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.
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.
The manifold which admits a genus-$2$ reducible Heegaard splitting is one of the $3$-sphere, $\mathbb{S}^2 \times \mathbb{S}^1$, lens spaces and their connected sums. For each of those manifolds except most lens spaces, the mapping class group of the genus-$2$ splitting was shown to be finitely presented. In this work, we study the remaining generic lens spaces, and show that the mapping class group of the genus-$2$ Heegaard splitting is finitely presented for any lens space by giving its explicit presentation. As an application, we show that the fundamental groups of the spaces of the genus-$…
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.
Haken showed that the Heegaard splittings of reducible 3-manifolds are reducible, that is, a reducing 2-sphere can be found which intersects the Heegaard surface in a single simple closed curve. When the genus of the "interesting" surface increases from zero, more complicated phenomena occur. Kobayashi showed that if a 3-manifold $M^3$ contains an essential torus $T$, then it contains one which can be isotoped to intersect a strongly irreducible Heegaard splitting surface $F$ in a collection of simple closed curves which are essential in $T$ and in $F$. In general there is no global bound on t…
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.
In this paper, we study reducible surgeries on knots in $S^3$. We develop thickness bounds for L-space knots that admit reducible surgeries, and lower bounds on the slice genus for general knots that admit reducible surgeries. The L-space knot thickness bounds allow us to finish off the verification of the Cabling Conjecture for thin knots, which was mostly worked out in \cite{DeY21b}. We also provide a new upper bound on reducing slopes for fibered, hyperbolic slice knots and on multiple reducing slopes for slice knots. Our techniques involve the $d$-invariants and mapping cone formula from H…
A manifold which admits a reducible genus-$2$ Heegaard splitting is one of the $3$-sphere, $S^2 \times S^1$, lens spaces or their connected sums. For each of those splittings, the complex of Haken spheres is defined. When the manifold is the $3$-sphere, $S^2 \times S^1$ or the connected sum whose summands are lens spaces or $S^2 \times S^1$, the combinatorial structure of the complex has been studied by several authors. In particular, it was shown that those complexes are all contractible. In this work, we study the remaining cases, that is, when the manifolds are lens spaces. We give a precis…
The structure of a solvmanifold's Heegaard splittings
我们对解流形的不可约Heegaard分裂的合痕类进行了分类。如果解流形的单值性可以表示为右下角为0的2 x 2矩阵（当迹的绝对值为3时总是如此），则任何不可约分裂都是强不可约的，并且是亏格2。此外，如果迹线的绝对值为4或更大，则任何两个这样的分裂是同位素的。如果迹的绝对值是3，那么直到合痕，恰好有两个不可约分裂，它们的相关超椭圆对合是可交换的，并且对合的乘积是解流形的中心对合。如果单值性不能表示为右下角为0的2 × 2矩阵，则分裂是弱可约的，亏格为3，并且直到合痕是唯一的。
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.
We consider finite group-actions on closed, orientable and nonorientable 3-manifolds; such a finite group-action leaves invariant the two handlebodies of a Heegaard splitting of M of some genus g. The maximal possible order of a finite group-action of an orientable or nonorientable handlebody of genus g > 1 is 24(g-1), and in the present paper we characterize the 3-manifolds M and groups G for which the maximal possible order |G| = 24(g-1) is obtained, for some G-invariant Heegaard splitting of genus g > 1. If M is reducible then it is obtained by doubling an action of maximal possible order 2…
Let $(\mathcal{V},\mathcal{W};F)$ be a weakly reducible, unstabilized, Heegaard splitting of genus at least three in an orientable, irreducible $3$-manifold $M$. Then $Mod(M,F)$ naturally acts on the disk complex $\mathcal{D}(F)$ as a group action. In this article, we prove if $F$ is topologically minimal and its topological index is two, then the orbit of any element of $\mathcal{D}(F)$ for this group action consists of infinitely many elements. Moreover, we prove there are at most two elements of $\mathcal{D}(F)$ whose orbits are finite if the genus of $F$ is three.
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 …
