On keen weakly reducbile Heegaard splittings

TL;DR

This paper introduces a construction method for keen weakly reducible Heegaard splittings of arbitrary genus (except 2) and explores conditions under which critical Heegaard splittings can be obtained.

Contribution

It provides a new construction technique for keen weakly reducible Heegaard splittings of arbitrary genus and discusses how to obtain critical splittings by modifying conditions.

Findings

Constructed keen weakly reducible Heegaard splittings for genus ≥ 3

Identified conditions leading to critical Heegaard splittings

Extended understanding of the structure of weakly reducible splittings

Abstract

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.

Equations

$${\mathcal{D}}_V={\mathcal{D}}_0\cup {\mathcal{D}}_1\cup {\mathcal{D}}_2\cup {\mathcal{D}}_3$$

$${\mathcal{D}}_W={\mathcal{D}}^0\cup {\mathcal{D}}^1\cup {\mathcal{D}}^2\cup {\mathcal{D}}^3$$

$$d_{\partial W^{\ast }}(f(\alpha ),{\mathcal{D}}_W^{\ast })\le d_{\partial W^{\ast }}(D_0,E_{{\gamma }_2})\le d_{\partial W^{\ast }}(D_0,\gamma )+d_{\partial W^{\ast }}(\gamma ,E_{{\gamma }_2})=2$$

$$d_{\partial W^{\ast }}(f(\alpha ),{\mathcal{D}}_W^{\ast })\le d_{\partial W^{\ast }}(D_0,E_{{\gamma }_2})\le d_{\partial W^{\ast }}(D_0,\gamma )+d_{\partial W^{\ast }}(\gamma ,E_{{\gamma }_2})\le 1+2=3$$

$$N=C_1{\cup }_S{C}_2=(V_1{\cup }_{S_1}{W}_1){\cup }_F(W_2{\cup }_{S_2}{V}_2)$$

$$M=V{\cup }_{S}W=(V_1{\cup }_{S_1}{W}_1){\cup }_F(W_2{\cup }_{S_2}{V}_2).$$

Full text

This research is supported by grants (No.11401069 and No.11671064) of NSFC and Fundamental Research Funds for the Central Universities (No. 3132015231)

1. Introduction

Let M be an oriented compact 3-manifold. If there exists a closed surface S which cuts M into two compression bodies V and W, such that $S=\partial_{+}V=\partial_{+}W$, then $M=V\cup_{S}W$ is called a Heegaard spitting of $M$ and S is called a Heegaard surface of M. It is well known that every compact connected orientable 3-manifold admits a Heegaard splitting.

A Heegaard splitting $M=V\cup_{S}$W is said to be reducible if there are two essential disks $D\subset V$ and $E\subset W$ such that $\partial D=\partial E$. In other words, there exists a 2-sphere which intersects $S$ in an essential curve; Otherwise, $M=V\cup_{S}W$ is said to be irreducible.

A Heegaard splitting $M=V\cup_{S}W$ is said to be weakly reducible[6] if there are two essential disks $D\subset V$ and $E\subset W$ such that $\partial D\cap\partial E=\emptyset$; Otherwise, it is said to be strongly irreducible.

Let $S$ be a closed surface whose genus is at least 2. The distance between two essential simple closed curves $\alpha$ and $\beta$ in $S$, denoted by $d_{S}(\alpha,\beta)$, is the smallest integer $n\geq 0$ such that there is a sequence of essential simple closed curves $\alpha=\alpha_{0},\alpha_{1}...,\alpha_{n}=\beta$ in $S$ such that $\alpha_{i-1}$ is disjoint from $\alpha_{i}$ for $1\leq i\leq n$. Let $A$ and $B$ be two sets of essential simple closed curves in $S$. The distance between $A$ and $B$, which is denoted by $d_{S}(A,B)$, is defined to be $min\{d_{S}(x,y)|x\in A,y\in B\}$.

If $D$ and $E$ are two compression disks on distinct sides of $S$, sometimes $d_{S}(\partial D,\partial E)$ is denoted simply by $d_{S}(D,E)$. For examples, suppose that $g(S)>1$ and $\partial D$ intersects $\partial E$ in one point then $d_{S}(D,E)=2$, since $\partial[\overline{N(\partial D\cup\partial E})]\subset S$ is essential and disjoint from $\partial D\cup\partial E$.

The distance of the Heegaard splitting $V\cup_{S}W$ is defined to be $d(S)=d_{S}(\mathscr{D}_{V},\mathscr{D}_{W})$ where $\mathscr{D}_{V}$ and $\mathscr{D}_{W}$ are sets of essential disks in $V$ and $W$, respectively. $d(S)$ was first defined by Hempel, see [9].

$M=V\cup_{S}W$ is said to be a keen Heegaard splitting, if its distance is realized by a unique pair of elements $\mathscr{D}_{V}$ and $\mathscr{D}_{W}$. For any integers $n\geq 2$ and $g\geqslant 3$, there exists a strongly keen Heegaard splitting of genus $g$ whose distance is $n$. For more details, see [10].

The distance of $M=V\cup_{S}W$ is equal to 1 if and only if it is irreducible and weakly reducible. By definition, $M=V\cup_{S}W$ is said to be a keen weakly reducible Heegaard splitting, if there exists only one pair of disjoint compression disks on opposite sides of the Heegaard surface.

The simplest example of keen weakly reducible Heegaard splittings is the genus 1 Heegaard splitting of $S^{2}\times S^{1}$ which is reducible. Moreover, it is well known that there is no keen weakly reducible Heegaard splitting of genus 2. In the following statement, we assume that the the genus of the Heegaard surface is at least 3.

Suppose that $M=V\cup_{S}W$ is a keen weakly reducible Heegaard splitting, $D_{0}\subset V$ and $E_{0}\subset W$ are disjoint compression disks. It is easy to observe that $\partial D_{0}$ and $\partial E_{0}$ are not isotopic, moreover, each is non-separating on $S$, and $\partial D_{0}\cup\partial E_{0}$ is separating on $S$. It follows that $V\cup_{S}W$ is irreducible and unstabilized.

Let $F$ be a properly embedded, separating surface with no torus components in a compact, orientable, irreducible 3-manifold $M$. Then the disk complex of $F$, denoted by $\mathscr{D}(F)$, is defined as follows: Vertices of $\mathscr{D}(F)$ are isotopy classes of compression disks for $F$, and a set of $m+1$ vertices forms an $m-$simplex if there are representatives for each that are pairwise disjoint.

David Bachman explored the information which is contained in the topology of $\mathscr{D}(F)$ by defining the topological index of $F$[5]. If $\mathscr{D}(F)$ is non-empty then the topological index of $F$ is the smallest $n$ such that $\pi_{n-1}(\mathscr{D}(F))$ is non-trivial. If $\mathscr{D}(F)$ is empty then $F$ will have topological index [math]. If $F$ has a well-defined topological index (i.e. $\mathscr{D}(F)=\emptyset$ or non-contractible) then we will say that $F$ is a topologically minimal surface.

This raises a question that which surfaces are topologically minimal surfaces, moreover, if a surface is topologically minimal, we may ask what the topological index is. By definition, $F$ has topological index 0 if and only if it is incompressible, and has topological index 1 if and only if it is strongly irreducible. So we only need to consider weakly reducible surfaces. If a weakly reducible surface is topologically minimal, the topological index is at least 2. Index 2 topologically minimal surface are called critical surfaces which are also defined by David Bachman, see[1]and[3].

If a weakly reducible Heegaard splitting is keen, the disk complex of the Heegaard surface is quite simple. By McCullough in[12], who showed that the disk complex of the boundary of a handlebody is contractible, the disk complex of the Heegaard surface is obtained by attaching these two together with a single edge. That will just create a larger contractible complex. Thus we have the following proposition.

Proposition 1.1**.**

Suppose that $V\cup_{S}W$ is a keen weakly reducible Heegaard splitting, then the disk complex of the Heegaard surface $S$ is contractible, in other words, $S$ is not topologically minimal.

In [8], The author proved that some self-amalgamated Heegaard surfaces are keen. We remark that the genera of all the Heegaard surfaces given in [8] are odd, and the main theorem contains strict assumptions. In this paper, we will provide an explicit construction of keen weakly reducible Heegaard splittings of arbitrary genus. Such splittings admit a “symmetrical structure” and there is a lot of flexibility in doing so. Moreover, we may construct critical Heegaard splittings if we change some conditions.

2. Preliminaries

Let $V^{*}$ be a handlebody. Denote its boundary by $S^{*}$. Let $V$ be the handlebody obtained by attaching one 1-handle $D_{0}\times I$ to $V^{*}$ along a pair of disjoint disks $D_{1},D_{2}\subset S^{*}$, where $D_{0}$ is a disk corresponding to the 1-handle. Thus $g(V)=g(V^{*})+1$. Recall $\mathscr{D}_{V}$ is defined to be the set of compression disks of $V$. There is a natural partition of $\mathscr{D}_{V}$ induced by $D_{0}$ up to isotopy, as follows:

[TABLE]

$\mathscr{D}_{0}=\{D_{0}\}$.

$\mathscr{D}_{1}=\{D~{}|D\cap D_{0}=\emptyset;~{}D\neq D_{0};D$ is inessential in $V^{*}\}$.

$\mathscr{D}_{2}=\{D~{}|D\cap D_{0}=\emptyset;~{}D$ is essential in $V^{*}\}$.

$\mathscr{D}_{3}=\{D~{}|D\cap D_{0}\neq\emptyset\}$.

Let $D$ be a compression disk of $V$. Then $D$ belongs to one and only one of the four subsets. If $D\subset\mathscr{D}_{1}$, then $\partial D,~{}\partial D_{1}$ and $\partial D_{2}$ co-bound a pair of pants on $S^{*}$. It follows that $D$ is isotopic to a band-sum of $D_{1},D_{2}$ along an arc and $D$ cuts $V$ into $V^{*}$ and a solid torus. $D\subset\mathscr{D}_{2}$ if and only if $D\subset\mathscr{D}_{V^{*}}$, obviously. See figure 1.

Now consider $D\subset\mathscr{D}_{3}$. Suppose $D$ is an essential disk in $V$ such that $D\cap D_{0}\neq\emptyset$. Furthermore, $D$ is isotoped in $V$ such that $|D\cap D_{0}|$ is minimal. Let $S_{1}$ be the surface $S^{*}-(int(D_{1})\cup int(D_{2}))$. Then $S_{1}$ is a sub-surface of $S^{*}$ with two boundary components $\partial D_{1}$ and $\partial D_{2}$. By standard arguments, we have some observations as follows. See also[15].

Lemma 2.1**.**

1. (1)

Each component of $D\cap D_{0}$ is a properly embedded arc in both $D$ and $D_{0}$. 2. (2)

Each component of $\partial D\cap S_{1}$ is essential on $S_{1}$. 3. (3)

Each component of $D\cap(\partial D_{0}\times I)$ is an arc with its two end points lying in distinct boundary components of the annulus $\partial D_{0}\times I$.

An essential arc $\gamma$ in $S_{1}$ is called strongly essential if both boundary points lie in $\partial D_{i}$ and $\gamma$ is an essential arc on $S_{1}\cup D_{j}$ , where $\{i,j\}=\{1,2\}$. Recall that $D\subset\mathscr{D}_{3}$. Let $\gamma$ be an outermost component of $D\cap(D_{1}\cup D_{2})$ on $D$. This means that $\gamma$, together with an arc $\gamma_{1}\subset\partial D$, bounds a sub-disk in $D$, say $D_{\gamma}$, such that $D_{\gamma}$ intersects $D_{\gamma}\cap(D_{1}\cup D_{2})=\gamma$. We call $\gamma_{1}$ an outermost arc related to $\gamma$ and call $D_{\gamma}$ an outermost disk related to $\gamma$.

Lemma 2.2**.**

[15]**

1. (1)

$\gamma_{1}$, whose end points lie in one of $D_{1}$ and $D_{2}$, is strongly essential in $S_{1}$. 2. (2)

$D_{\gamma}\subset V^{*}$* and is essential in $V^{*}$.*

3. Construction of keen weakly reducible Heegaard splittings

In this section, we will provide an explicit construction of keen weakly reducible Heegaard splittings of genus g.

Let $V^{*}$ be a genus $(g-1)$ handlebody. We choose an essential curve $\beta\subset\partial V^{*}$, such that $\beta$ is separating on $\partial V^{*}$ and $d_{\partial V^{*}}(\beta,\mathscr{D}_{V^{*}})\geq 4$ . Then we attach a 2-handle $E_{0}\times I$ to $V^{*}$ along $\beta$ so that $\beta$ bounds a disk $E_{0}$. This yields a 3-manifold denoted by $M_{1}$.

Next, we attach a 1-handle $D_{0}\times I$ to $M_{1}$ such that the gluing disks denoted by $D_{1},D_{2}$ are on distinct components of $\partial V^{*}-\beta$, where $D_{0}$ is an disk corresponding to the 1-handle and we denote $\partial D_{0}$ by $\alpha$. The resulted 3-manifold denoted by $M_{2}$. $\partial M_{2}$ is connected and $g(\partial M_{2})=g-1$.

Next, let $W^{*}$ be another genus $(g-1)$ handlebody. We glue $W^{*}$ to $M_{2}$ via an orientation preserving homeomorphism $f:\partial W^{*}\rightarrow\partial M_{2}$ such that $d_{\partial W^{*}}(f(\alpha),\mathscr{D}_{W^{*}})\geq 4$. Since $\alpha$ is separating on $\partial M_{2}$, $f(\alpha)$ is separating on $\partial W^{*}$. Recall $D_{1}\cup D_{2}=V^{*}\cap(D_{0}\times I)$. Similarly, $W^{*}\cap(E_{0}\times I)$ are two disks, denoted by $E_{1}$,$E_{2}$.

The result is a closed 3-manifold denoted by $M$. Denote $V=V^{*}\cup(D_{0}\times I)$, $W=W^{*}\cup(E_{0}\times I)$ and $S=V\cap W$. Then both $V$ and $W$ are genus $g$ handlebodies and $M=V\cup_{S}W$ is a Heegaard splitting. $\alpha$ bounds the disk $D_{0}$ in $V$ and $\beta$ bounds the disk $E_{0}$ in $W$. Since the 2-handle $E_{0}\times I$ is attached before the 1-handle $D_{0}\times I$, $D_{0}\cap E_{0}=\alpha\cap\beta=\emptyset$. It follows that $M=V\cup_{S}W$ is weakly reducible.

Theorem 3.1**.**

$M=V\cup_{S}W$* is a keen weakly reducible Heegaard splitting.*

Proof.

In order to show $V\cup_{S}W$ is keen, we firstly divide $\mathscr{D}_{V}$ and $\mathscr{D}_{W}$ as mentioned in Section 2:

[TABLE]

and

[TABLE]

such that:

$\mathscr{D}_{0}=\{D_{0}\}$.

$\mathscr{D}_{1}=\{D~{}|D\cap D_{0}=\emptyset;~{}D\neq D_{0};D$ is inessential in $V^{*}\}$.

$\mathscr{D}_{2}=\{D~{}|D\cap D_{0}=\emptyset;~{}D$ is essential in $V^{*}\}=\mathscr{D}_{V^{*}}$.

$\mathscr{D}_{3}=\{D~{}|D\cap D_{0}\neq\emptyset\}$.

$\mathscr{D}^{0}=\{E_{0}\}$.

$\mathscr{D}^{1}=\{E~{}|E\cap E_{0}=\emptyset;~{}E\neq E_{0};D$ is inessential in $W^{*}\}$.

$\mathscr{D}^{2}=\{E~{}|E\cap E_{0}=\emptyset;~{}E$ is essential in $W^{*}\}=\mathscr{D}_{W^{*}}$.

$\mathscr{D}^{3}=\{E~{}|E\cap E_{0}\neq\emptyset\}$.

In order to show that ($D_{0}$, $E_{0}$) is the unique disjoint pair of compression disks on distinct sides of $S$, we will show that for each $D\subset\mathscr{D}_{i}$, $E\subset\mathscr{D}^{j}$, where $i,j=0,1,2,3$ and $(i,j)\neq(0,0)$, $D\cap E=\partial D\cap\partial E\neq\emptyset$ holds.

Claim 3.2**.**

For each $D\subset\mathscr{D}_{0}$, $E\subset\mathscr{D}^{1}$, $D\cap E\neq\emptyset$.

In this case, $D=D_{0}$ and $E$ is a band-sum of two copies of $E_{0}$ along an arc. Recall that $\alpha=\partial D_{0}$ and $\beta=\partial E_{0}$ and the union separates $S$. $\alpha$ separates the two copies of $E_{0}$. Any arc connected to the two copies intersects $\alpha$. It follows that $D\cap E\neq\emptyset$.

Claim 3.3**.**

For each $D\subset\mathscr{D}_{0}$, $E\subset\mathscr{D}^{2}$, $D\cap E\neq\emptyset$.

In this case, $D=D_{0}$ and $E$ is essential both in handlebodies $W$ and $W^{*}$. Notice that $W^{*}=\overline{W\backslash(E_{0}\times I)}$. Recall that $d(f(\alpha),\mathscr{D}_{W^{*}})\geq 3$, thus $D_{0}$ intersects each compression disk of $W^{*}$.

Remark 3.4**.**

By the above two claims, if $E$ is an essential disk in $W$ which is not isotopic to $E_{0}$ and disjoint from $E_{0}$, then $D_{0}\cap E\neq\emptyset$. Similarly, if $D$ is an essential disk in $V$ which is not isotopic to $D_{0}$ and disjoint from $D_{0}$, then $D\cap E_{0}\neq\emptyset$.

In the following argument, we assume that $|D\cap D_{0}|+|E\cap E_{0}|$ is minimal in the isotopy classes of $D$ and $E$.

Claim 3.5**.**

For each $D\subset\mathscr{D}_{0}$, $E\subset\mathscr{D}^{3}$, $D\cap E\neq\emptyset$.

In this case, $D=D_{0}$ and $E\cap E_{0}\neq\emptyset$. Assume that $D_{0}\cap E=\emptyset$. By applying Lemma 2.2 to $W$ and $W^{*}$, there is an outermost disk of $E$, say $E_{\gamma}$, which is essential in $W^{*}$. By Claim 3.3, $D_{0}\cap E_{\gamma}\neq\emptyset$. Notice that $D_{0}\cap E_{\gamma}\subset D_{0}\cap E$. This contradicts the assumption $D_{0}\cap E=\emptyset$.

Claim 3.6**.**

For each $D\subset\mathscr{D}_{1}$, $E\subset\mathscr{D}^{1}$, $D\cap E\neq\emptyset$.

In this case, $D$ is a band-sum of two copies of $D_{0}$ and $E$ is a band-sum of two copies of $E_{0}$. $\partial D$ bounds a once-punctured torus $T_{D}$ on $S$ and $\partial E$ bounds a once punctured torus $T_{E}$. Suppose to the contrary that $D\cap E=\emptyset$. First we assume that $\partial E$ is isotopic to $\partial D$. In this case, since $g(S)>2$, we have $T_{D}=T_{E}$. It follows that $\alpha$ and $\beta$ are isotopic because $\alpha\cap\beta=\emptyset$, a contradiction. Hence $\partial E\neq\partial D$. In this case $T_{D}\cap T_{E}=\emptyset$ and $\alpha\cup\beta$ does not separate the Heegaard surface, a contradiction.

Claim 3.7**.**

For each $D\subset\mathscr{D}_{1}$, $E\subset\mathscr{D}^{2}$, $D\cap E\neq\emptyset$.

In this case, $D$ is a band-sum of two copies of $D_{0}$ and $E$ is an essential disk in $W^{*}$. $\partial D$ bounds a once-punctured torus $T_{D}$ on $S$ and $\alpha\subset T_{D}$. Suppose to the contrary that $D\cap E=\emptyset$. If $\partial E\nsubseteq T_{D}$ then $E\cap D_{0}=\emptyset$ which contradicts Claim 3.3. Thus $\partial E\subset T_{D}\subset S$. Moreover, $\partial E\subset N(\alpha)\subset S$ since $E\cap E_{0}=\emptyset$. In this case $\partial E$ is isotopic to $\alpha$ which implies that $d_{\partial W^{*}}(f(\alpha),~{}\mathscr{D}_{W^{*}})=0$, a contradiction.

Claim 3.8**.**

For each $D\subset\mathscr{D}_{1}$, $E\subset\mathscr{D}^{3}$, $D\cap E\neq\emptyset$.

In this case $E\cap E_{0}\neq\emptyset$. By applying Lemma 2.2 to $W$ and $W^{*}$, there is an outermost arc, say $\gamma$, whose two endpoints lie in one of $\partial E_{1}$ and $\partial E_{2}$. Furthermore, there is an outermost disk of $E$, say $E_{\gamma}$, such that $E_{\gamma}$ is essential in $W^{*}$. Without loss of generality, we assume that $E_{\gamma}\cap E_{1}$ is an arc denoted by $e_{1}$, that is to say, $\partial\gamma\subset\partial E_{1}$.

$D$ is a band-sum of two copies of $D_{0}$ and the two copies lie distinct sides of $\beta$. Hence there is a sub-arc $\gamma_{1}$ of $\partial D$ such that $\partial\gamma_{1}\subset\partial E_{2}$. $\gamma_{1}$, together with a sub-arc of $\partial E_{2}$, say $e_{2}$, bounds a disk isotopic to $D_{0}$.

$D\cap E=\emptyset$ implies that $\gamma\cap\gamma_{1}=\emptyset$, thus $(\gamma\cup e_{1})\cap(\gamma_{1}\cup e_{2})=\emptyset$ which means that $E_{\gamma}\cap D_{0}=\emptyset$. This contradicts Claim 3.3.

Claim 3.9**.**

For each $D\subset\mathscr{D}_{2}$, $E\subset\mathscr{D}^{2}$, $D\cap E\neq\emptyset$.

In this case, $D$ is essential in $V^{*}$ and $E$ is essential in $W^{*}$. Suppose to the contrary that $D\cap E=\emptyset$. We remark that $\partial D\cap\beta\neq\emptyset$ and $\beta$ separates $\partial V^{*}$. Hence there is a sub-arc $\gamma_{1}$ of $\partial D$ such that $\partial\gamma_{1}\subset\partial E_{1}$ and $\gamma_{1}$ together with a arc of $E_{1}$ forms an essential closed curve $\gamma$ on $\partial W^{*}$. Moreover, $D_{0}\cap D=\emptyset$ and $D_{0}\cap\beta=\emptyset$ mean that $D_{0}\cap\gamma=\emptyset$. $E\cap D=\emptyset$ and $E\cap\beta=\emptyset$ mean that $E\cap\gamma=\emptyset$. Hence $d_{\partial W^{*}}(f(\alpha),\mathscr{D}_{W}^{*})\leq d_{\partial W^{*}}(D_{0},E)\leq d_{\partial W^{*}}(D_{0},\gamma)+d_{\partial W^{*}}(\gamma,E)=1+1=2$, a contradiction.

Claim 3.10**.**

For each $D\subset\mathscr{D}_{2}$, $E\subset\mathscr{D}^{3}$, $D\cap E\neq\emptyset$.

In this case, $D$ is essential in $V^{*}$ and $E\cap E_{0}\neq\emptyset$. Suppose to the contrary that $D\cap E=\emptyset$.

By Lemma 2.2, there is an outermost arc $\gamma_{2}$ of $\partial E$ and an outermost disk of $E$ , say $E_{\gamma_{2}}$ which is essential in $W^{*}$. Without loss of generality, we may assume that $\partial\gamma_{2}\subset\partial E_{2}$.

Since $D\subset\mathscr{D}_{2}$, by the argument mentioned in Claim3.9, there is a sub-arc $\gamma_{1}$ of $\partial D$ such that $\partial\gamma_{1}\subset\partial E_{1}$ and $\gamma_{1}$ together with an arc of $E_{1}$ forms an essential closed curve $\gamma$ on $\partial W^{*}$ such that $D_{0}\cap\gamma=\emptyset$.

$D\cap E=\emptyset$ implies that $\gamma_{1}\cap\gamma_{2}=\emptyset$. According to our choice, $\partial\gamma_{1}$ and $\partial\gamma_{2}$ lie on distinct disks. Hence $E_{\gamma_{2}}\cap\gamma=\emptyset$. Thus

[TABLE]

a contradiction.

Claim 3.11**.**

For each $D\subset\mathscr{D}_{3}$, $E\subset\mathscr{D}^{3}$, $D\cap E\neq\emptyset$.

In this case, $D\cap D_{0}\neq\emptyset$ and $E\cap E_{0}\neq\emptyset$. Suppose to the contrary that $D\cap E=\emptyset$.

By Lemma 2.2, there exists an outermost arc $\gamma_{1}$ of $\partial D$ and an outermost disk of $D$ , say $D_{\gamma_{1}}$ which is essential in $V^{*}$. Since $D_{\gamma_{1}}\cap\beta\neq\emptyset$ and $\beta$ separates $\partial V^{*}$, there is an outermost arc say of $\gamma_{11}$ such that $\gamma_{11}\subset\gamma_{1}$, moreover, $\partial\gamma_{11}\subset\partial E_{1}$ or $\partial\gamma_{11}\subset\partial E_{2}$. See Figure 2.

Without loss of generality, we may assume that $\partial\gamma_{11}\subset\partial E_{1}$. Thus $\gamma_{11}$ together with an arc of $E_{1}$ forms an closed curve, say $\gamma$, which is essential in $\partial W^{*}$. We remark that $\gamma=\gamma_{11}\cup e_{1}$, where $e_{1}$ is an arc in $E_{1}$. Since the outermost arc of $D$ and $E_{2}$ are disjoint from $\alpha$, $D_{0}\cap\gamma=\emptyset$.

By applying Lemma 2.2 to $W$ and $W^{*}$, there is an outermost arc of $\partial E$, say $\gamma_{2}$, where $\partial\gamma_{2}\subset\partial E_{1}$ or $\partial\gamma_{2}\subset\partial E_{2}$.

$D\cap E=\emptyset$ means that $\gamma_{11}$ and $\gamma_{2}$ are disjoint.

If $\partial\gamma_{2}\subset\partial E_{2}$, $\gamma_{2}$, together with an arc in $E_{2}$, bounds an outermost disk, say $E_{\gamma_{2}}$, which is essential in $\partial W^{*}$. In this case $\gamma$ and $E_{\gamma_{2}}$ are disjoint. It follows that:

[TABLE]

which is a contradiction.

If $\partial\gamma_{2}\subset\partial E_{1}$, $\gamma_{2}$, together with an arc $e_{2}$ in $E_{1}$, bounds an outermost disk, say $E_{\gamma_{2}}$, which is essential in $\partial W^{*}$. In this case $\gamma$ intersects $E_{\gamma_{2}}$ in at most one point, since $e_{1}$ intersects $e_{2}$ at most one point.

It follows that:

[TABLE]

which is also a contradiction.

If for some $(i,j)$, each $D\subset\mathscr{D}_{i}$, $E\subset\mathscr{D}^{j}$, $D\cap E=\partial D\cap\partial E\neq\emptyset$ holds, then by the symmetric construction of the Heegaard splitting and the same partitions of the $\mathscr{D}_{V}$ and $\mathscr{D}_{W}$, it also holds for $(j,i)$. This completes the proof. ∎

4. on topologically minimal amalgamated Heegaard surfaces

Let $N$ be a 3-manifold. Suppose $F$ is an incompressible separating surface which cuts $N$ into $N_{1}$ and $N_{2}$. Then $N$ is obtained by gluing two 3-manifolds, say $N_{1}$ and $N_{2}$, along a homeomorphism $f$. If $N_{i}$ admits a Heegaard splitting $V_{i}\cup_{S_{i}}W_{i}$ for $i=1,~{}2$, $N$ has a natural Heegaard splitting called the amalgamation of $V_{i}\cup_{S_{i}}W_{i}$ for $i=1,~{}2$, as:

[TABLE]

The amalgamation of two unstabilized Heegaard splittings may be stabilized, see[13]. However, if the gluing map $f$ is complicated enough, then the amalgamation of two minimal Heegaard splittings is unstabilized[11]. On the other hand, the amalgamation of two high distance Heegaard splittings is unstabilized[14].

David Bachman showed that if the gluing map $f$ is complicated enough, then the amalgamated Heegaard surface is not topologically minimal[2]. We apply the idea of [14] to conjecture that the amalgamated Heegaard surface of two high distance Heegaard splittings is not topologically minimal. Our construction of keen weakly reducible Heegaard splittings prsents positive examples.

Theorem 4.1**.**

Let $M$ be a 3-manifold. Suppose that $M$ admits an amalgamated Heegaard splitting:

[TABLE]

such that $g(F)=g(S_{1})=g(S_{2})$ and $d(S_{i})\geq 4$, for $i=1,~{}2$, then $M=V\cup_{S}W$ is keen weakly reducible hence $S$ is not topologically minimal.

Proof.

For each $i$, $d(S_{i})\geq 4$ implies that $V_{i}\cup_{S_{i}}W_{i}$ is not an trivial Heegaard splitting. Since $g(\partial_{+}W_{i})=g(S_{i})=g(F)=g(\partial_{-}W_{i})$, $F$ is disconnected and there exists only on separating compression disk embedded in $W_{i}$.(See figure 3) If $d(S_{i})\geq 4$, it is easy to check that $V\cup_{S}W$ coincides with our construction. ∎

We remark that if $d(S_{i})\geq 2$ then $S$ may not be keen and be topologically minimal. One example is that the standard Heegaard surface of $F^{*}\times S^{1}$, where $F^{*}\times S^{1}$ is a $S^{1}$ -bundle of a connected closed surface. In this case $V_{i}\cup_{S_{i}}W_{i}$ is the standard type 2 Heegaard splitting of $F^{*}\times I$ and $d(S_{i})=2$. the standard Heegaard surface of $F^{*}\times S^{1}$ is not keen and critical. In fact, if $d(S_{i})\geq 2$ we show no higher index topologically minimal Heegaard surfaces exist.

Theorem 4.2**.**

Let $M$ be a 3-manifold. Suppose that $M$ admits an amalgamated Heegaard splitting:

[TABLE]

such that $g(F)=g(S_{1})=g(S_{2})$ and $d(S_{i})\geq 2$, for $i=1,~{}2$, then $S$ is either critical or not topologically minimal.

Proof.

We use notions as defined in Section 3. $d(S_{i})\geq 2$ means that $d_{\partial V^{*}}(\beta,\mathscr{D}_{V^{*}})\geq 2$ and $d_{\partial W^{*}}(f(\alpha),\mathscr{D}_{W^{*}})\geq 2$. By Claim 3.2, Claim 3.3 and Claim 3.5, if $d(S_{i})\geq 2$, any disk in $\mathscr{D}_{W}\backslash E_{0}$ intersects $D_{0}$, equally, any disk in $\mathscr{D}_{V}\backslash D_{0}$ intersects $E_{0}$. Hence if there exist $D\subset\mathscr{D}_{V}\backslash D_{0}$ and $E\subset\mathscr{D}_{W}\backslash E_{0}$, such that $D\cap E=\emptyset$ then $\pi_{1}(\mathscr{D}(S))\neq 1$ and $S$ is a critical Heegaard surface. if no such pair of disks exists, $S$ is keen weakly reducible and not topologically minimal. ∎