Homeomorphisms of the 3-sphere that preserve a Heegaard splitting of genus two

Let ${\mathcal H}$ be the group of isotopy classes of orientation preserving homeomorphisms of $S^3$ that preserve a Heegaard splitting of genus two. In this paper, we use a tree in the barycentric subdivision of the disk complex of a handlebody of the splitting to obtain a finite presentation of ${\mathcal H}$.


Introduction
Let H g be the group of isotopy classes of orientation-preserving homeomorphisms of S 3 that preserve a Heegaard splitting of genus g, for g ≥ 2. It was shown by Goeritz [3] in 1933 that H 2 is finitely generated. Scharlemann [7] gave a modern proof of Goeritz's result, and Akbas [1] refined this argument to give a finite presentation of H 2 . In arbitrary genus, first Powell [6] and then Hirose [4] claimed to have found a finite generating set for the group H g , though serious gaps in both arguments were found by Scharlemann. Establishing the existence of such generating sets appears to be an open problem.
In this paper, we recover Akbas's presentation of H 2 by a new argument. First, we define the complex P (V ) of primitive disks, which is a subcomplex of the disk complex of a handlebody V in a Heegaard splitting of genus two. Then we find a suitable tree T , on which H 2 acts, in the barycentric subdivision of P (V ) to get a finite presentation of H 2 . In the last section, we will see that the tree T can be identified with the tree used in Akbas's argument [1].
Throughout the paper, (V, W ; Σ) will denote a Heegaard splitting of genus two of S 3 . That is, S 3 = V ∪ W and V ∩ W = ∂V = ∂W = Σ, where V and W are handlebodies of genus two. For essential disks D and E in a handlebody, the intersection D ∩ E is always assumed to be transverse and minimal up to isotopy. In particular, if D intersects E (indicated as D ∩ E = ∅), then D ∩ E is a collection of pairwise disjoint arcs that are properly embedded in both D and E.
Finally, Nbd(X) will denote a regular neighborhood of X and cl(X) the closure of X for a subspace X of a polyhedral space, where the ambient space will always be clear from the context.

Acknowledgement
The author would like to thank his advisor D. McCullough for his consistent encouragement and sharing his enlightening ideas on the foundations of this project. The author also would like to thank M. Scharlemann for his valuable suggestions and corrections. Finally, the author is grateful to the referee for carefully reading the paper and suggesting many improvements.

Primitive disks in a handlebody
We call an essential disk D in V primitive if there exists an essential disk D d in W such that ∂D and ∂D d have a single transverse intersection in Σ. Such a D d is called a dual disk of D. Notice that any primitive disk is nonseparating. We call a pair of disjoint, nonisotopic primitive disks in V a reducing pair of V . Similarly, a triple of pairwise disjoint, nonisotopic primitive disks is a reducing triple.
A 2-sphere P in S 3 is called a reducing sphere for (V, W ; Σ) if P intersects Σ transversally in a single essential circle and so intersects each handlebody in a single essential disk. It is clear that V ∩ P and W ∩ P are essential separating disks in V and W respectively. Proof. Let D be a primitive disk with a dual disk D d . Then the boundary P of a regular neighborhood of D ∪ D d is a reducing sphere that is disjoint from D (see Fig. 1). The sphere P splits V into two solid tori, X and Y , and we may assume that D is a meridian disk for X. Cutting S 3 along P , we get two 3-balls, B 1 and B 2 , where X and Y are contained in B 1 and B 2 respectively. Since the handlebody W is the boundary connected sum of cl(B 1 − X) and cl(B 2 − Y ) along the disk W ∩ P , we have Thus π 1 (S 3 − X) = π 1 (S 3 − Y ) = Z, and consequently X and Y are unknotted. Since D is a meridian of X, so cl(V − Nbd(D)) is ambient isotopic to Y , and is thus unknotted. The converse is a special case of Theorem 1 in [3].
Let E and D be nonseparating disks in V such that E ∩D = ∅, and let C ⊂ D be a disk cut off from D by an outermost arc α of D ∩E in D such that C ∩E = α. The arc α cuts E into two disk components, say G and H. The two disks E 1 = G ∪ C and E 2 = H ∪ C are called the disks obtained from surgery on E along C. Since E and D are assumed to intersect minimally, E 1 and E 2 are isotopic to neither E nor D, and moreover have fewer arcs of intersection with D than E had.
Notice that E 1 and E 2 are not isotopic to each other, otherwise E would be a separating disk. Finally, observe that both E 1 and E 2 are isotopic to a meridian disk of the solid torus cl(V − Nbd(E)), otherwise one of them would be isotopic to E. Thus E 1 and E 2 are all nonseparating disks in V .
Denote the arc C ∩ Σ by δ. The disks obtained from surgery on E ′ d along C are isotopic to a meridian disk of the solid torus cl(W − Nbd(E ′ d )). Hence δ must intersect ∂E which is a longitudinal circle of the solid torus.
LetΣ be the 2-holed torus obtained by cutting Σ along E ′ d . Denote by l ± the boundary circles ofΣ that came from ∂E ′ d . Then δ is an essential arc inΣ with endpoints in a single boundary circle l + or l − , say l + . Moreover, since ∂F d ∩ l + and ∂F d ∩ l − contain the same number of points, we also have an essential arc δ ′ of ∂F d ∩Σ with endpoints in the boundary circle l − . The arc δ ′ also intersects ∂E, otherwise δ ′ would be inessential. Thus ∂F d meets ∂E in at least two points and this contradicts that F d is a dual disk of E. Therefore, we conclude that  Proof. First, we consider the case when there exists a primitive disk E ′ such that {E, E ′ } is a reducing pair and E ′ is disjoint from C. (The existence of such an E ′ will be established in the final paragraph.) Let E d and E ′ d be the dual disks of E and E ′ respectively given by Lemma 2.2. By isotopy of D, we may assume that ∂D intersects ∂E d and ∂E ′ d minimally in Σ. Denote the arc C ∩ Σ by δ. It suffices to show that δ intersects ∂E ′ d in a single point, since then the resulting disks from surgery are both primitive with common dual disk E ′ d . Let Σ ′ be the 4-holed sphere obtained by cutting Σ along ∂E ∪ ∂E ′ . Denote by ∂E ± (resp. ∂E ′ ± ) the boundary circles of Σ ′ that came from ∂E (resp. ∂E ′ ). Then δ is an essential arc in Σ ′ with endpoints in a single boundary circle ∂E + or ∂E − , say ∂E − , and δ cuts off an annulus from Σ ′ . The boundary circle of the annulus that does not contain δ cannot be ∂E + otherwise one of the disks obtained from surgery on E along C would be isotopic to E. Thus it must be ∂E ′ + or ∂E ′ − , say ∂E ′ + . Let γ be a spanning arc of the annulus connecting ∂E − and ∂E ′ + . Then δ can be regarded as the frontier of Nbd(∂E ′ + ∪ γ) in Σ ′ (see Fig. 2(a)). In Σ ′ , the boundary circle ∂E d (resp. ∂E ′ d ) appears as an arc connecting ∂E + and ∂E − (resp. ∂E ′ + and ∂E ′ − ). We observe that δ intersects ∂E ′ d in at least one point. Suppose, for contradiction, that δ intersects ∂E ′ d in more than one point. Then γ intersects ∂E d (and ∂E ′ d ) at least once. In particular, there exists an arc component Fig. 2(b)). Now, ∂E and ∂E ′ represent the two generators x and y, respectively, of the free group π 1 (W ) = x, y , and ∂D represents a word w in terms of x and y. Each such word w can be read off from the intersections of ∂D with ∂E d and ∂E ′ d . In particular, the arc δ ∩ ∂A determines a sub-word of a word w of the form xy ±1 x −1 or x −1 y ±1 x, and hence each word w contains both x and x −1 (see Fig. 2 We claim that each word w is reduced, and therefore cyclically reduced. Since ∂D ∩ ∂E + and ∂D ∩ ∂E − contain the same number of points, we also have an essential arc δ ′ of ∂D ∩ Σ ′ whose endpoints lie in ∂E + . The arc δ ′ cuts off an annulus from Σ ′ , and the boundary circle of the annulus that does not contain δ ′ must be ∂E ′ − (see Fig. 2(a)). Since δ ′ also intersects ∂E ′ d in more than one point, the above argument holds for δ ′ . In particular, δ ′ cuts off an annulus A ′ from Σ ′ − ∂E d having one boundary circle ∂E ′ − . Observe that the annuli A and A ′ meet ∂E d on opposite sides from each other, as in Fig. 2 Then ∂A ∪ ∂A ′ cuts Σ ′′ into two disks, and each disk meets each boundary circle of Σ ′′ in a single arc. Consequently, we see there exists no arc component of ∂D in Σ ′′ that meets only one of ∂E d and ∂E ′ d in the same side. Thus we conclude that w contains neither x ±1 x ∓1 nor y ±1 y ∓1 . Since this is true for each word w, so each is cyclically reduced.
It is well known that a cyclically reduced word w in the free group x, y of rank two cannot be a generator if w contains x and x −1 simultaneously. Therefore, π 1 (W ∪ Nbd(D)) = x, y | w cannot be the infinite cyclic group, and consequently W ∪ Nbd(D) is not a solid torus. This contradicts, by Lemma 2.1, that D is primitive in V .
It remains to show that such a primitive disk E ′ does exist. Choose a primitive disk E ′ so that {E, E ′ } is a reducing pair. If C is disjoint from E ′ , we are done. Thus suppose C intersects E ′ . Then we have a disk F ⊂ C cut off from C by an outermost arc β of C ∩ E ′ in C such that F ∩ E ′ = β. By the above argument, the disks obtained from surgery on E ′ along F are primitive. One of them, say Figure 3.
at least β no longer counts. Then repeating the process for finding E ′′ , we get the desired primitive disk.

A sufficient condition for contractibility
In this section, we introduce a sufficient condition for contractibility of certain simplicial complexes. Let K be a simplicial complex. A vertex w is said to be adjacent to a vertex v if equal to v or if w spans a 1-simplex in K with v. The star st(v) of v is the maximal subcomplex spanned by all vertices adjacent to v.
A multiset is a pair (A, m), typically abbreviated to A, where A is a set and m : A → N is a function. In other words, a multiset is a set with multiplicity. An adjacency pair is a pair (X, v), where X is a finite multiset whose elements are vertices of K which are adjacent to v.
A remoteness function on K for a vertex v 0 is a function r from the set of vertices of K to N ∪ {0} such that r −1 (0) ⊂ st(v 0 ).
A function b from the set of adjacency pairs of K to N ∪ {0} is called a blocking function for the remoteness function r if it has the following properties whenever (X, v) is an adjacency pair with r(v) > 0.
(1) if b(X, v) = 0, then there exists a vertex w of K which is adjacent to v such that r(w) < r(v) and (X, w) is also an adjacency pair (see Fig. 3(a)), and Fig. 3(b)). A simplicial complex K is said to be flag if any collection of k+1 pairwise distinct vertices of K spans a k-simplex whenever any two span a 1-simplex. The proof of the following proposition is based on the proof of Theorem 5.3 in [5].
Proposition 3.1. Let K be a flag complex with base vertex v 0 . If K has a remoteness function r for v 0 that admits a blocking function b, then K is contractible.
Proof. It suffices to show that the homotopy groups are all trivial. Let f : S q → K, q ≥ 0, be a map carrying the base point of S q to v 0 . We may assume that f is simplicial with respect to a triangulation ∆ of S q . If r(f (u)) = 0 for every vertex u of ∆, then the image of f lies in st(v 0 ). Since K is a flag complex, f is null-homotopic.
Let us assume then that there exists a vertex u of ∆ such that r(f (u)) > 0. Choose such a vertex u so that r(f (u)) = n is maximal among all vertices of ∆. Our goal is to find a simplicial map g : S q → K with respect to some subdivision ∆ ′ of ∆ such that: • g is homotopic to f , • r(g(u)) < n, • r(g(z)) = r(f (z)) for every vertex z of ∆ with z = u, and • r(g(u ′ )) < n for every vertex u ′ of ∆ ′ − ∆. Then, repeating the process for other vertices whose values by r • f are also n, we obtain a simplicial map h with respect to some subdivision of ∆ such that h is homotopic to f and r(h(u)) < n for every vertex u. We thus complete the proof inductively.
Let u 1 , u 2 , · · · , u s be the vertices in the link of u in ∆ and let f (u) = v, f (u j ) = v j for j = 1, · · · , s, and X = {v 1 , v 2 , · · · , v s }. Then X is a finite multiset and (X, v) is an adjacency pair. If b(X, v) = 0, then there exists a vertex w in st(v) such that r(w) < r(v) and (X, w) is an adjacency pair. Define a simplicial map g : S q → K with respect to the same triangulation ∆ by g(u) = w and g(z) = f (z) for all vertices z = u. Since K is a flag complex, g is homotopic to f and we have r(g(u)) < n. Now suppose b(X, v) > 0. Then there exist v j ∈ X, and a vertex w j of K which is adjacent to v j such that (1) r(w j ) < r(v j ), (2) every element of X that is adjacent to v j is also adjacent to w j , and Construct a subdivision ∆ ′ of ∆ by introducing the barycenter u ′ j of the simplex u, u j as a vertex, and replacing each simplex of the form u, u j , z 1 , z 2 , · · · , z r by the two simplices u, u ′ j , z 1 , z 2 , · · · , z r and u ′ j , u j , z 1 , z 2 , · · · , z r . Define a simplicial map f ′ : S q → K with respect to ∆ ′ by f ′ (u ′ j ) = w j and f ′ (z) = f (z) for every vertex z of ∆. Since K is a flag complex, f ′ is homotopic to f . Now Y is the image of the vertices of the link of u in ∆ ′ , and r(f ′ (u ′ j )) = r(w j ) < r(v j ) ≤ r(v) = n. Repeating finitely many times, we obtain a subdivision ∆ ′′ of ∆ and a simplicial map f ′′ with respect to ∆ ′′ so that f ′′ is homotopic to f and b(Y ′′ , v) = 0, where Y ′′ is the image of the vertices of the link of u in ∆ ′′ . Observe that r(f ′′ (u ′′ )) < n for every vertex u ′′ of ∆ ′′ − ∆ and f ′′ (z) = f (z) for every vertex z of ∆. Since b(Y ′′ , v) = 0, we obtain a simplicial map g : S q → K with respect to ∆ ′′ as above such that g is homotopic to f , r(g(u)) < n, and r(g(u ′′ )) < n for every vertex u ′′ of ∆ ′′ − ∆.

The complex of primitive disks
The disk complex D(V g ) of a handlebody V g of genus g, for g ≥ 2, is a simplicial complex defined as follows. The vertices of D(V g ) are isotopy classes of essential disks in V g , and a collection of k + 1 vertices spans a k-simplex if and only if it admits a collection of representative disks which are pairwise disjoint. When V g is a handlebody in a Heegaard splitting (V g , W g ; Σ g ) of S 3 , the complex of primitive disks P (V g ) of V g is defined to be the full subcomplex of D(V g ) spanned by vertices whose representatives are primitive disks in V g . As before, we write V and (V, W ; Σ) for V 2 and (V 2 , W 2 ; Σ 2 ) respectively. Notice that D(V ) and P (V ) are 2-dimensional.
It is a standard fact that any collection of isotopy classes of essential disks in V g can be realized by a collection of representative disks that have pairwise minimal intersection. One way to see this is to choose the disks in their isotopy classes so that their boundaries are geodesics with respect to some hyperbolic structure on the boundary surface Σ g and then remove simple closed curve intersections of the disks by isotopy. In particular, for a collection {v 0 , v 1 , · · · , v k } of vertices of D(V g ), if v i and v j bound a 1-simplex for each i < j, then {v 0 , v 1 , · · · , v k } is realized by a collection of pairwise disjoint representatives. Thus we have Theorem 4.2. If K is a full subcomplex of D(V g ) satisfying the following condition, then K is contractible.
• Suppose E and D are any two disks in V g which represent vertices of K such that E ∩ D = ∅. If C ⊂ D is a disk cut off from D by an outermost arc α of D ∩ E in D such that C ∩ E = α, then at least one of the disks obtained from surgery on E along C also represents a vertex of K.
Proof. Since K is a flag complex, by Lemma 4.1, it suffices to find a remoteness function that admits a blocking function as in Proposition 3.1. Fix a base vertex v 0 of K. Define a remoteness function r on the set of vertices of K by putting r(w) equal to the minimal number of intersection arcs of disks representing v 0 and w. Let (X, v) be an adjacency pair in K where r(v) > 0 and X = {v 1 , v 2 , · · · , v n }. Choose representative disks E, E 1 , · · · , E n and D of v, v 1 , · · · , v n and v 0 respectively so that they have transversal and pairwise minimal intersection. Since r(v) > 0, so D ∩ E = ∅. Let C ⊂ D be a disk cut off from D by an outermost arc α of D ∩ E in D such that C ∩ E = α. Observe that each E i is disjoint from the arc α.
Let b 0 = b 0 (E, E 1 , · · · , E n , D) be the minimal number of arcs {C ∩ E 1 } ∪ {C ∩ E 2 } ∪ · · · ∪ {C ∩ E n } in C as we vary over such disks C cut off from D. Define b = b(X, v) to be the minimal number b 0 as we vary over such representative disks of v, v 1 , · · · , v n and v 0 . We verify that b is a blocking function for the remoteness function r as follows.
Suppose, first, b(X, v) = 0. There exist then representative disks E ′ , E ′ 1 , · · · , E ′ n and D ′ of v, v 1 , · · · , v n and v 0 respectively and exists, as above, a disk C ′ ⊂ D ′ cut off from D ′ so that C ′ ∩ (E ′ 1 ∪ · · · ∪ E ′ n ) = ∅. By the assumption, a disk obtained from surgery on E ′ along C ′ represents a vertex w ′ of K again, and (X, w ′ ) is an adjacent pair. We have r(w ′ ) < r(v) since the arc C ′ ∩ E ′ no longer counts.
Next, suppose b(X, v) > 0. Choose then representative disks E ′′ , E ′′ 1 , · · · , E ′′ n and D ′′ of v, v 1 , · · · , v n and v 0 respectively, and, similarly, a disk C ′′ ⊂ D ′′ cut off from D ′′ so that they realize b(X, v). Choose an outermost arc β of C ′′ ∩ E ′′ k in C ′′ that cuts off a disk F ′′ ⊂ C ′′ such that (1) F ′′ ∩ E ′′ k = β, (2) F ′′ is disjoint from the arc C ′′ ∩ E ′′ , and (3) F ′′ contains no arc of intersection of any E ′′ i which is disjoint from E ′′ k (note that β may still intersect an arc of some C ′′ ∩ E ′′ i ). Then a disk obtained from surgery on E ′′ k along F ′′ represents a vertex w ′′ of K by the assumption. By the construction, every element of X that is adjacent to v k is also adjacent to w ′′ . We have r(w ′′ ) < r(v k ) and b(Y, v) < b(X, v), where Y = (X − {v k }) ∪ {w ′′ }, since β no longer counts. Theorem 4.2 shows that D(V g ) is contractible since a disk obtained from surgery on an essential disk is also essential. Now we return to the genus two case.

A finite presentation of H 2
In this section, we will use our description of T to recover Akbas's presentation of H 2 given in [1]. The tree T is invariant under the action of H 2 . In particular, H 2 acts transitively on the set of vertices of T which are represented by reducing triples. The disks in a reducing triple are permuted by representative homeomorphisms of H 2 . It follows that the quotient of T by the action of H 2 is a single edge. For convenience, we will not distinguish disks and homeomorphisms from their isotopy classes in their notations.
Fix two vertices P = {D 1 , D 2 } and Q = {D 1 , D 2 , D 3 } that are endpoints of an edge e of T . Denote by H P , H Q and H e the subgroups of H 2 that preserve P , Q and e respectively. By the theory of groups acting on trees due to Bass and Serre [8], H 2 can be expressed as the free product of H P and H Q with amalgamated subgroup H e .
It is not difficult to describe these subgroups of H 2 . We sketch the argument as follows. Let D ′ 1 and D ′ 2 be the disjoint dual disks of D 1 and D 2 respectively, and let D ′ 3 be the dual disk of both D 1 and D 2 which is disjoint from D ′ 1 , D ′ 2 and D 3 (see Fig. 4(a)). The reducing triple {D ′ 1 , D ′ 2 , D ′ 3 } of W is uniquely determined by Lemma 2.2. Denote ∂D i and ∂D ′ i by d i and d ′ i , for i ∈ {1, 2, 3}, respectively. Consider the group H P . The elements of H P also preserve D ′ 1 ∪ D ′ 2 . Let H ′ P be the subgroup of elements of H P which preserve each of D 1 and D 2 . Since the union of Σ with the four disks D i and D ′ i , for i ∈ {1, 2}, separates S 3 into (two) 3-balls, H ′ P can be identified with the group of isotopy classes of orientation preserving homeomorphisms of the annulus obtained by cutting Σ along , which preserve each of d 1 and d 2 . This group is generated by two elements β and β ′ (π-twists of each boundary circle) and has relations (ββ ′ ) 2 = 1 and ββ ′ = β ′ β. As generators of H ′ P , β and β ′ are shown in Fig. 4(b) and Fig. 4(c). Let γ be the order two element of H P which interchanges D 1 and D 2 , as shown in Fig. 4(d). Then H P is an extension of H ′ P by γ with relations γβγ = β ′ −1 and γβ ′ γ = β −1 .
Next, let V ′ be a 3-ball cut off from V by D 1 ∪ D 2 ∪ D 3 , and let H ′ Q be the subgroup of elements of H Q which preserve V ′ . Since the union of Σ with the six disks D i and D ′ i , for i ∈ {1, 2, 3}, separates S 3 into (four) 3-balls, H ′ Q can be identified with the group of isotopy classes of orientation preserving homeomorphisms of the

3-holed sphere ∂V
respectively. This group is a dihedral group generated by γ and δ, where δ is the order three element shown in Fig. 4(e).
Let α be the order two element of H Q which preserves each D 1 , D 2 and D 3 but takes V ′ to another component cut off from V by D 1 ∪ D 2 ∪ D 3 , as shown in Fig. 4(f). Then H Q is an extension of H ′ Q by α with relations αγα = γ and αδα = δ. Finally, it is easy to see that H e is generated by γ and α. Observing that β ′ = γβ −1 γ and α = ββ ′ , we get a finite presentation of H 2 with generators β, γ, and δ.

The Scharlemann-Akbas Tree
Scharlemann [7] constructed a connected simplicial 2-complex Γ which deformation retracts to a certain graphΓ on which H 2 acts, and gave a finite generating set of H 2 . In this section, we describe Γ andΓ briefly and compare them with P (V ) and T . For details about Γ andΓ, we refer the reader to [7] and [1].
Let P and Q be reducing spheres for a genus two Heegaard splitting (V, W ; Σ) of S 3 . Then P ∩ V and Q ∩ V are essential separating disks in V and each of them cuts V into two unknotted solid tori. Define the intersection number P · Q to be the minimal number of arcs of P ∩ Q ∩ V up to isotopy. Then we observe that P · Q ≥ 2 if P ∩ V and Q ∩ V are not isotopic to each other in V . The simplicial complex Γ for (V, W ; Σ) is defined as follows. The vertices are the isotopy classes of reducing spheres relative to V and a collection P 0 , P 1 , · · · , P k of k + 1 vertices bounds a k-simplex if and only if P i · P j = 2 for all i < j. It turns out that Γ is a 2-complex and each edge of Γ lies on a single 2-simplex. Thus Γ deformation retracts naturally to a graphΓ in which each 2-simplex in Γ is replaced by the cone on its three vertices, and Akbas [1] showed thatΓ is a tree.
We observe that the reducing spheres for (V, W ; Σ) correspond exactly to the reducing pairs in V up to isotopy. That is, a reducing sphere cuts V into two unknotted solid tori, whose meridian disks are primitive in V . Conversely, a reducing pair has a unique pair of disjoint dual disks, as in Lemma 2.2, and the corresponding reducing sphere is the boundary of a small regular neighborhood of the union of either of the disks with its dual.
It is routine to check that two reducing spheres represent two vertices connected by an edge in Γ if and only if the corresponding reducing pairs are contained in a reducing triple. So the correspondence from reducing spheres to reducing pairs determines a natural embedding of Γ into P (V ) so that the image ofΓ is identified with the tree T (see Fig. 5). Therefore, Corollary 4.4 can be considered as an alternate proof thatΓ is a tree.