三维流形Heegaard曲面的稳定化距离与拓扑指数

The theory of Heegaard splittings of 3-manifolds is one of important parts of low dimensional topology. In this research project, we will use a new tool called "stability distance" combined with Hempel's Heegaard distance to discuss strongly irreducible Heegaard splittings; and will use "topological index theory" which was introduced by Bachman combined with the "untelescoping-amalgamation method" to discuss weakly reducible Heegaard splittings. It includes: We will use both stability distance and Hempel's distance to describe the complexity of Heegaard splittings and to discuss the properties of 3-manifods (and their splittings); We will detect the change of stability distance after Dehn surgery which keep Heegaard structure, then show how such surgery affects 3-manifolds. We will construct topologically minimal surfaces by amalgamation and try to find their topological indice; We will also study the relation between Hempel distance of the splittings of the factors and the topologically minimal property of amalgamated Heegaard surfaces. The results of this research are significant to describe properties of 3-manifolds and promote the solution of the problems on the classification of 3-manifolds.

三维流形的Heegaard分解理论是低维拓扑的重要研究领域之一。本项目利用新工具“稳定化距离”结合Hempel的Heegaard距离来研究强不可约的Heegaard分解；利用Bachman提出的“拓扑指数理论”结合“细化-融合方法”来研究弱可约的Heegaard分解。内容包括：通过稳定化距离与Hempel距离共同刻画分解的复杂度并反映流形（以及分解）的性质；利用稳定化距离的改变来研究保持Heegaard结构的Dehn手术对流形的影响；利用融合的方法构造拓扑极小曲面并确定其拓扑指数；研究因子Heegaard分解的Hempel距离与融合Heegaard分解的拓扑极小性质的联系等。 研究结果对于刻画三维流形的性质有重要意义，从而推动三维流形分类问题的解决。