# Critical Heegaard surfaces obtained by self-amalgamation

###### Abstract.

Critical surfaces can be regarded as topological index 2 minimal surfaces which was introduced by David Bachman. In this paper we give a sufficiently condition and a necessary condition for self-amalgamated Heegaard surfaces to be critical.

###### Key words and phrases:

critical Heegaard surface, self-amalgamation, surface bundle###### 2000 Mathematics Subject Classification:

Primary 57M50## 1. Introduction

Let be a properly embedded, separating surface with no torus components in a compact, orientable, irreducible 3-manifold , dividing into two submanifolds. Then the disk complex, , is defined as follows:

(1) Vertices of are isotopy classes of compressing disks for .

(2) A set of vertices forms an simplex if there are representatives for each that are pairwise disjoint.

David Bachman explored the information which is contained in the topology of by defining the topological index of [3]. If is non-empty then the topological index of is the smallest such that is non-trivial. If is empty then will have topological index . If has a well-defined topological index (i.e. or non-contractible) then we will say that is a topologically minimal surface.

By definition, has topological index 0 if and only if it is incompressible, and has topological index 1 if and only if it is strongly irreducible. Critical surfaces, which are also defined by David Bachman[1][4], can be regarded as topological index 2 minimal surfaces[4].

###### Definition 1.1.

[4] is critical if the compressing disks for can be partitioned into two sets and , such that

(1) for each , there is at least one pair of disks on opposite sides of such that ;

(2) if and are on opposite sides of then .

Some critical Heegaard surfaces have been constructed by Jung Hoon Lee.

###### Theorem 1.2.

[9] The standard minimal genus Heegaard splitting of (closed orientable surface) is a critical Heegaard splitting.

Jung Hoon Lee also showed that some critical Heegaard surfaces can be obtained by amalgamating two strongly irreducible Heegaard splittings.

###### Theorem 1.3.

[9] Let be an amalgamation of two strongly irreducible Heegaard splittings and along homeomorphic boundary components of and . Assume that is constructed from by attaching only one 1-handle. If there exist essential disks and which persist into disjoint essential disks in and respectively, then is critical.

The following theorem is the main result of this paper, which states that some critical Heegaard surfaces can be obtained by self-amalgamating strongly irreducible Heegaard splittings. Terms in the theorems will be defined in Section 2.

###### Theorem 1.4.

Suppose is an irreducible 3-manifold with two homeomorphic boundary components and , and is a strongly irreducible Heegaard splitting of M such that . Suppose admits an essential disk in and two spanning annuli , in , such that , , are disjoint curves on , and for . Let = be the self-amalgamation of , such that is identified with . Then is a critical Heegaard surface of .

As a corollary, we show a generalized result of Theorem 1.2.

###### Corollary 1.5.

Let F be a closed, connected, orientable surface, and let be a surface diffeomophism which preserves orientation. If , then the standard Heegaard surface of the surface bundle is critical.

We also show a necessary condition for self-amalgamated Heegaard surfaces to be critical.

###### Theorem 1.6.

Suppose that is the self-amalgamation of . If is a critical Heegaard surface of , then .

## 2. Preliminaries

An essential annulus properly embedded in a compression body is called a spanning annulus if one component of denoted by lies in , while the other denoted by lies in .

Let M be a compact orientable 3-manifold. If there is a closed surface S which cuts M into two compression bodies V and W with , then we say M has a Heegaard splitting, denoted by ; and S is called a Heegaard surface of M.

A Heegaard splitting W is said to be reducible if there are two essential disks and such that ; otherwise, it is irreducible. A Heegaard splitting W is said to be weakly reducible if there are two essential disks and such that ; otherwise , it is strongly irreducible.

The distance between two essential simple closed curves and in , denoted by , is the smallest integer such that there is a sequence of essential simple closed curves in such that is disjoint from for .

The distance of the Heegaard splitting is where bounds an essential disk in and bounds an essential disk in . was first defined by Hempel, see [7].

Let be a compact orientable 3-manifold with homeomorphic boundary components and , and be a Heegaard splitting such that . Let be the manifold obtained from by gluing and via a homeomorphism . Then has a natural Heegaard splitting called the self-amalgamation of as follows:

Let be a point on such that . Note that is obtained by attaching 1-handles to . Let , be the regular neighborhood of for . We may assume that is disjoint from the 1-handles , and .

Now, in the closure of , the arc has a regular neighborhood which intersects in two disks and . We denote by the point , the disk , and the surface in . Let and be the closure of . and are compression bodies. Let be , then is a Heegaard splitting called the self-amalgamation of . It is clear that (Fig.1).

###### Lemma 2.1.

[11] is incompressible in .

Let be the surface . Then is a sub-surface of with two boundary components and . An essential arc in is called strongly essential if both two boundary points lie in and is an essential arc on , where {i, j}={1, 2}.

###### Lemma 2.2.

[11] Suppose that is an essential disk in or and . Then there exist an arc such that is strongly essential in .

###### Lemma 2.3.

[11] Suppose that is an essential disk in and is minimal up to isotopy. Let be any outermost disk of cut by . Then is strongly essential in .

A surface bundle, denoted by , is a 3-manifold obtained from by gluing its boundary components via a surface diffeomorphism . When is the identity, .

Let be a closed orientable surface with genus . Suppose that is a homeomorphism of . The translation distance of is , where is an essential simple closed curve on . was first defined by Bachman and Schleimer [5].

## 3. Proofs of Theorem 1.4 and Corollary 1.5

Now we give the proof of Theorem 1.4. It shows a sufficient condition for a self-amalgamated Heegaard surface to be critical.

###### Proof.

(of Theorem 1.4.) Since is strongly irreducible, it follows from Casson and Gordon’s theorem [6] that F is incompressible. Since , it follows that is an essential annulus in , denoted by A. Take a spanning arc in A, and let and be the closure of . Then is obtained by self-amalgamation of . Now we prove that is a critical Heegaard surface of .

Let be a compressing disk of corresponding to the 1-handle . We give a partition of the compressing disks for , , as follows: (For the sake of convenience, in the following statement, “ a disk in ” means “ a compressing disk in which belongs to ”.)

consists of compressing disks in that could be be isotoped into but inessential in ;

consists of compressing disks in that are disjoint from ;

consists of compressing disks in that do not belong to ;

consists of compressing disks in that are not disjoint from .

Each compressing disk of must be contained in or . Now we need to show satisfies the definition of criticality.

Claim 1. contains a disjoint pair of disks on opposite sides of .

Note that belongs to . Since has two components, there exists at least one essential disk in disjoint from . Hence there exists at least one essential disk in which is disjoint from . This means contains disjoint compressing disks for on opposite sides.

Claim 2. contains a disjoint pair of disks on opposite sides of .

Since is contained in the annulus , is an essential disk in intersecting in at least two points, so it belongs to . The essential disk in persists as an essential disk in and belongs to . By assumption, is disjoint with . This means also contains disjoint compressing disks for on opposite sides.

Claim 3. Any disk in intersects any disk in .

Let be any disk in that intersect with . Let be any disk essential in , but inessential in . Recall . If is isotopic to one of and , then and there is nothing to prove. So we suppose that bounds a pair of pants together with and . By Lemma 2.2, there is an arc such that is strongly essential in . Note that a strongly essential arc in must intersect with . Hence .

Claim 4. Any disk in intersects any disk in .

Let be an essential disk in that is disjoint from . After isotopy, can be made disjoint from . By Lemma 2.1 and can be made disjoint by a standard innermost disk argument. This means that can be regarded as an essential disk in . Let be an essential disk in that belongs to . For proving Claim 4, we need to show .

Suppose to the contrary that . We assume that is chosen so that the number of components of intersection is minimal up to isotopy of , satisfying . First, we suppose . Then can be regard as an essential disk in . Then since is strongly irreducible, a contradiction. Hence . By a standard innermost disk argument, we can assume consists of arc components. Let be an outermost arc component in and be the corresponding outermost disk in . Since the disk cut into , after a small isotopy lies in V.

By Lemma 2.3, is strongly essential in , hence is essential in V. Since is strongly irreducible, . It is easy to see that . However, we have assumed , a contradiction. Claim 4 follows.

Hence satisfies the definition of criticality. This completes the proof of Theorem 1.4. ∎

###### Proof.

## 4. A necessary condition for self-amalgamated Heegaard surfaces to be critical

The following result could be found in the proof for the main theorem in[11]. Recall we suppose is the self-amalgamation of and is a meridian disk of corresponding to the 1-handle attached to .

###### Lemma 4.1.

[11] If , for each pair of disks and such that is not isotopic to and , we have .

###### Proof.

(of Theorem 1.6.) Since is critical, the compressing disks for can be partitioned into two sets and satisfying the definition of criticality.

Assume that . Each disk in is not isotopic to and each disk in intersects with . Since is critical, there exists at least one disjoint pair of disks and . By Lemma 4.1, we have . ∎

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