辫Floer同调及其推广

Braid Floer homology is a new theory which is introduced several years ago. It is a topological invariant of braids on unit disc. How to generalize braid Floer homology and study its property is an impotant question in low dimensional topology and symplectic geometry. In this project, we will focus on braid on closed surface and study its property, investigate the moduli space of braid on closed surface, study the compactness of the moduli space, then define braid Floer homology of closed surface. Furthermore, we can find its influence on Hamiltonian dynamics on closed surface and Poincare-Hopf theorem of braid Floer homology of closed surface.

辫Floer同调是最近几年提出的一种新的Floer同调，它是圆盘上辫的拓扑不变量。如何推广辫Floer同调的构造并且研究它的的性质，是低维拓扑和辛拓扑中的一个重要课题。在本项目中，我们打算研究闭曲面上的辫以及辫的性质，考察闭曲面上辫的模空间及其紧致性，进而将辫Floer同调推广到闭曲面上。在本项目的基础上，可以继续考察闭曲面上辫Floer同调对闭曲面上哈密尔顿周期点的影响，以及相应的Poincare-Hopf定理。