本文考虑具有H_1中的本质圆盘D和H_2中的本质曲面F的三维流形M的Heegaard分裂M=H_1 \cup_S H_2，|D \cap F| =1。(We当H_2是具有非空“负”边界的压缩体时，要求F的边界在S中。设F是具有n个边界分支的亏格g曲面.从S出发，通过沿F切割H_2并将F \times [0，1]连接到H_1上，得到了一个亏格g（S）+2g+n-2 Heegaard分裂M= H '_1 \cup_S ' H '_2.作为应用，利用Casson和Gordon的一个定理，我们给出了具有两个不同亏格的Heegaard分裂的三维流形的例子，其中两个Heegaard分裂之一是强不可约的非极小亏格分裂，它是由另一个Heegaard分裂得到的.
英文摘要：
We consider a Heegaard splitting M=H_1 \cup_S H_2 of a 3-manifold M having an essential disk D in H_1 and an essential surface F in H_2 with |D \cap F|=1. (We require that boundary of F is in S when H_2 is a compressionbody with non-empty "minus" boundary.) Let F be a genus g surface with n boundary components. From S, we obtain a genus g(S)+2g+n-2 Heegaard splitting M=H'_1 \cup_S' H'_2 by cutting H_2 along F and attaching F \times [0,1] to H_1. As an application, by using a theorem due to Casson and Gordon, we give examples of 3-manifolds having two Heegaard splittings of distinct genera where one of the two Heegaard splittings is a strongly irreducible non-minimal genus splitting and it is obtained from the other by the above construction.
A Heegaard splitting of a $3$-manifold is flippable if there is an isotopy that interchanges the two sides of the Heegaard splitting. We explore which Heegaard splittings of Seifert fibered spaces are flippable.
A Heegaard splitting which admits a unique pair of disjoint compression disks on distinct sides is said to be keen weakly reducible. This paper provides an construction of keen weakly reducible Heegaard splittings of arbitrary genus except 2. Furthermore, critical Heegaard splittings may yield if we change some conditions.
A closed orientable 3-manifold with distinct distance three genus two Heegaard splittings
本文给出了一个闭可定向三维流形的例子，它具有不同距离的三亏格两Heegaard分裂.这证明了Rubinstein和Scharlemann在1998年的论文Genus Two Heegaard Splittings of Orientable 3-Manifolds中描述的闭可定向3-流形的交替亏格2 Heegaard分裂的构造，并不产生所有的交替亏格2分裂，必须增广。
In this paper, we introduce a new concept of {\it strongly keen} for Heegaard splittings, and show that, for any integers $n\geq 2$ and $g\geq 3$, there exists a strongly keen Heegaard splitting of genus $g$ whose Hempel distance is $n$.
We give a necessary and sufficient condition for a simple closed curve on the boundary of a genus two handlebody to decompose the handlebody into (torus with one boundary component times [0,1]. We use this condition to decide whether a simple closed curve on a genus two Heegaard surface is a GOF-knot (genus one fibered knot) which induces the Heegaard splitting. By using this, we determine the number and the positions with respect to the Heegaard splittings of GOF-knots in the 3-manifolds with reducible genus two Heegaard splittings. This is another proof of results of Morimoto and Baker.
3-manifolding admitting locally large distance 2 Heegaard splittings
从Heegaard分裂的观点出发，我们知道，如果一个闭的可定向3-流形允许至少三个Heegaard分裂的距离，那么它是双曲的。然而，对于一个封闭的可定向3-流形只允许至多两个Heegaard分裂的距离，有例子表明它可以是可约的，Seifert的，环面的或双曲的。根据瑟斯顿的几何化猜想，八种几何中最重要的一种是双曲几何。因此，从远处读出双曲三维流形的两个Heegaard分裂对于研究Heegaard分裂至关重要。受距离为2的双曲3-流形构造的启发[Qiu，Zou and Guo，Pacific J. Math. 275（2015），no. 1，231-255]，我们引入了曲线复形中局部大测地线的定义，并进一步引入了局部大距离两个Heegaard分裂。然后证明了如果一个3-流形允许局部大距离2 Heegaard分裂，则它是一个沿不可压缩环面的双曲流形或一个双曲流形和一个seifert流形的合并，即，几乎双曲，而第3节的例子表明，在这种情况下，有一个非双曲3-流形。在考察了这些非双曲情形后，我们给出了双曲3-流形允许局部大距离2 Heegaard分裂的一个充分和必要条件.
Colding and Gabai have given an effective version of Li's theorem that non-Haken hyperbolic 3-manifolds have finitely many irreducible Heegaard splittings. As a corollary of their work, we show that Haken hyperbolic 3-manifolds have a finite collection of strongly irreducible Heegaard surfaces $S_i$ and incompressible surfaces $K_j$ such that any strongly irreducible Heegaard surface is a Haken sum $S_i + \sum_j n_j K_j$, up to one-sided associates of the Heegaard surfaces.
The Goeritz group of the standard genus-g Heegaard splitting of the three sphere, $G_g$, acts on the space of isotopy classes of reducing spheres for this Heegaard splitting. Scharlemann MR2199366 (2007c:57020) uses this action to prove that $G_2$ is finitely generated. In this article, we give an algorithm to construct any reducing sphere from a standard reducing sphere for a genus-2 Heegaard splitting of the $S^3$. Using this we give an alternate proof of the finite generation of $G_2$ assuming the finite generation of the stabilizer of the standard reducing sphere.
In this paper, by putting a separating incompressible surface in a 3-manifold into Morse position relative to the height function associated to a strongly irreducible Heegaard splitting, we show that an incompressible subsurface of the Heegaard splitting can be found, by decomposing the 3-manifold along the separating surface. Further if the Heegaard surface is of Hempel distance at least 4, then there is a pair of such subsurfaces on both sides of the given separating surface. This gives a particularly simple hierarchy for the 3-manifold.
