Cut-and-paste contact topology

Bennequin和Eliashberg在20世纪80年代和90年代的开创性工作表明，在所有的接触结构中，紧密的接触结构最能反映环境空间的拓扑结构。只是在过去的十年里，才开发出足够的工具来攻击该主题中的许多基本分类问题。PI建议与本田和Mati‘c合作，使用剪切和粘贴技术分析Haken歧管的接触拓扑。这一研究的重要成果包括在Haken同调球面上构造紧接触结构，将与叶层理论的关系推广到层片和双曲流形，以及理解次表面复合体与乘积上接触结构之间的相互作用。接触和辛拓扑最初是由从事经典力学和热力学的物理学家提出的问题引起的。它们出现在令人惊讶的广泛的应用场合，从流体流动到投影屏幕电视的制造。拟议的研究有可能产生适用于这些领域的新结果。

The pioneering works of Bennequin and Eliashberg from the 1980s and the 1990s have shown that, among all contact structures, the tight ones best reflect the topology of the ambient space. Only in the past decade have sufficient tools been developed to allow many of the basic classification questions in the subject to be attacked. The PI proposes joint work with Honda and Mati\'c to analyze the contact topology of Haken manifolds using cut-and-paste techniques. Important outgrowths of this investigation include constructions of tight contact structures in Haken homology spheres, extensions of the relationship with foliation theory to laminations and hyperbolic manifolds, and an understanding of the interplay between the subsurface complex and contact structures on a product.Three-dimensional manifolds are modeled on the three-dimensional space we live in. Contact and symplectic topology were originally motivated by questions of physicists working in classical mechanics and thermodynamics. They arise in a surprisingly broad array of applied situations ranging from fluid flows to the manufacture of projection screen televisions. The proposed research has the potential to produce new results applicable to these fields.