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

在渐近连接概念的基础上，提出了将接触结构研究与叶状结构研究统一起来的框架，并开始了研究。在叶理方面，证明了外来类和第一叶理上同调与这个框架密切相关。另一方面，接触结构的扭转不变量与此有很深的关系。对于代数Anosov叶理，我们也建立了它的第一叶理上同调的计算，并找到了它与局部轨道刚性的关系。包括Miyoshi和Mitsumatsu在内的一个研究小组研究了紧化Stein曲面边界上叶形的Thurston不等式，并在某些情况下建立了该不等式的绝对版本。相对版本和更一般的情况留给未来作为重要课题。Tsuboi和Mitsumatsu研究了差分同态群保持某些几何结构的完备性。特别是Tsuboi证明了某些流形的接触微分同态和解析微分同态的完备性。Tsuboi还在Seifert纤维空间上分类了规则的双接触结构。另一组包括Ono和Ohta，主要研究接触/对称拓扑，表征了简单和超椭圆奇点连接填充的辛不同类型。他们还开始了拉格朗日花同调理论中阻碍理论的建构。

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.