Contact manifolds and Heegaard Floer homology

该项目致力于深化低维拓扑中两个看似不同的子域之间的联系，即接触拓扑和Heegaard Floer同调。Heegaard Floer同调中的Legendrian纽结的不变量给出了纽结理论中新现象的几个首次证明。在这个项目中，PI希望了解已定义的Legendrian纽结不变量之间的关系，了解这些不变量如何与协调的Legendrian纽结相关，并将这些不变量与其他工具一起使用来区分并有望对广泛类别的纽结的Legendrian表示进行分类。接触几何为证明Heegaard Floer同调的结果提供了一种工具。作为拟议项目的一部分，PI希望使用接触几何的工具来给出包含不可压缩曲面的流形的Heegaard Floer同调的秩下界。低维拓扑描述了我们的物理世界；我们生活的空间是三维流形，而时空是四维流形。提出的项目旨在加深低维拓扑的两个子场之间的联系，这两个子场以不同的方式受到物理的启发。第一种是接触流形理论，它作为运动物体的相空间非常自然地产生。另一种是Heegaard Floer同调，它是通过将一个约束函数的配置空间结合到一个对象上，并计算该空间的代数拓扑不变量而得到的。

The proposed project is devoted to deepen the connection of two seemingly different subfields of low dimensional topology, namely contact topology and Heegaard Floer homologies. The invariants of Legendrian knots in Heegaard Floer homology gave several first proofs of new phenomenons in knot theory. During this project the PI would like to understand relations between the already defined knot invariants for Legendrian knots, see how these invariants are related for concordant Legendrian knots and use these invariants together with other tools to distinguish and hopefully classify Legendrian representations of a wide class of knots. Contact geometry provided a tool for proving results in Heegaard Floer homology. As part of the proposed projects the PI would like to use tools from contact geometry to give a lower bound for the rank of Heegaard Floer homology for a manifold that contains an incompressible surface.Low dimensional topology describes our physical world; the space we live in is a 3-dimensional manifold while space-time is a 4-dimensional manifold. The proposed project is aimed to deepen the connection between two subfields of low dimensional topology that are inspired by physics in different ways. The first is the theory of contact manifolds, which arise quite naturally as phase spaces of moving objects. The other is Heegaard Floer homology which is obtained by associating to an object a configuration space of constrained functions and computing the space's algebro-topological invariants.