Study on contac structures and foliations on 3 and 4 dimensional manifolds

3 维和 4 维流形上的接触结构和叶状结构研究

• 批准号: 13440026
• 负责人: MITSUMATSU Yoshihiko
• 金额: $ 5.5万
• 依托单位: Chuo University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-13440026/
• 关键词: contact structures; foliations; symplectic structures; bi-contact structures; Ansov flows; foliated cohomology; asymptotic linking; Stein surface; Asymptotic linking; 擬正則曲線; 概複素構造; 射影的Anosov流

Based on the notion of asymptotic linking, the head investigator proposed the framework where the study on contact structures and that on foliations would be unified, and began the research. Op the side of foliations, it turned out that exotic classes and the 1st foliated cohomology are strongly related to this framework. On the other side, the torsion invariant of contact structures has a deep relation with it. For algebraic Anosov foliations, we also established the computation of its 1st foliated cohomology and found its relation to local orbit rigidity.A research group including Miyoshi and Mitsumatsu investigated Thurston's inequality for foliations on the boundary of compact Stein surfaces and established the absolute version of the inequality for certain cases. The relative ersion and more general case are left for the future as important subject. Tsuboi and Mitsumatsu worked on the perfectness of groups of diffeomorphisms preservein certain geometric structures. Especially Tsuboi provednthe perfectness for contact diffeomorphisms and analytic diffeomorphisms of certain manifolds. Tsuboi also classified regular bi-contact structures on Seifert fibered spaces.Another group including Ono and Ohta, mainly working on contact/symlectic topology, characterized the symplectic diffeo-types of the filling of the link of simple and hyper-elliptic singularities. Also they got started the construction of obstruction theory for Lagrangian Floer homology theory.

基于渐近联系的概念，首席研究员提出了将接触构造研究与叶理研究统一起来的框架，并开始了研究。在叶理方面，我们发现奇异类和第一叶理上同调都与这个框架密切相关。对于代数Anosov叶理，我们也建立了它的一阶叶上同调的计算，并发现了它与局部轨道刚性的关系。Miyoshi和Mitsumatsu等研究小组研究了紧致Stein曲面边界上的叶理的Thurston不等式，并在一定情况下建立了该不等式的绝对形式。相对的版本和更一般的情况作为重要的课题留给未来。坪井和光松工作的完善性群体的contimorphisms contimorphisms静脉某些几何结构。特别是Tsuboi证明了某些流形的切触同态和解析同态的完备性。坪井还分类经常双接触结构的塞弗特纤维空间。另一组包括小野和太田，主要工作接触/辛拓扑，特点是辛的双接触型填充的联系简单和超椭圆奇点。他们还开始了拉格朗日-弗洛尔同调理论的障碍理论的建立。

项目成果

佐藤 肇: "2階偏微分方程式系の接触幾何学"数学. 55. 155-165 (2003)

Hajime Sato：“二阶偏微分方程系统的接触几何” 数学 55. 155-165 (2003)

Kaoru ONO: "Simple singularities and symplectic fillings"Contemporary Mathematics. 391. 195-197 (2002)

小野薰：“简单奇点和辛填充”当代数学。

Yusuke HAGIWARA, Tadayoshi MIZUTANI: "Leibniz Algebras Associated with Foliations"Koda : Mathematical Journal. Vol.25No.2. 151-165 (2002)

Yusuke HAGIWARA，Tadayoshi MIZUTANI：“莱布尼兹代数与叶状结构”Koda：数学杂志。

Hitoshi MORIYOSHI: "Operator algebras and the index theorem on foliated manifolds"Proceedings of Foliations : Geometry and Dynamics. 127-155 (2002)

Hitoshi MORIYOSHI：“算子代数和叶流形上的指数定理”叶状流形论文集：几何与动力学。

Takashi TSUBOI: "Regular projectively Anosov flows on the Seifert fibered spaces"Journal of the Mathematical Society of Japan. (to appear).

Takashi TSUBOI：“Seifert 纤维空间上的正则投影阿诺索夫流”日本数学会杂志。