三维流形加柄的性质研究

The Heegaard genus is an elementary 3-manifold invariant. The research of Heegaard genus is an important topic in 3-manifold theory. With the development of curve complex, many research achievements have been made in the past few decades. But there are still many basic important questions to be solved. The main object in this project is studying the affections of handle additions to Heegaard genera of 3-manifolds. The research contents are as following: the sufficient condition, necessary condition, and sufficient and necessary condition for the handle addition to be genus non-degenerating; the existence and boundedness of genus degenerating handle additions; the condition for the genus degeneration of a handle addition to be arbitrarily large. The project uses the subsurface projection to generalize the Heegaard distance, and then finds the key point of the affections of handle additions to the genera. The results of this project will conduce to the study of handle additions to 3-manifolds, and have theoretic meaning for the study of 3-manifolds with Heegaard splittings.

三维流形的亏格（Heegaard亏格）是三维流形的一个基本不变量，其研究一直是三维流形拓扑学的一个重要课题。近年来，随着曲线复形等工具的运用，对于三维流形亏格及相关问题的研究不断取得重要突破，很多困难的问题相继得到解决，取得了丰硕的研究成果。但该领域仍有大量基本并且重要的问题有待解决。众所周知，三维流形的亏格在加柄操作下不增加。本项目主要研究三维流形的亏格在加柄操作下何时不变（非退化），以及减少（退化）时影响减量的因素。研究内容包括：给出亏格非退化加柄的充分条件、必要条件和充分必要条件，给出亏格退化加柄的存在性、有界性的特征刻画，确定亏格退化可任意大的条件。主要研究方法是利用曲线复形中子曲面投影对Heegaard分解距离进行细化，进而找出加柄操作对亏格影响的关键要素。研究结果对于深入透彻地了解加柄操作对三维流形及性质的改变以及通过Heegaard分解来了解三维流形有重要的理论意义。

三维流形的亏格是一种十分重要的不变量，加柄操作是构造三维流形的一种基本方法，流形的亏格以及加柄操作下流形性质的研究一直是低维拓扑领域十分重要的研究课题。本项目主要研究了加柄操作对于三维流形亏格的影响。利用曲线复形中的子曲面投影，找到了弱可约不可约Heegaard分解距离退化的加柄有界的充分条件；给出了Heegaard分解是keen且实现Heegaard距离的测地线具备局部唯一性的充分条件；给出了带边流形沿边界上本质子曲面自融合unstabilized且uncritical的充分条件。