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组件的3个manifold上的叶子享有瑟斯顿的不平等，符合切线束的Euler类满足不平等的特性。另一方面，在这三个领域的reeb展叶子满足了瑟斯顿的不平等和先前的研究，随后进行了这项研究，这表明每种叶子都具有reeb成分，并满足瑟斯顿的不平等。在2006年的研究中，对于一类可支付的叶子，我们可以在一类可支付的叶子中获得一类，以使我们获得了一个足够的条件，以使其达成一定的条件。此外，我们揭示了瑟斯顿不平等不存在的方面。它们是通过单次构造差异性的特性来描述的，这些差异决定了研究对研究的可旋转观点，即在接触结构与叶子的融合方面，研究了较细的不平等，瑟斯顿的相对版本，这是瑟斯顿的不平等现象的相对版本，这使研究加深了研究，直到2006年，我们在2006年都暗示了相对版本。绝对的版本是绝对的。绝对的版本。但是，已知相同的接触结构陈述，它通常不适合叶子。同样在2007年，我们发现了构建叶面的方法，该叶面满足了瑟斯顿的不平等，无限顺序。在此之前，满足瑟斯顿不平等的所有叶子都具有微不足道的欧拉类。的确，我们可以找到一个可旋转的叶面，其欧拉的阶级是2006年的研究。然后，我们可以沿着Reeb的组件进行Dehn手术，并且在某些条件下，我们可以得出结论，我们可以得出结论，在有限的许多例外中，生成的刻有墨西哥人的不平等能够满足Infinite Order of Infinite by Virtie firt virt of D. gabai sut of D. Gabai的效率。

项目成果

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

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

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

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

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