曲线复形是低维拓扑研究的重要对象. 近些年, 曲线复形的一些研究结果被广泛应用于低维流形的研究, 例如曲面映射类群, 三维流形的Heegaard 分解以及双曲三维流形. .由 Thurston几何化定理及特殊类型三维流形的 Heegaard 分解的结果可知, 有一个距离至少为3的 Heegaard 分解的三维流形是双曲的. 自然地, 拓扑学家关心具有距离至多为2的 Heegaard 分解的三维流形的几何与拓扑性质. 此外三维流形的基本群是低维流形的重要研究对象之一. 因此拓扑学家也关心如何从 Heegaard 分解读出流形基本群的一些性质..本项目主要研究以下问题: 构造与比较曲线复形的一系列度量; 刻画具有距离不大于2的 Heegaard 分解的三维流形几何及拓扑性质; 研究 Heegaard分解的一般条件使得三维流形基本群的秩等于其最小Heegaard亏格.

As an important subject in low dimensional topology, the curve complex is widely used in study of low dimensional manifolds, such as mapping class groups, Heegaard splitting and hyperbolic 3-manifolds.. By Thurston's Geometrization conjecture, fully proved by Perelman, and some results of Heegaard splittings of some special 3-manifolds, a 3-manifold admitting a distance at least 3 Heegaard splitting is hyperbolic. Naturally, topologists concern about the topological and geometrical properties of a 3-manifold admitting a distance at most 2 Heegaard splitting. In addition, there is one presentation of the fundamental group, as one important subject in low dimensional topology, of a 3-manifold from a Heegaard splitting. Thus it is interesting to know that how to read out some algebraic properties of the fundamental group of a 3-manifold from a Heegaard splitting.. We focus on the following problems: (1) constructing some metrics on the curve complex and making some comparison between all those metrics ; (2) making some classifications of 3-manifolds admitting distance at most 2 Heegaard splitting; (3) try to give some general and sufficient condition from Heegaard splitting so that the rank of fundamental group equals the minimal Heegaard genus of a 3-manifold.