Abstract
Abstract. A manifold which admits a reducible genus-2 Heegaard splitting is one of the 3-sphere, S 2 × 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 × S 1 or the connected sum whose summands are lens spaces or S 2 × 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 precise description of each of the complexes for the genus-2 Heegaard splittings of lens spaces. A remarkable fact is that the complexes for most lens spaces are not contractible and even not connected.
Introduction
Let (V, W ; Σ) be a genus-2 Heegaard splitting of a closed orientable 3-manifold M . That is, M = V ∪ W and V ∩ W = ∂V = ∂W = Σ, where V and W are genus-2 handlebodies. A Haken sphere S for the splitting (V, W ; Σ) is a separating sphere in M that intersects Σ transversely in a single essential circle. The circle S ∩ Σ is necessarily separating in Σ. Two Haken spheres S and T are said to be equivalent if S ∩ Σ is isotopic to T ∩ Σ in Σ. We note that the study of Haken spheres of (V, W ; Σ) corresponds to the study of tunnel number-1 links in M .
Introduction
Given two Haken spheres S and T for the splitting (V, W ; Σ), the intersection number S · T is defined to be the minimal number of points of S ∩ T ∩ Σ up to isotopy in Σ. Then S and T are equivalent if and only if S · T = 0, and otherwise we have S · T ≥ 4 by Scharlemann-Thompson [15] . When S · T = 4, we say that S and T are joined by a 4-gon replacement or by an elementary move. The notion of 4-gon replacement proposes a natural simplicial complex called the complex of Haken spheres, or simply the sphere complex, for the splitting (V, W ; Σ). The sphere complex is defined as follows. The vertices of the sphere complex are the equivalence classes of Haken spheres, and a collection of distinct k+1 vertices S 0 , S 1 , . . . , S k spans a k-simplex if and only if S i ·S j = 4 for all 0 i < j k.
Introduction
If a genus-2 Heegaard splitting of a manifold admits a Haken sphere, then the manifold is one of the 3-sphere, S 2 × S 1 , lens spaces and their connected sums. We note The second-named author is supported in part by the Grant-in-Aid for Young Scientists (B), JSPS KAKENHI Grant Number 26800028.
Introduction
that the genus-2 Heegaard splittings of these manifolds are completely classified up to isotopy by Waldhausen [16] , Bonahon-Otal [2] and [6] . In fact, among them, each prime manifold admits a unique genus-2 Heegaard splitting, while each non-prime one admits at most two genus-2 Heegaard splittings. It is natural to ask whether the structures, such as dimension, connectivity and contractibility, of the sphere complex of each genus-2 Heegaard splitting reflect the topological structure of the corresponding 3-manifold. In [14] , Scharlemann showed that the sphere complex for the genus-2 Heegaard splitting of the 3-sphere is connected, and subsequently a complete description of its shape was given in Akbas [1] and Cho [3] . In fact, this sphere complex is 2-dimensional contractible complex and each edge is contained in a single 2-simplex. In Lei [10] and Lei-Zhang [11] , it was shown that the sphere complex is connected for a genus-2 Heegaard splitting of the connected sum whose summands are lens spaces or S 2 × S 1 . In [6] later, this result is refined that they are all contractible. In fact, the sphere complex is a tree if the summands are both lens spaces or both S 2 × S 1 , and is a contractible 3-dimensional complex if one summand is S 2 × S 1 and the other one a lens space. In [7] , it is shown that the sphere complex is a 3-dimensional contractible complex when the maninfold is S 2 × S 1 .
Introduction
In this paper, we investigate the sphere complexes for the remaining cases, that is, for the genus-2 Heegaard splittings of lens spaces, and give a description of each of those complexes. We state the main result as follows.
Introduction
Theorem 0.1. Let (V, W ; Σ) be the genus-2 Heegaard splitting of a lens space L = L(p, q) with 1 ≤ q ≤ p/2, and let S(V, W ; Σ) be the sphere complex for (V, W ; Σ).
Introduction
(1) S(V, W ; Σ) contains no 3-cycle, that is, it is a 1-dimensional complex. Each vertex of S(V, W ; Σ) has infinite valency.
Introduction
is not connected, and it consists of infinitely many tree components. In the theorem, an n-cycle, n ≥ 3, means a 1-dimensional subcomplex of S(V, W ; Σ) consisting of exactly n distinct edges and n distinct vertices such that each vertex has valency two. It is remarkable that the sphere complex is not connected for every lens space other than L(p, 1), and the complexes for L(2, 1) and L(3, 1) are connected but not contractible. The idea of the proof is based on the following fact. Given the genus-2 Heegaard splitting of each of lens spaces, there is a one-to-one correspondence between the collection of Haken spheres and the collection of pairs of a primitive disk and one of its dual disks up to isotopy. Then the 4-gon replacement of a Haken sphere to another one can be translated in terms of such pairs in a simple way, and then we can use the known results on the primitive disks developed in [4] and [8] .
Introduction
We use the standard notation L = L(p, q) for a lens space in standard textbooks. For example, we refer [13] to the reader. Throughout the paper, (V, W ; Σ) will denote the genus-2 Heegaard splitting of a lens space L(p, q), and we will always assume 1 ≤ q ≤ p/2. We will denote by S(V, W ; Σ) the sphere complex for the splitting (V, W ; Σ). For convenience, we will not distinguish (sub)spaces and homeomorphisms from their isotopy classes in their notation.
The primitive disks and their dual disks
An essential disk D in V is said to be primitive if there exists an essential disk D ′ in W such that ∂D intersects ∂D ′ transversely in a single point. Such a disk D ′ is called a dual disk of D, and we call the ordered pair (D, D ′ ) simply a dual pair. We note that the disk D ′ is also primitive in W with a dual disk D, and that W ∪ Nbd(D) and V ∪ Nbd(D ′ ) are solid tori. Primitive disks are necessarily non-separating. We call a pair {D, E} of disjoint primitive disks in a handlebody simply a primitive pair, and similarly a triple of pairwise disjoint primitive disks a primitive triple. When the two disks D and E in a primitive pair {D, E} have a common dual disk, we simply say that the pair {D, E} admits a common dual disk. Of course, there exist infinitely many primitive pairs of the handlebodies V and W in the genus-2 Heegaard splitting of a lens space. In fact, any primitive disk is contained in infinitely many primitive pairs. However not every genus-2 Heegaard splitting of a lens space admits a primitive triple. See Lemma 1.3 in the following. • {D, E} is a primitive pair of V , and D ′ is equal to E ′ and is a common dual disk of {D, E}, or • {D ′ , E ′ } is a primitive pair of W , and D is equal to E and is a common dual disk of {D ′ , E ′ }.
The primitive disks and their dual disks
Proof. The "if" part is clear. For the "only if" part, we assume S · T = 4. The sphere S cuts off a 3-ball B from the lens space, and we have four solid tori
The primitive disks and their dual disks
to S, D and D ′ are meridian disks of V 1 and W 1 respectively. Since S · T = 4, the disk S ∩ V divides the disk T ∩ V into two bigons ∆ 1 and ∆ 2 with a rectangle R. Similarly, S ∩ W divides the disk T ∩ W into two bigons ∆ ′ 1 and ∆ ′ 2 with a rectangle R ′ . Both of the bigons ∆ 1 and ∆ 2 are contained in either
The primitive disks and their dual disks
First, consider the case that ∆ 1 and ∆ 2 are contained in V 1 (see Figure 2 ). Then they are meridian disks of V 1 , which are disjoint from the meridian disk D of V 1 up to isotopy. Since the circle ∂D is a longitude of the solid torus W 1 , the rectangle R ′ of the disk T ∩ W is contained in the solid torus W 1 , and is disjoint from ∂D, and intersects D ′ in a single arc. Consequently, the two bigons ∆ ′ 1 and ∆ ′ 2 are contained in W 2 , and hence the disk T ∩ W is disjoint from ∂D and intersects D ′ in a single arc. The disk T ∩ W divides W into two solid tori. Let W ′ 1 be one of them containing ∂D as a longitude. Then the disk E ′ of the dual pair (E, E ′ ) is a meridian disk of W ′ 1 , and, up to isotopy, E ′ is disjoint from D ′ and intersects the longitude ∂D in a single point. Thus we have that {D ′ , E ′ } is a primitive pair of W , and D is equal to E and is a common dual disk of {D ′ , E ′ }. If ∆ 1 and ∆ 2 are contained in V 2 , we conclude that {D, E} is a primitive pair of V , and D ′ is equal to E ′ and is a common dual disk of {D, E} by a similar argument.
The primitive disks and their dual disks
By the lemma, we can translate the properties of Haken spheres into those of dual pairs. The properties of primitive disks and their dual disks that we need were already developed in [8] , which are summarized in the following two lemmas. Lemma 1.2 (Theorem 4.3 [8] ). Given a lens space L = L(p, q), 1 ≤ q ≤ p/2, with a genus-2 Heegaard splitting (V, W ; Σ), each primitive pair in V has a common dual disk if and only if q = 1. In this case, if p ≥ 3, the pair has a unique common dual disk, and if p = 2, the pair has exactly two disjoint common dual disks, which form a primitive pair in W .
If
in a unique or in no primitive triple, and each having no common dual disk is contained in a unique primitive triple. (4) Further, if p = 3, then each of the three primitive pairs in any primitive triple in V has a unique common dual disk, which form a primitive triple in W . If p ≥ 5, then exactly one of the three primitive pairs in any primitive triple has a common dual disk, which is unique.
If
Recalling the fact that V and W are isotopic in L(p, q) shown by Bonahon-Otal [2] , it is clear that the above two lemmas still hold when we exchange V and W in the statements.
The complex of primitive disks
Given an irreducible 3-manifold with compressible boundary, the disk complex of the manifold is a simplicial complex defined as follows. The vertices are the isotopy classes of essential disks in the manifold, 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. It is well known from McCullough [12] , the disk complex for any irreducible 3-manifold with compressible boundary is contractible, and further, in Cho [3] , a useful criterion was developed to determine whether a given subcomplex of a disk complex is contractible or not. We will not introduce the details here, but just summarize the results we need.
The complex of primitive disks
Consider the case that the manifold is a genus-2 handlebody V . Then the disk complex of V is a 2-dimensional contractible complex. We denote by D(V ) the non-separating disk complex of V , which is the full subcomplex of the disk complex spanned by the vertices of non-separating disks. It is easy to see that D(V ) is also 2-dimensional and every edge of D(V ) is contained in infinitely but countably many 2-simplices. Further, the complex D(V ) is contractible, and the link of any vertex of D(V ) is also contractible, i.e. the link is a tree, see [12] , [3] . Thus, we can describe the structure of the non-separating disk complex D(V ); a portion of D(V ) is described in Figure 3 . We observe that D(V ) deformation retracts to a tree in the barycentric subdivision of it. Actually, this tree is a dual complex of D(V ). From the structure of D(V ), every connected component of any full subcomplex of D(V ) is contractible.
The complex of primitive disks
Remark. Here is a simple but important observation on the structure of D(V ). If C is any n-cycle in D(V ) with n ≥ 4, then there is no 2-simplex of which the three edges are all contained in C. But there exist at least two 2-simplices in D(V ) such that exactly two edges of each of the two 2-simplices are contained in C and further the two 2-simplices have no common edges contained in C. [5] , [6] , [8] and [9] . The complete combinatorial structure of the primitive disk complex P(V ) for each lens space has been well understood (see [8, Theorem 4.5 and Figure 11] ). In particular, P(V ) is contractible if p ≡ ±1 (mod q), and otherwise it consists of infinitely many tree components. By Lemma 1.3, the complex P(V ) has a 2-simplex if and only if q = 2 or p = 2q + 1.
The complex of primitive disks
We define two special subcomplexes, one is of P(V ) and the other is of P(W ). First, P ′ (V ) is the subcomplex of P(V ) defined as follows. The vertices of P ′ (V ) are the vertices of P(V ), and distinct k + 1 vertices D 0 , D 1 , · · · , D k of P ′ (V ) span a k-simplex if and only if the primitive pair {D i , D j } admits a common dual disk for each 0 ≤ i < j ≤ k. Next, given a primitive disk D in V , we denote by P D (W ) the full subcomplex of the primitive disk complex P(W ) spanned by the vertices of dual disks of D.
The complex of primitive disks
each of whose edge is contained in a unique 2-simplex.
The complex of primitive disks
is not connected, and it consists of infinitely many tree components. 
Proof of the main theorem
In this section, we prove Theorem 0.1. Let (V, W ; Σ) be the genus-2 Heegaard splitting of a lens space L(p, q) with 1 ≤ q ≤ p/2, and let S(V, W ; Σ) be the sphere complex for (V, W ; Σ). Assume that L is L(p, 1), and let S and T be any two distinct vertices of S(V, W ; Σ). We may assume that the Haken spheres S and T correspond to the dual pairs (D, D ′ ) and (E, E ′ ), where D and E are primitive disks in V , and D ′ and E ′ are their dual disks respectively. By Lemma 2.1 (1) and (2), the complex P ′ (V ) is connected, and hence there exists a sequence 
Proof of the main theorem
} is a primitive pair of W for each j ∈ {0, 1, · · · , n i }. Thus, by Lemma 1.1, we obtain a sequence of dual pairs (
Proof of the main theorem
, which realizes a path in S(V, W ; Σ) from the vertex of (D i , D ′ i ) to of (D i , D ′ i+1 ) for each i ∈ {0, 1, . . . , n}. Moreover, the two vertices corresponding to the dual pairs (D i , D ′ i+1 ) and (D i+1 , D ′ i+1 ) are joined by an edge in S(V, W ; Σ) for each i ∈ {0, 1, . . . , n}. Consequently, we obtain a sequence of dual pairs from (D, D ′ ) to (E, E ′ ) which realizes a path from S to T in S(V, W ; Σ).
Proof of the main theorem
Conversely, assume that L is not L(p, 1). Then, by Lemma 1.2, there exists a primitive pair {D, E} of V which has no common dual disk. Choose any dual disks D ′ and E ′ of D and E, and let S and T be the vertices of S(V, W ; Σ) corresponding to the dual pairs (D, D ′ ) and (E, E ′ ) respectively. We will show that there is no path joining S and T . Suppose, for contradiction, that there exists a sequence of dual pairs (D, 
Proof of the main theorem
Let e be the edge in P(V ) whose end vertices are D and E. Since {D, E} has no common dual disk, the edge e is not contained in the path Φ V (L). Thus, the path Φ V (L) together with the edge e in P(V ) is itself a cycle or contains at least one cycle containing the edge e, which we denote by C. The cycle C is an n-cycle for some n ≥ 3. Further, among the primitive pairs which determines the edges of C, {D, E} is the only one having no common dual disk.
Proof of the main theorem
Suppose first that C is a 3-cycle (in this case, we have q = 2 or p = 2q + 1 necessarily, by Lemma 1.3). Then C bounds a 2-simplex in P(V ). Among the three primitive pairs in the primitive triple representing the vertices of the 2-simplex, {D, E} is the only one having no common dual disk, which contradicts Lemma 1.3 (4) .
Proof of the main theorem
Suppose that C is an n-cycle with n ≥ 4 (in this case, we have (p, q) = (5, 2) necessarily by Lemma 1.3). Then, as mentioned in the remark in Section 2, there exist two 2-simplices in D(V ) such that exactly two edges of each of them are contained in C and further they have no common edges contained in C. Thus one of them, say ∆, does not contain the edge e. Two edges of ∆ are contained in the cycle C, but the other one is not contained in C and not even in P ′ (V ) since P ′ (V ) consists of tree components by Lemma 2.1 (3). Consequently, among the three primitive pairs of the primitive triple representing the vertices of ∆, exactly one pair has no common dual disk, which contradicts Lemma 1.3 (4) again.
Proof of the main theorem
(3) If L is L(2, 1), then every edge in S(V, W ; Σ) is contained in a unique cycle, which is a 4-cycle.
Proof of the main theorem
Let S and T be the end vertices of an edge in S(V, W ; Σ) whose corresponding dual pairs are (D, E ′ ) and (E, E ′ ) respectively. By Lemma 1.2, there exists exactly one more common dual disk D ′ of the pair {D, E}, and {D ′ , E ′ } is a primitive pair in W . Thus we have a 4-cycle containing the edge joining S and T whose vertices correspond to (D, E ′ ), (E, E ′ ), (E, D ′ ) and (D, D ′ ) (see Figure 4 (a) ). Figure 4 Now suppose that there exists a cycle Z in S(V, W ; Σ) which shares edge(s) with the 4-cycle. Suppose first that Z shares a single edge with the 4-cycle. We may assume that the end vertices of the edge are S and T . Then we write all the dual pairs of the vertices of Z consecutively, (D, E ′ ) = (D 0 , E ′ 0 ), (D 1 , E ′ 1 ), · · · , (D n , E ′ n ) = (E, E ′ ). For each j ∈ {0, 1, · · · , n}, the pair {D j−1 , D j } cannot be {D, E} since the only common dual disks of {D, E} are D ′ and E ′ . Thus the image Φ V (Z) contains at least a cycle in P ′ (V ), which contradicts Lemma 2.1 (1). In case that Z shares more edges with the 4-cycle, a similar argument holds to have the same contradiction.

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