三维流形的Heegaard亏格及不同的Heegaard分解

We will study Heegaard splitting on 3-manifold and unknotting number of knot fron the view of Heegaard splitting. We will mainly pay attention to the following questions:.1. Let M be a closed Haken 3-manifold. Are there non-equivalent unstabilized Heegaard splittings on M? It have been conjectured that most of Haken closed 3-manifolds admit non-equivalent unstabilized Heegaard splittings for many years. For example, see Problem 3.85 in “Kirby, Problems in low dimensional topology”..2. Let M be a closed Haken 3-manifold. How can we determine the Heegaard genus of M? It is an importtant and still open question in Heegaard theory though if there are many part results ..3. Does the unuknotting number of the connected sum of two knots equal to the sum of the unknotting numbers of the two knots?

我们将研究三维流形的Heegaard分解, 并从Heegaard分解的角度去研究纽结的解结数。我们将主要关注以下一些问题： .1. 设M是一个闭Haken流形， 是否M上有互不等价的不可稳定化的Heegaard分解？ 人们猜测绝大多数Haken流形上有互不合痕的不可稳定化的Heegaard分解， 参见Kirby的“Problems in low dimensional topology”一文中问题3.85。.2. 设M是一个闭Haken流形，如何去确定M的Heegaard亏格？尽管已有部分结论，但这个问题依然是Heegaard分解理论中重要而公开的问题。.3. 两个纽结连通和的解结数是否等于两个这两个纽结的解结数的和？

（1）我们给出了两个Heegaard分解的融合积是极小Heegaard分解的充分条件。（2）证明了局部大的距离至少为3的Heegaard分解的边界稳定化是不可稳定化的。进而如果一个带边三维流形具有局部大的距离至少为3的Heegaard分解，则它具有至少两个不同的Heegaard结构。（3）如果一个闭三维流形具有局部大的距离为2的Heegaard分解，则这个流形是双曲的或是一个极小seifert流形与一个双曲流形的融合积。（4）如果一个闭三维流形具有局部大的距离大于1的Heegaard分解，则这个流形的映射类群是有限的。