把柄添加对Heegaard距离的影响

The main objects in this project are the Heegaard splittings and the curve complexes of 3-manifolds.We'll study the affection of handle-addition to the Heegaard distance of the resulting manifold.The goal is to reduce the Heegaard distance by adding 2-handles and then construct 3-manifolds with smaller Heegaard distance. We'll also try to find the deeper connections between the curve complex and the topological properties of a 3-manifold.The results of the project will be combined with some classical theorems of the Heegaard spliting, hoping to make some new discoveries. In 2001, Hempel defined the Heegaard distance by relating the geometric and combinatorial properties of the curve complex with topological properties of the manifold. The introduction of Heegaard distance brings a lot breakthrough results on 3-manifolds.Three manifolds with small distance (especially, with distance less than or equal to 1) usually possesses intuitive interpretation.The project uses a combination of some classical theories, such as Heegaard distance,handle-addition theorem, and their internal relationship. The whole process not only just follows the traditional research direction, but also studies the Heegaard distance from a new point of view.The results of this project will give breakthrough results to the study of the Heegaard distance, then break new ground of the study.

本项目主要研究三维流形的Heegaard分解和曲线复形，讨论把柄添加对Heegaaard距离的影响，试图通过添加把柄而降低进而控制Heegaard距离，得到Heegaard距离小的流形，找到曲线复形与3-流形的性质之间深层次的联系，并将此结果与Heegaard分解理论经典结果相结合。2001年Hempel结合曲线复形的几何和组合性质与3-流形的拓扑性质定义了Heegaard距离。Hee-gaard距离的引入带来了很多的3-流形研究上的突破。小距离（尤其是小于等于1）的3-流形具有很好的直观刻画。本项目，结合了传统的Heegaard距离、把柄添加原理等经典理论，找到他们之间的内在联系。研究过程在遵循经典的研究方法的基础之上，从新的视角研究Heegaard距离，研究成果将得到Heeggard距离研究的突破性结果，为Heegaard距离的研究开辟新的途径。