Examples of unstabilized critical Heegaard surfaces
Jung Hoon Lee
证明了（闭可定向曲面）\times S^1的标准极小亏格Heegaard分裂是临界Heegaard分裂.
We show that the standard minimal genus Heegaard splitting of (closed orientable surface)\times S^1 is a critical Heegaard splitting.
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Parity criterion and Dehn twists for unstabilized Heegaard splittings
给出了Heegaard图的一个宇称条件，证明了它是不稳定的.这改进了[5]的结果。作为应用，我们在任意给定的Heegaard分裂上利用Dehn扭曲构造了不稳定的Heegaard分裂.
Let (V,W;F) be a weakly reducible, unstabilized, genus three Heegaard splitting in an orientable, irreducible 3-manifold M. In this article, we prove that either the disk complex D(F) is contractible or F is critical. Hence, the topological index of F is two if F is topologically minimal.
Connected sums of unstabilized Heegaard splittings are unstabilized
设M_1和M_2是闭的可定向三维流形。设H_i表示M_i中的Heegaard曲面。证明了如果H_1 # H_2来自于稳定M_1 # M_2的下亏格分裂，则H_1或H_2来自于稳定下亏格分裂.如果H_i和G_i是M_i中的非同位素Heegaard曲面，并且H_i是不稳定的，则我们证明了M_1 # M_2中的H_1 # H_2与G_1 # G_2不是同位素的。前一个结果回答了C. Gordon（Kirby's problem list中的问题3.91）。
Heegaard splittings of sufficiently complicated 3-manifolds II: Amalgamation
设M_1和M_2是紧致的可定向的3-流形，M是由\bdy M_1的某个分支F通过同胚同构F与\bdy M_2的某个分支粘合而成的流形。我们证明了当M是“充分复杂的”时，（1）M_i的低亏格、不稳定、边界不稳定的Heegaard分裂的合并是M的不稳定分裂，（2）M的每一个低亏格、不稳定的Heegaard分裂都可以表示为M_i的不稳定、边界不稳定分裂的合并，也可能表示为F \times I的II型分裂，以及（3）如果在这样的表达式中不存在II型分裂，则它是唯一的。
Stabilizations of Heegaard splittings of sufficiently complicated 3-manifolds (Preliminary Report)
我们构造的流形，有对属$g$ Heegaard分裂，必须稳定大约$g$倍，成为等价的家庭。我们还表明，当两个不稳定，边界不稳定的Heegaard分裂合并的“足够复杂”的地图，所产生的分裂是不稳定的。作为一个推论，我们产生一个流形，具有距离1 Heegaard分裂任意高的亏格。最后，我们表明，在一个3-流形形成的一个足够复杂的胶合，低属，不稳定的Heegaard分裂可以表示在一个独特的方式作为合并的胶合面。
Let (V,W;F) be a weakly reducible, unstabilized, genus three Heegaard splitting in an orientable, irreducible 3-manifold M and DVW(F) the subset of the disk complex D(F) consisting of simplices having at least one vertex from V and at least one vertex from W. In this article, we describe the shape of DVW(F) and prove that there is a function from the components of DVW(F) to the isotopy classes of the generalized Heegaard splittings obtained by weak reductions from (V,W;F), where this function describes how the thick levels are embedded in the relevant compression bodies.
Let $M$ be an orientable, irreducible $3$-manifold admitting a weakly reducible genus three Heegaard splitting as a minimal genus Heegaard splitting. In this article, we prove that if $[f]$, $[g]\in Mod(M)$ give the same correspondence between two isotopy classes of generalized Heegaard splittings consisting of two Heegaard splittings of genus two, say $[\mathbf{H}]\to[\mathbf{H}']$, then there exists a representative $h$ of the difference $[h]=[g]\cdot[f]^{-1}$ such that (i) $h$ preserves a suitably chosen embedding of the Heegaard surface $F'$ obtained by amalgamation from $\mathbf{H}'$ whic…
On unstabilzed genus three critical Heegaard surfaces
设$M$是紧致可定向不可约3-流形，$H$是$M$的亏格为3的不稳定Heegaard分裂.本文将定义$H$的弱约化对的单纯复形，并得到该复形的若干性质。利用这种方法，我们将证明亏格为3的不稳定Heegaard分裂在一定条件下是临界的。此外，我们将表明，标准的亏格3 Heegaard分裂的$T^3$和诱导的Heegaard分裂的三个组件链外部由一定的隧道系统的主要定理的例子是关键的。
Destabilizing amalgamated Heegaard splittings
我们构造了一个具有环面边界的3-流形对序列，该序列具有以下两个性质：1）对于第n对3-流形沿其边界环面的一个精心选择的胶合结果，所得到的3-流形的亏格与这对3-流形的亏格之和之比小于1/2. 2)合并这对三维流形的某些不稳定的Heegaard分裂以形成所得三维流形的Heegaard分裂的结果产生了可以连续n次不稳定的稳定Heegaard分裂。
On critical Heegaard splittings of tunnel number two composite knot exteriors
本文证明了：如果存在两个弱约化对，使得每个弱约化对包含每个隧道的共核圆盘，则隧道2号纽结在其外部诱导临界Heegaard分裂.此外，作为主要定理的例子，我们证明了两个2-桥纽结的连通和或更一般的两个$(1,1)$ -纽结的连通和可以在其外部诱导临界Heegaard分裂.最后，当流形是闭的、不可约的、Heegaard分裂是亏格3的且不稳定时，我们给出了弱约化对由压缩圆盘唯一确定的一个等价条件.
Generalized Heegaard splittings and the disk complex
本文研究3-流形的弱可约Heegaard分解，定义广义Heegaard分解的等价关系与圆盘复形的等价簇，构造典范函数并证明其在亏格3时为双射，还给出拓扑极小Heegaard分解的相关2-单形存在性结论。
Critical Heegaard surfaces obtained by amalgamation
临界曲面是Bachman定义的拓扑指数为2的曲面，它是不可压缩曲面和强不可约曲面的推广。本文给出了一个用合并法得到临界Heegaard曲面的条件。作为一种特殊情况，我们得到临界Heegaard曲面的边界稳定。它给出了非极小亏格的临界Heegaard曲面，对于不允许不同Heegaard分裂的3-流形（直到合痕）。
On the weak reducing pairs in critical Heegaard splitting
本文研究临界 Heegaard 分裂中的弱约化对，分析其性质，并给出判定 Heegaard 曲面为临界曲面的必要条件。
Critical Heegaard surfaces obtained by self-amalgamation
临界曲面是由David Bachman提出的拓扑指数为2的极小曲面。本文给出了自拼合Heegaard曲面临界的一个充分条件和一个必要条件。
Critical Heegaard Surfaces
在本文中，我们介绍了“临界曲面”，这是通过1复杂的定义是让人想起的曲线复杂。我们的主要结果是：如果3-流形的一对强不可约Heegaard分裂的最小亏格公共稳定性不是临界的，则该流形包含一个不可压缩曲面.相反，我们还证明了，如果一个非哈肯3-流形允许最多一个Heegaard分裂的每种，那么它不包含一个临界Heegaard表面。在最后一节中，我们讨论这项工作如何导致一个自然的度量空间的强不可约Heegaard分裂，以及许多新的和有趣的开放问题。
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.
Suppose $V\cup_S W$ is a weakly reducible Heegaard splitting of a closed 3-manifold which admits only $n$ pairs of disjoint compression disks on distinct sides and $g>2$. We show $V\cup_S W$ admits an untelescoping: $(V_1\cup_{S_1}W_1)\cup_F(W_2\cup_{S_2}V_2)$ such that $W_i$ has only one separating compressing disk and $d(S_i)\geq 2$, for $i=1,~2$. If $n>1$, at least one of $d(S_i)$ is 2 and $S$ is a critical Heegaard surface.
A Distance for Circular Heegaard Splittings
本文基于圆周值Morse函数定义了$E(K)$的循环Heegaard分裂及其循环距离，并证明当循环距离过大时，$E(K)$无法包含低亏格不可压缩曲面且极小亏格Seifert曲面合痕唯一。
From the view of Heegaard splitting, it is known that if a closed orientable 3-manifold admits a distance at least three Heegaard splitting, then it is hyperbolic. However, for a closed orientable 3-manifold admitting only distance at most two Heegaard splittings, there are examples shows that it could be reducible, Seifert, toroidal or hyperbolic. According to Thurston's Geometrization conjecture, the most important piece of eight geometries is hyperbolic. Thus to read out a hyperbolic 3-manifold from a distance two Heegaard splittings is critical in studying Heegaard splittings. Inspired by …
Critical Heegaard Surfaces and Index 2 Minimal Surfaces
本文包含了研究临界曲面的动机。在以前的工作中，唯一的理由给出的定义，这类新的表面是强度的结果。然而，当被看作是拓扑类似指数2极小曲面，临界曲面变得非常自然。
Given a Heegaard splitting of a three-manifold Y, we consider the SL(2,C) character variety of the Heegaard surface, and two complex Lagrangians associated to the handlebodies. We focus on the smooth open subset corresponding to irreducible representations. On that subset, the intersection of the Lagrangians is an oriented d-critical locus in the sense of Joyce. Bussi associates to such an intersection a perverse sheaf of vanishing cycles. We prove that in our setting, the perverse sheaf is an invariant of Y, i.e., it is independent of the Heegaard splitting. The hypercohomology of this sheaf …
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