的可旋转叶状结构。然后我们可以沿着Reeb分量进行Dehn手术，并且在原始可旋转叶状结构的一定条件下，我们可以得出结论，除了有限的许多例外，凭借D. Gabai的缝合流形理论，结果满足瑟斯顿无限阶欧拉级不等式。

项目成果

Multiplicities and topology of symplectic quotients in tensor product representations

张量积表示中辛商的重数和拓扑

• 发表时间: 2007
• 作者: Tatsuru;TAKAKURA
• 通讯作者: TAKAKURA

2-Dimensional foliations on 4-manifolds and self-intersection of compact leaves

4 流形上的二维叶状结构和紧凑叶的自相交

• 发表时间: 2006
• 作者: Yoshihiko;MITSUMATSU
• 通讯作者: MITSUMATSU

葉層構造に対するThurstonの不等式3

叶状结构的瑟斯顿不等式 3

• 发表时间: 2007
• 作者: Tatsuru;TAKAKURA;三好 重明
• 通讯作者: 三好 重明

Asymptotic linking pairing and foliated cohomology

渐近连接配对和叶状上同调

• 发表时间: 2006
• 作者: Yoshihiko;MITSUMATSU;Yoshihiko MITSUMATSU
• 通讯作者: Yoshihiko MITSUMATSU

On Thurston's inequality for spinnable foliations

关于可旋转叶状体的瑟斯顿不等式

• 发表时间: 2006
• 作者: Tatsuru;TAKAKURA;Yoshihiko MITSUMATSU
• 通讯作者: Yoshihiko MITSUMATSU