The mapping class groups of reducible Heegaard splittings of genus two
Abstract.
The manifold which admits a genus reducible Heegaard splitting is one of the sphere, , lens spaces and their connected sums. For each of those manifolds except most lens spaces, the mapping class group of the genus 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 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 Heegaard splittings of lens spaces are all finitely presented.
2000 Mathematics Subject Classification:
Primary 57N10; 57M60.Introduction
It is well known that every closed orientable manifold can be decomposed into two handlebodies and of the same genus for some . That is, and , a genus closed orientable surface. We call such a decomposition a Heegaard splitting for the manifold and denote it by . The surface is called the Heegaard surface of the splitting, and the genus of is called the genus of the splitting. The sphere admits a Heegaard splitting of each genus , and a lens space a Heegaard splitting of each genus .
The mapping class group of a Heegaard splitting for a manifold is the group of isotopy classes of orientationpreserving diffeomorphisms of the manifold that preserve each of the two handlebodies of the splitting setwise. We call such a group the Goeritz group of the splitting, or a genus Goeritz group when the splitting has genus as well. Further, when a manifold admits a unique Heegaard splitting of genus up to isotopy, we call the Goeritz group of the splitting simply the genus Goeritz group of the manifold without mentioning a specific splitting. We note that the mapping class group of a Heegaard splitting is a subgroup of the mapping class group of the Heegaard surface.
It is important to understand the structure of a Goeritz group, in particular, a finite generating set or a finite presentation of it if any. For example, using a finite presentation of the genus Goeritz group of the 3sphere given in [11], [23], [1] and [4], it is constructed a new theory on the collection of the tunnel number knots in [10]. Further, it has been an open problem whether the fundamental groups of the spaces of genus Heegaard splittings of lens spaces are finitely generated/presented or not, see [17]. If the genus Goeritz groups are shown to be finitely presented, then so are those fundamental groups.
In [6] a finite presentation of the genus Goeritz group of was obtained, and in [7] finite presentations of the genus Goeritz groups were obtained for the connected sums whose summands are or lens spaces. We refer the reader to [15], [16], [24], [18] and [9] for finite presentations or finite generating sets of the Goeritz groups of several Heegaard splittings and related topics.
For the genus Goeritz groups of lens spaces, finite presentations are obtained only for a small class of lens spaces in [5] and [8]. That is, for the lens spaces , , under the condition . In this work, we study the remaining generic lens spaces, the case of . We show that the genus Goeritz group of each of those lens spaces is again finitely presented and obtain an explicit presentation, which is introduced in Theorem 4.8 in Section 4. The manifold which admits a genus reducible Heegaard splitting is one of the sphere, , lens spaces and their connected sums. Therefore, Theorem 4.8 together with the previous results mentioned above implies the following.
Theorem 0.1.
The mapping class group of each of the reducible Heegaard splittings of genus is finitely presented.
In other words, the theorem says that the mapping class groups of genus Heegaard splittings of Hempel distance are all finitely presented. It is shown in [20] and [16] that the mapping class groups are all finite for the Heegaard splittings of Hempel distance at least . The mapping class groups of the splittings of Hempel distances and still remain mysterious. (Here note that there are no genus splittings of Hempel distance .)
To obtain a presentation of the Goeritz group, we have constructed a simply connected simplicial complex on which the group acts “nicely”, in particular, so that the quotient of the action is a simple finite complex. And then we calculate the isotropy subgroups of each of the simplices of the quotient, and express the Goeritz group in terms of those subgroups.
For the genus Heegaard splitting of a lens space with , we have constructed the primitive disk complex, denoted by , whose vertices are defined to be the isotopy classes of the primitive disks in the handlebody . In [8], the combinatorial structure of the complex are fully studied and it was shown that is simply connected, in fact contractible, under the condition , and is used to obtained the presentation of the Goeritz group. In the case of , the complex is no longer simply connected. In fact, it consists of infinitely many tree components isomorphic to each other. In the present paper, we will construct a new simplicial complex for this case, which we will call the “tree of trees”, whose vertices are the tree components of .
In Section 1, it will be briefly reviewed the primitive disk complex for the genus Heegaard splitting of each lens space. In Section 2, we construct the complex “tree of trees” for the case of and develop some related properties that we need. In the main section, Section 4, the action of the Goeritz group on the tree of trees will be investigated to obtain the presentation of the group. Right before Section 4, the simplest example of our case , the lens space , will be studied in detail in Section 3 as a motivating example. In the final section, we show that the fundamental groups of the spaces of genus Heegaard splittings of lens spaces are all finitely presented (up to the Smale Conjecture for ).
We use the standard notation for a lens space. We refer [22] to the reader. The integer can be assumed to be positive. It is well known that two lens spaces and are diffeomorphic if and only if and . Thus, we will assume for the lens space . Note that each lens space admits a unique Heegaard splitting of each genus up to isotopy by [3]. Throughout the paper, any disks in a handlebody are always assumed to be properly embedded, and their intersection is transverse and minimal up to isotopy. In particular, if a disk intersects a disk , then is a collection of pairwise disjoint arcs that are properly embedded in both and . For convenience, we will not distinguish disks (or union of disks) and diffeomorphisms from their isotopy classes in their notation. Finally, will denote a regular neighborhood of and the closure of for a subspace of a space, where the ambient space will always be clear from the context.
1. The primitive disk complexes
1.1. The nonseparating disk complex for the genus handlebody
Let be a genus handlebody. The nonseparating disk complex, denoted by , of is a simplicial complex whose vertices are the isotopy classes of nonseparating disks in such that a collection of vertices spans a simplex if and only if it admits a collection of representative disks which are pairwise disjoint. We note that the disk complex is dimensional and every edge of is contained in infinitely but countably many simplices. In [19], it is proved that and the link of any vertex of are all contractible. Thus, the complex deformation retracts to a tree in its barycentric subdivision spanned by the barycenters of the simplices and simplices, which we call the dual tree of . See Figure 1. We note that each component of any full subcomplex of is contractible.
Let and be nonseparating disks in and suppose that the vertices of the disks in , which we denote by and again, are not adjacent to each other, that is, . In the barycentric subdivision of , the links of the vertices and are disjoint trees. Then there exists a unique shortest path in the dual tree of connecting the two links. Let , , , be the sequence of vertices of this path. We note that each is trivalent while each has infinite valency in the dual tree. Let , be the 2simplices of whose baricenters are the trivalent vertices , respectively. We call the full subcomplex of spanned by the vertices of , the corridor connecting and and we denote it by . See Figure 2. Let and be the two vertices of other than . We call the pair the principal pair of with respect to for the corridor .
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Let and be nonseparating disks in . We assume that intersects transversely and minimally. Let be an outermost subdisk of cut off by , that is, is a disk cut off from by an arc of in such that . The arc cuts into two disks, say and . Then we have two disjoint disks and which are isotopic to the disks and respectively. We note that each of and is isotopic to neither nor , and each of and has fewer arcs of intersection with than had since at least the arc no longer counts, as and are assumed to intersect minimally. Further, it is easy to check that both and are nonseparating, and these two disks are determined without depending on the choice of the outermost subdisk of cut off by (this is a special property of a genus handlebody). We call the disks and the disks from surgery on along the outermost subdisk or simply the disks from surgery on along .
Let , and be nonseparating disks in . Assume that and are nonseparating disks in which are disjoint and are not isotopic to each other, and that intersects transversely and minimally. In the same way to the above, we can consider the surgery on along an outermost subdisk of cut off by . In fact, one of the two resulting disks from the surgery is isotopic to either or , and the other, denoted by , is isotopic to none of and (this is also a special property of a genus handlebody). We call the disk the disk from surgery on along . (If is already disjoint from , then define simply to be .)
Lemma 1.1.
Let be the corridor connecting and . Then the disks of the principal pair are exactly the disks from surgery on along .
Proof.
We use the induction on the number of the simplices of the corridor. If , the conclusion holds immediately since each of and is disjoint from , and so any outermost subdisk of cut off by is also disjoint from . If , choose a vertex, say , of other than that is not adjacent to . Then we have the (sub)corridor connecting and for some . See Figure 2. By the assumption of the induction, the disks and are exactly the disks from surgery on along . Since is disjoint from , the disks from surgery on along are the same to those from surgery on along . ∎
Consider any two consecutive simplices and , , of a corridor connecting and . When we write and as triples of vertices, we see that are the principal pair of with respect to for the (sub)corridor . Further, the followings are immediate from Lemma 1.1.

and are the disks from surgery on along .

is the disk from surgery on along .
This observation implies the following lemma.
Lemma 1.2.
Let be the corridor connecting and . For each , we write the edge and the simplex . Let be a vertex that is not adjacent to . If is the disk from surgery on along for each , then the corridor connecting and contains the corridor .
1.2. The primitive disk complexes for lens spaces
Let be the genus Heegaard splitting of a lens space . A disk properly embedded in is said to be primitive if there exists a disk properly embedded in such that the two loops and intersect transversely in a single point. Such a disk is called a dual disk of , which is also primitive in having a dual disk . Note that both and are solid tori. Primitive disks are necessarily nonseparating.
The primitive disk complex for the genus splitting is defined to be the full subcomplex of spanned by the primitive disks in . If a genus Heegaard splitting admits primitive disks, then the manifold is one of the sphere, or a lens space, and so we can define the primitive disk complex for each of those manifolds. The combinatorial structure of the primitive disk complexes for each of the sphere and has been well understood in [4] and [6]. For the lens spaces, we have the following results from [8].
Theorem 1.3 (Theorems 4.2 and 4.5 in [8]).
For a lens space with , the primitive disk complex for the genus Heegaard splitting of is contractible if and only if . If , then is not connected and consists of infinitely many tree components.
In the case of , each vertex of any tree component of has infinite valency, that is, for each primitive disk in there exist infinitely many nonisotopic primitive disks disjoint from . Thus, all the tree components of are isomorphic to each other.
2. The tree of trees
2.1. The primitive disks
Let be the genus Heegaard splitting of a lens space . In this subsection, we will develop several properties of the primitive disks in and we need, in particular, some sufficient conditions for the nonprimitiveness. Each simple closed curve on the boundary of the genus handlebody represents an element of the free group of rank 2. The following is a well known fact.
Lemma 2.1 (Gordon [12]).
Let be a nonseparating disk in . Then is primitive if and only if represents a primitive element of .
Here, an element of a free group is said to be primitive if it is a member of a generating set. Primitive elements of the rank free group has been well understood. In particular, we have the following property.
Lemma 2.2 (OsborneZieschang [21]).
Given a generating pair of the free group of rank , a cyclically reduced form of any primitive element can be written as a product of terms each of the form or , or else a product of terms each of the form or , for some and some .
Therefore, we see that no cyclically reduced form of a primitive element in terms of and can contain and and and simultaneously.
Let be a complete meridian system of the genus handlebody . Assign symbols and to the oriented circles and respectively. Then any oriented simple closed curve on intersecting transversely and minimally represents an element of the free group , whose word in can be read off from the intersections with and . Let be an oriented simple closed curve on that meets transversely and minimally. The following lemma is given in [8].
Lemma 2.3 (Lemma 3.3 in [8]).
With a suitable choice of orientations of and , if a word in corresponding to contains one of the pairs of terms

both of and , or

both of and for ,
then the element of represented by cannot be a positive power of a primitive element.
We introduce three more sufficient conditions for nonprimitiveness as follows.
Lemma 2.4.
With a suitable choice of orientations of and , if a word in corresponding to contains one of the pairs of terms

both of and ,

both of and , or

both of and ,
then the element of represented by cannot be a positive power of a primitive element.
Proof.
Let be the holed sphere cut off from along . Denote by and (by and , respectively) the boundary circles of coming from (from , respectively).
Suppose first that determines a word containing both and . We can assume that there are two subarcs and of such that connects and , and connects and as in Figure 3.
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Since and , we must have two other arcs and of such that connects and , and connects and . See Figure 3. Consequently, there exists no arc component of whose endpoints lines on the same boundary component of . That is, any word corresponding to contains neither nor , and hence, it is cyclically reduced. Since that word contains both and (or and ), cannot represent (a positive power of) a primitive element of . The case where determines a word containing both and can be proved in the same way.
Next suppose that determines a word containing both and . Then there are two arcs and of such that connects and , and connects and . By a similar argument to the above, we must have two other arcs and of such that connects and , say , and connects and . See Figure 4.
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Then we see again that the word corresponding to is cyclically reduced. Since that word contains both of and , cannot represent (a positive power of) a primitive element. ∎
Lemma 2.5.
With a suitable choice of orientations of and , if a word in corresponding to contains a term of the form , where and , then the element of represented by cannot be a positive power of a primitive element.
Proof.
Set . Let the subarc of corresponding to the subword . By cutting the Heegaard surface along , we get a 4holed sphere . Let and be the holes coming from and respectively. Without loss of generality we can assume that . If , then we get the conclusion by Lemma 2.4. Thus, we assume that , this implies that the word contains the term . Let , , be the subarcs of corresponding to the term , , . Note that on the surface the arc connects the two circles and , connects the circle to itself, and connects the two circles and . Let be the band sum of and along , that is, is the frontier of a regular neighborhood of the union . Then and form a new complete meridian system of . See Figure 5.
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We assign the same symbol to . Then the arc determines a word of the form , where , and . Applying this argument finitely many times, we end with the case where is reduced (in particular ), thus, the conclusion follows. ∎
Lemma 2.6.
With a suitable choice of orientations of and , if a word in corresponding to contains both of terms of the forms and , where for , for and , then the element of represented by cannot be a positive power of a primitive element.
Proof.
Set , , and . Let and be the subarcs of corresponding to the subwords and respectively. By cutting the Heegaard surface along , we get a 4holed sphere . Let and be the holes coming from and respectively.
Suppose first that both subwords and are reduced, so and . If both of and are nonzero and these have different signs, then we get the conclusion by Lemma 2.4. Thus, we assume that and have the same sign or one of them is zero. Without loss of generality we can assume that . Suppose that . Then and () and thus, by Lemma 2.4, cannot represent a primitive element of . Suppose that . Let be the subarc of corresponding to the term . Then connects two circles and in . Let be the band sum of and along . Then and form a new complete meridian system of . See Figure 6.
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Assigning the same symbol to , the arc determines a word of the form while determines . Applying this argument times, we finally end with the case of , thus, the conclusion follows by induction.
Next suppose that at least one of or is reducible. Without loss of generality we can assume that is reducible and . Since contains the term , there is no arc in connecting and . See Figure 7.
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This implies that the word represented by cannot contain the term , thus, . Further if one of , and is , then cannot represent a primitive element of by Lemma 2.4. Thus, we can assume that . Let and be the subarcs of corresponding to the subwords and respectively. Then consits of arcs shown in Figure 8. Let be the subarc of correspoinding to the word and let be the band sum of and along .
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Then and form a new complete meridian system of . Assigning the same symbol to , the arc determines a word of the form (), while determines (). Here note that we have , , and . We repeat this argument until both words correspoinding to and become reduced. We claim that this process finishes in finitely many times. Suppose not. Then after repeating this process finitely many times, we finally end with the case where the word correspoinding to one of and (say ) is . Then again by Figure 7 the word corresponding to must be reduced. This is a contradiction. Now the conclusion follows from the argument of the case where both and are reduced. ∎
2.2. Shells
Let be the genus Heegaard splitting of a lens space with . We briefly review the definition of a shell, which is a special subcomplex of the star neighborhood of a vertex of a primitive disk in the nonseparating disk complex , introduced in Section 3.3 of [8]. We call a pair of disjoint, nonisotopic primitive disks in a primitive pair in . A nonseparating disk properly embedded in is said to be semiprimitive if there is a primitive disk in disjoint from . A primitive pair and a semiprimitive disks in can be defined in the same way.
Let be a primitive disk in . Choose a dual disk of . Then we have unique semiprimitive disks and in and respectively that are disjoint from . We denote the solid torus simply by . By a suitable choice of the oriented longitude and meridian of , the circle can be assumed to be a curve on the boundary of , where and is the unique integer satisfying and . We say that is of type if is a curve on .
Suppose first that is of type. Then the circle is a curve on the boundary of the solid torus . We construct a sequence of disks , starting at the semiprimitive disk as follows. Choose an arc on so that meets each of and in exactly one point of , and is disjoint from . Let be a disk in which is the band sum of and along . Then the disk is disjoint from and intersects in a single point and in points. Then inductively, we construct from until we get by taking the band sum with along an arc on such that meets each of and in exactly one point of , and is disjoint from . Figure 9 illustrates the case when is of type, and so is a curve on the boundary of the solid torus in the lens space .
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We call the full subcomplex of spanned by the vertices , and a shell centered at the primitive disk and denote it simply by , see Figure 10 for example.
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A shell is a 2dimensional subcomplex of . It is straightforward from the construction that for , consists of arcs. We note that there are infinitely many choice of such an arc , and so of the disk . But once we choose , the arcs for are uniquely determined and so are the successive disks , .
Assign symbols and to the oriented circles and respectively. Then the oriented boundary circles , , from the shell represent elements of the free group . We observe that, with a suitable choice of orientation, the circles , , determine the words of the form , , respectively.
Lemma 2.7 (Lemmas 3.8 and 3.13 in [8]).
Let be a shell centered at a primitive disk in . Then we have

and are semiprimitive.

is primitive if and only if where is the unique integer satisfying and .

and are of type while and are of type.
We have constructed a shell by assuming is of a type. If is of type, then is a shell, and Lemma 2.7 still holds by exchanging and in the conclusion.
Lemma 2.8 (Lemma 3.6 in [8]).
Let be a semiprimitive disk in , and let be a primitive disk in disjoint from . If a primitive or semiprimitive disk in intersects transversely and minimally, then the disk from surgery on along is a primitive disk, which has a common dual disk with .
By Lemma 2.8, given a primitive disk and a semiprimitive disk in disjoint from , any primitive disk in determines a unique shell such that if is disjoint from , and is the disk from surgery on along if intersects . The following is a generalization of Lemma 2.8.
Lemma 2.9 (Lemma 3.10 in [8]).
Let be a shell centered at a primitive disk in , and let be a primitive or semiprimitive disk in . For ,

if is disjoint from and is isotopic to none of and , then is isotopic to either or , and

if intersects , then the disk from surgery on along is isotopic to either or .
2.3. Bridges
Let be the genus Heegaard splitting of a lens space with and . Throughout the subsection, we fix the followings.

An integer where is the unique integer satisfying and ; and

The integers and satisfying with .
We recall from Theorem 1.3 that consists of infinitely many isomorphic tree components in this case. The key of the disconnectivity is that we can find nonadjacent primitive disks and in such that the corridor connecting and contains no vertices of primitive disks except and . Then and are contained in different components of since the dual complex of is a tree.
We call a corridor in a bridge when it connects vertices and of primitive disks, and contains no vertices of primitive disks except and . In this case, we denote the bridge by instead of . We recall that the two vertices of the