Reducing spheres of genus-2 Heegaard splitting of $S^{3}$

The genus-$g$ Goeritz group of $S^{3}$, denoted $G_{g}$, is the group of isotopy classes of homeomorphisms of $S^{3}$ which preserve a genus-$g$ Heegaard surface of $S^{3}$. By a theorem of Waldhausen [9], every Heegaard splitting of $S^{3}$ of genus greater than $1$ is stabilized. So the group $G_{g}$ can be taken to be the group of isotopy classes of homeomorphisms of $S^{3}$ which preserve the standard genus-$g$ Heegaard splitting of $S^{3}$. Scharlemann [7] proved that the genus-2 Goeritz group, $G_{2}$, of $S^{3}$ is finitely generated by four generators. He did so by considering a certain simplicial complex whose vertices are the isotopy classes of reducing spheres of the standard genus-$2$ Heegaard splitting of $S^{3}$. Akbas [1] used Scharlemann’s complex to give a finite presentation for $G_{2}$. Cho [2] independently used a certain disk complex to give a finite presentation for $G_{2}$. For $g\geq 3$, Zupan [10] generalized Scharlemann’s work and defined a reducing sphere complex for genus-$g$ Heegaard splittings of $S^{3}$ and showed that the group $G_{g}$ is finitely generated if and only if this complex is connected. Freedman and Scharlemann [4] proved that $G_{3}$ is also finitely generated. They conjectured that the five generators mentioned in Powell [5] generate $G_{g}$ for all $g\geq 3$. Much is unknown about $G_{g}$ for $g\geq 4$.

The literature on the Goeritz group of $S^{3}$ shows the importance of describing an arbitrary reducing sphere for reducible Heegaard splittings of $S^{3}$, and for reducible Heegaard splittings of 3-manifolds in general. In this article, we look at the set of all the reducing spheres for the genus-2 Heegaard splitting of $S^{3}$. We give structure theorems for such spheres. We then give an algorithm to construct any reducing sphere for the standard genus-2 Heegaard splitting of $S^{3}$, upto isotopy, from the standard reducing sphere.

The layout of this article is as follows. In section 2, we prove Theorem 2.2 which gives some intersection conditions satisfied by the curve of intersection of a reducing sphere with the standard genus-2 Heegaard surface of $S^{3}$. In section 3, we use the fundamental group of a genus-2 handlebody to deduce some conditions which apply to the arcs of intersection of a reducing sphere with a certain component pair of pants of the genus-2 Heegaard surface of $S^{3}$. We then use these conditions to prove Theorem 3.6 for such arcs of a reducing sphere. In section 4, we prove Theorem 4.1 which is a key tool to de-construct a given reducing sphere. In section 5, we give an algorithm to de-construct and transform any reducing sphere into the standard reducing sphere. Using this algorithm and assuming that the stabilizer of the standard reducing sphere of the genus-2 Heegaard splitting of $S^{3}$ is finitely generated, we show that the Goeritz group $G_{2}$ is finitely generated.

For preliminaries regarding Heegaard splittings of 3-manifolds, one can consult [6] and for preliminaries on mapping class groups and simple closed curves on surfaces, one can consult Farb and Margalit [3]. We start by considering the standard genus-2 Heegaard splitting of $S^{3}$, written $S^{3}=V\cup_{{\Sigma}}W$ and denoted by ${\mathcal{H}}$, where both $V$ and $W$ are unknotted genus-2 handlebodies embedded in $S^{3}$ with disjoint interiors glued along their common boundary ${\Sigma}$. The Heegaard surface, ${\Sigma}$, of the above splitting is a closed orientable surface of genus two embedded in $S^{3}$. Since ${\mathcal{H}}$ is stabilized, one can find a reducing sphere for ${\mathcal{H}}$ i.e. a sphere which intersects each of $V$ and $W$ in an embedded disk with a common boundary circle on ${\Sigma}$. Such a reducing sphere intersects ${\Sigma}$ in an essential separating simple closed curve, which we call a reducing curve for ${\Sigma}$. A reducing curve on ${\Sigma}$ uniquely determines a reducing sphere, upto isotopy in $S^{3}$. Two reducing spheres of ${\mathcal{H}}$ are said to be isotopic with respect to ${\mathcal{H}}$, if any of their isotopy in $S^{3}$ restricts to an isotopy of the corresponding reducing curves on the surface ${\Sigma}$. Likewise, if two reducing curves for ${\Sigma}$ are isotopic on ${\Sigma}$, this isotopy can be extended to an isotopy of the corresponding reducing spheres with respect to ${\mathcal{H}}$. This allows one to study reducing curves on ${\Sigma}$ in order to study reducing spheres of ${\mathcal{H}}$.

The Heegaard splitting ${\mathcal{H}}$ of $S^{3}$ admits the following four automorphisms, as described in Scharlemann [7], which preserve the structure of ${\mathcal{H}}$. The automorphisms $\alpha$ and $\gamma$ are the automorphisms of $S^{3}$ induced by the $\pi$-rotations of ${\Sigma}$ about the axes indicated in Figures 1(a) and 1(c). The automorphism $\beta$ is the half twist of the ${\Sigma}$ about the central separating curve as indicated in Figure 1(b).

The automorphism $\delta$ is of order three which can be described as follows. Consider a graph on the unit sphere in $\mathbb{R}^{3}$ whose vertices are the north and the south pole and whose edges are three distinct half meridians on the sphere connecting the north pole and the south pole such that the rotation of the sphere by an angle $\frac{2\pi}{3}$ about the axis connecting the north and south pole permutes the three edges cyclically. If $V^{\prime}$ is a regular neighborhood in $S^{3}$ of this graph such that $V^{\prime}$ is homeomorphic to the handlebody $V$, then $\delta$ is the order three automorphism which cyclically permutes the 1-handles of $V^{\prime}$ by rotation by an angle $\frac{2\pi}{3}$ as shown in Figure 1(d).

Figure 2 shows an embedding of the Heegaard surface ${\Sigma}$ of ${\mathcal{H}}$ in $\mathbb{R}^{3}$. $P$ is the reducing sphere whose reducing curve, $P_{\Sigma}$, is shown in Figure 2. We call $P$ as the standard reducing sphere. Consider the non-separating curves $A,B,C,X,Y,Z$ on ${\Sigma}$ as shown in Figure 2.

$A\cup B\cup C$ separates ${\Sigma}$ into two thrice boundered spheres, call them ${\Sigma_{1}}$ and ${\Sigma_{2}}$. We will refer to the curves in Figure 2 throughout this article.

Any reducing sphere $Q$ of ${\mathcal{H}}$ intersects ${\Sigma}$ in the corresponding reducing curve, which we denote by $Q_{{\Sigma}}$. Such a ${Q_{{\Sigma}}}$ is an essential separating simple closed curve on ${\Sigma}$ which separates ${\Sigma}$ into two surfaces, each of which is a surface of genus $1$ with one boundary component. Since ${Q_{{\Sigma}}}$ cannot be null-homotopic on $\Sigma$, ${Q_{{\Sigma}}}$ will intersect both ${\Sigma_{1}}$ and ${\Sigma_{2}}$ and will also intersect at least one of $A,B$ or $C$. For any $Q$, we assume that ${Q_{{\Sigma}}}$ intersects $A,B,C,X,Y$ and $Z$ minimally and transversally (refer to Figure 2). ${Q_{{\Sigma}}}\cap{\Sigma_{1}}$ or ${Q_{{\Sigma}}}\cap{\Sigma_{2}}$ is a collection of simple essential arcs on the thrice boundered spheres ${\Sigma_{1}}$ or ${\Sigma_{2}}$, respectively. For $D,E\in\{A,B,C\}$, an arc of type $DE$, or a $DE$ arc, for short, on ${\Sigma_{1}}$ (or on ${\Sigma_{2}}$) is a connected arc of ${Q_{{\Sigma}}}$ which is contained in ${\Sigma_{1}}$ (or in ${\Sigma_{2}}$) and which has its end-points on the curves $D$ and $E$. So there are six possible types of arcs of ${Q_{{\Sigma}}}$ on ${\Sigma_{1}}$ or ${\Sigma_{2}}$ namely an $AA$ arc, a $BB$ arc, a $CC$ arc, an $AB$ arc, an $AC$ arc and a $BC$ arc. Some arc types intersect the others. So a reducing curve, ${Q_{{\Sigma}}}$, cannot have intersecting arc types.

Table 1 shows arc types which necessarily intersect and hence cannot be components of the same reducing curve on ${\Sigma_{1}}$ and likewise on ${\Sigma_{2}}$:

For an arbitrary reducing sphere $Q$, the number of arcs of ${Q_{{\Sigma}}}\cap{\Sigma_{1}}$ and of ${Q_{{\Sigma}}}\cap{\Sigma_{2}}$ have to be equal, as a mismatch will leave an open end which is impossible for a closed curve. For $D,E\in\{A,B,C\}$, let $|DE|_{Q,1}$ and $|DE|_{Q,2}$ denote the number of arcs of ${Q_{{\Sigma}}}$ of type $DE$ on ${\Sigma_{1}}$ and ${\Sigma_{2}}$, respectively. For instance, $|AB|_{Q,1}$ denotes the number of arcs of ${Q_{{\Sigma}}}$ of type $AB$ on ${\Sigma_{1}}$ and $|BB|_{Q,2}$ denotes the number of $BB$ arcs of ${Q_{{\Sigma}}}$ on ${\Sigma_{2}}$. When the context is clear, we will drop $Q$ in the subscript. For instance, we will just use $|BB|_{2}$, instead of $|BB|_{Q,2}$. We define $a_{Q}:=|{Q_{{\Sigma}}}\cdot A|,b_{Q}:=|{Q_{{\Sigma}}}\cdot B|$ and $c_{Q}:=|{Q_{{\Sigma}}}\cdot C|$. Since ${Q_{{\Sigma}}}$ is a separating curve, $a_{Q},b_{Q}$ and $c_{Q}$ are non-negative even numbers.

The four automorphisms of ${\mathcal{H}}$ described above, namely, $\alpha,\beta,\gamma$ and $\delta$ affect the numbers $a_{Q},b_{Q}$ and $c_{Q}$ of a reducing sphere $Q$ of ${\mathcal{H}}$ as follows:

1. 1.

   $\delta$ cyclically permutes $a_{Q},b_{Q}$ and $c_{Q}$,

2. 2.

   $\gamma$ permutes $b_{Q}$ and $c_{Q}$ and keeps $a_{Q}$ unchanged,

3. 3.

   $\alpha$ keeps $a_{Q},b_{Q}$ and $c_{Q}$ unchanged and

4. 4.

   $\beta$ or its inverse increase or decrease $a_{Q}$, while keeping $b_{Q}$ and $c_{Q}$ unchanged.

Table 2 lists the contribution of each arc type to $a_{Q}$, $b_{Q}$ and $c_{Q}$.

The proof of Theorem 2.2 also shows the following:

We assume the setup of the previous section. By Theorem 2.2, and by invoking the order three symmetry, $\delta$, of ${\mathcal{H}}$, in order to describe an arbitrary reducing sphere, $Q$, of ${\mathcal{H}}$, it is enough to describe a reducing sphere in the case $a_{Q}>b_{Q}+c_{Q}$. Also, by Theorem 2.2, the inequality $a_{Q}>b_{Q}+c_{Q}$ implies the existence of an $AA$ arc on ${\Sigma_{1}}$ and an $AA$ arc on ${\Sigma_{2}}$. Further, by Remark 2.3, there must be an $AA$ arc on ${\Sigma_{1}}$ or on ${\Sigma_{2}}$ which is as shown in Figure 4.

So, throughout this section, we assume that a reducing sphere $Q$ satisfies the inequality $a_{Q}>b_{Q}+c_{Q}$ and describe how the $AA$, $AB$ and $AC$ arcs of the corresponding reducing curve ${Q_{{\Sigma}}}$ can be presented on ${\Sigma_{1}}$ and on ${\Sigma_{2}}$.

Let the two distinct arcs of $A$ each of which has one end-point on $Y$ and another end-point on $Z$ be denoted by $A_{1}$ and $A_{2}$.

We can arrange the standard embedded genus-2 surface, $\partial V$, in $\mathbb{R}^{3}$ such that: (i) $A$ is the intersection of the $y-z$ plane with ${\Sigma}$ (ii) the points $Y\cap A$ and $Z\cap A$ lie along the $z$-axis with the $z$-coordinate of points on $A$ increasing from $Y\cap A$ to $Z\cap A$, (iii) no two points of $A_{1}$ or no two points of $A_{2}$ have the same $z$-coordinate. We will refer to this arrangement throughout this section. With this arrangement the $z$-coordinate can be thought of as a height function on $A_{1}$ and likewise on $A_{2}$. See Figure 5. Owing to this observation, if $x_{1},x_{2}\in A_{1}$ and if the height of $x_{1}$ is greater than the height of $x_{2}$, then we say $x_{1}$ is above $x_{2}$ or that $x_{2}$ is below $x_{1}$. If $x_{3}\in A_{1}$ such that the height of $x_{3}$ is between the heights of $x_{1}$ and $x_{2}$ then we say that $x_{3}$ is in between $x_{1}$ and $x_{2}$.

By an isotopy of ${Q_{{\Sigma}}}$, we can assume that it does not intersect the points $Z\cap A,Z\cap B,Y\cap A$ or $Y\cap C$. Since $|{Q_{{\Sigma}}}\cap A|$ is finite, there are only finitely many $AA$ arcs on ${\Sigma_{1}}$ or on ${\Sigma_{2}}$. These arcs are essential on ${\Sigma_{1}}$ or on ${\Sigma_{2}}$ as inessential arcs on a thrice boundered sphere are boundary reducible and a boundary reducing disk gives an isotopy of ${Q_{{\Sigma}}}$ reducing the intersection number of ${Q_{{\Sigma}}}$ with $A$. Also these $AA$ arcs do not intersect each other as ${Q_{{\Sigma}}}$ is a simple curve. Since every essential arc on a thrice-boundered sphere from a boundary component to itself is separating, these $AA$ arcs are separating arcs of ${\Sigma_{1}}$ and ${\Sigma_{2}}$.

The curve ${Q_{{\Sigma}}}$ is a finite collection of $2n$ arcs on ${\Sigma_{1}}$ and ${\Sigma_{2}}$ for some natural number $n$. Fix an orientation of ${Q_{{\Sigma}}}$ and number these arcs sequentially as $\rho_{1},\rho_{2},\ldots,\rho_{2n}$ following the orientation of ${Q_{{\Sigma}}}$ such that $\rho_{i}\cap\rho_{j}\in A\cup B\cup C$ if and only if $i,j\in\{1,2,...,2n\}$ and $i-j\equiv\pm 1\;\textrm{mod}\;2n$. Without loss of generality we assume that for each $k\in{\mathbb{N}}$, $\rho_{2k-1}$ is on ${\Sigma_{1}}$ and $\rho_{2k}$ is on ${\Sigma_{2}}$.

We now use the fundamental group of the handlebody $V$ to get more information about the $AA$, $AB$ and $AC$ arcs of ${Q_{{\Sigma}}}$ when $a_{Q}>b_{Q}+c_{Q}$. In particular we will show that an $AA$ arc of ${Q_{{\Sigma}}}$ on ${\Sigma_{1}}$ or on ${\Sigma_{2}}$ can wind around the circles $B$ or $C$ at most once.

Let $D_{X},D_{Y}$ and $D_{Z}$ be the essential disks in $V$ bounded by the curves $X,Y$ and $Z$ respectively. Consider the core curves of the handlebody $V$, dual to the disks $D_{Z}$ and $D_{Y}$, based at a point $x_{0}$ lying in the interior of $D_{X}$ and denote them by $B_{x_{0}}$ and $C_{x_{0}}$ respectively. Note that $B_{x_{0}}$ and $C_{x_{0}}$ are freely isotopic in $V$ to $B$ and $C$ respectively. The element in $\pi_{1}(V,x_{0})$ whose representative is a loop which traces $B_{x_{0}}$ once in the direction shown in Figure 7 will be denoted by $b$ and the element which traces $C_{x_{0}}$ once in the direction shown in Figure 7 will be denoted by $c$.

We consider the presentation $\langle b,c|-\rangle$ of the fundamental group of $V$ based at $x_{0}$ and henceforth refer to this presentation simply as $\pi_{1}(V)$. With a slight abuse of notation, we use the same letters to indicate the elements in $\pi_{1}(V)$ and the loops in $V$. By this abuse of notation the loop $B_{x_{0}}$ in Figure 7 is also denoted by $b$, for instance.

Now, let $Q$ be a reducing sphere which is in minimal position with the curves $A,B,C,X,Y$ and $Z$ and which satisfies $a_{Q}>b_{Q}+c_{Q}$. Consider some orientation of ${Q_{{\Sigma}}}$ and view it as a union of oriented arcs $\rho_{1},\rho_{2},\ldots,\rho_{2n}$ as mentioned earlier, where the orientation on $\rho_{i}$, $1\leq i\leq 2n$, is the one induced from the orientation of ${Q_{{\Sigma}}}$. Let $i$ denote an integer in the set $\{1,2,...,2n\}$ throughout the following discussion with addition and subtraction performed modulo $2n$ when indices are added or subtracted. Each $\rho_{i}$ is an $AA$, $AB$ or an $AC$ arc with a starting point indicated by its orientation. Since $\rho_{i}$ is an oriented arc we would distinguish between an $AC$ arc and a $CA$ arc, and likewise between an $AB$ and a $BA$ arc depending upon the orientation of $\rho_{i}$. Let $\delta_{i}$ be an arc contained in the interior of $V$ oriented from $x_{0}$ to the starting point of $\rho_{i}$ such that $\delta_{i}$ does not intersect $D_{Y}$ or $D_{Z}$. By $\overline{\delta_{i}}$, we mean $\delta_{i}$ with the reversed orientation. Now, $\sigma_{i}:=\delta_{i}\cdot\rho_{i}\cdot\overline{\delta_{i+1}}$ is a loop in $V$ based at $x_{0}$. This loop, $\sigma_{i}$, represents a word in the in the generators $b$ and $c$ in $\pi_{1}(V)$, which we call the word of $\rho_{i}$.

Also, ${Q_{{\Sigma}}}$ is freely homotopic to the product loop $\prod_{i=1}^{2n}\sigma_{i}$ and correspondingly represents a product word in the generators $b$ and $c$ in $\pi_{1}(V)$. Since ${Q_{{\Sigma}}}$ bounds a disk in $V$, the corresponding reduced word must be cyclically trivial. This last requirement imposes restrictions on the arcs $\rho_{i}$’s which we discuss in the following.

The union of curves $A\cup B\cup C\cup X\cup Y\cup Z$ divides ${\Sigma}$ into four disks, which we call the component hexagons of ${\Sigma}$, owing to the fact that the boundary of each of these disks contain exactly one arc each of the six curves $A,B,C,X,Y$ and $Z$. In this context, an arc of $\rho_{i}$ on a component hexagon of ${\Sigma}$ is of the type $DE$ if it starts on the curve $D$ and ends on the curve $E$ in the orientation of $\rho_{i}$, where $D,E\in\{A,B,C,X,Y,Z\}$. A sub-arc of $\rho_{i}$ is called its initial-arc if it is completely contained in one of the component hexagons of ${\Sigma}$ and is of the type $DE$ where $D\in\{A,B,C\}$ and $E\in\{X,Y,Z,A,B,C\}$. Similarly, a sub-arc of $\rho_{i}$ is called its terminal-arc if it is completely contained in one of the component hexagons of ${\Sigma}$ and is of the type $DE$ where $D\in\{X,Y,Z,A,B,C\}$ and $E\in\{A,B,C\}$. Note that neither the terminal nor the initial arc of $\rho_{i}$ of type $DE$ can have $D=E$.