Generators for the handlebody subgroup of the Torelli group

A small normal generating set for the handlebody subgroup of the Torelli group

Genki Omori (Genki Omori) Department of Mathematics, Tokyo Institute of Technology, Oh-okayama, Meguro, Tokyo 152-8551, Japan omori.g.aa@m.titech.ac.jp
July 15, 2019
Abstract.

We prove that the handlebody subgroup of the Torelli group of an orientable surface is generated by genus one BP-maps . As an application, we give a normal generating set for the handlebody subgroup of the level mapping class group of an orientable surface.

1. Introduction

Let be an oriented 3-dimensional handlebody of genus and let be a disk on the boundary of . We fix a model of and as in Figure 1 and set . The mapping class group of is the group of isotopy classes of orientation preserving self-diffeomorphisms on fixing pointwise and the handlebody group is the subgroup of which consists of elements that extend to .

For a simple closed curve on , denote by the right-handed Dehn twist along . A pair of simple closed curves and on is a bounding pair (BP) on if and are disjoint, non-isotopic and their integral homology classes are non-trivial and the same. A BP on is a genus- bounding pair (genus- BP) on if the union of and bounds a subsurface of of genus with two boundary components. For a BP (resp. genus- BP) on , we call a BP-map (resp. genus- BP-map).

The Torelli group of is the the kernel of a homomorphism induced by the action of on the integral first homology group of . Genus- BP-maps are elements of . For a group , a normal subgroup of and elements of , is normally generated in by if is the normal closure of in . By an argument of Powell [14], is normally generated in by a genus-1 BP-map and Dehn twists along separating simple closed curves (actually, Powell proved that the Torelli group of an closed oriented surface is generated by genus-1 BP-maps and Dehn twists along separating simple closed curves by using Birman’s finite presentation [3] for the symplectic group ). Johnson showed that is normally generated in by a genus-1 BP-map in [7] and gave an explicit finite generating set for in [8]. A smaller finite generating set for is given by Putman [15].

Denote by the set of diffeomorphism classes of connected closed oriented 3-manifolds and by the set of diffeomorphism classes of integral homology 3-spheres. Let be a 3-dimensional handlebody of genus such that and the union is diffeomorphic to the 3-sphere , and let be the subgroup of which consists of elements that extend to . For each , we denote by the closed oriented 3-manifold obtained by gluing the disjoint union of and along . We regard as a subgroup of by a natural injective stabilization map . Then we have a bijection

by to (see for instance [2]). The above bijection induces the following bijection [12]:

Hence any integral homology 3-sphere is represented by an element of . Note that and are not subgroups of , and for , , means there exist elements and such that . We denote by (resp. ) the intersection of and (resp. ). Pitsch [13] gave the following theorem.

Theorem 1.1 ([13]).

For , , if and only if there exist elements , and such that

For these reasons, it is important for the classification of integral homology 3-spheres to give a simple generating set for .

For a genus- BP on , is a genus- homotopical bounding pair (genus- HBP) on if each doesn’t bound a disk on and the disjoint union bounds an annulus on . We remark that such an annulus is unique up to isotopy by the irreducibility of . For example, a pair of simple closed curves and on as in Figure 1 is a genus-1 HBP on . For a genus- HBP on , we call a genus- HBP-map. Hence is a genus- HBP-map. Remark that genus- HBP-maps are elements of . The main theorem in this paper is as follows:

Theorem 1.2.

For , is normally generated in by . In particular, for , is generated by genus-1 HBP-maps.

We prove Theorem 1.2 in Section 2.1. In Section 2.2, we give a necessary and sufficient condition that a genus-1 HBP-map is conjugate to in .

For , we define . The level mapping class group is the kernel of a homomorphism induced by the action of on . Denote by the intersection of and . Let and be simple closed curves on as in Figure 1. Each of bounds a disk in . We define and denote by the diffeomorphism on which is described as the result of the half rotation of the first handle of as in Figure 2. Note that , , and a genus- HBP-maps are elements of and is an element of . As an application of Theorem 1.2, we obtain the following theorem. The proof is given in Section 3.1.

Theorem 1.3.

For , is normally generated in by , and .

For and , is normally generated in by , and .

Let (resp. ) be the the kernel of the natural homomorphism (resp. ). As a corollary of Theorem 1.2 and Theorem 1.3, we have the following result.

Corollary 1.4.

For , is normally generated in by and .

For , is normally generated in by , and .

For and , is normally generated in by , and .

We prove Corollary 1.4 in Section 4.1. Luft [9] proved that is normally generated in by disk twists and a map whose action on the fundamental group of is the same as the action of . An action of on is non-trivial, however, an action of a BP-map on is trivial. As a corollary of Corollary 1.4, we also have the following corollary. The proof is given in Section 4.2.

Corollary 1.5.

For , is normally generated in by , and .

Figure 1. The model of and simple closed curves , , and on .
Figure 2. The element of .

2. Generators for the handlebody subgroup of the Torelli group

2.1. Proof of main theorem

In this section, we prove Theorem 1.2. Let be a point of and let be generators for the fundamental group of represented by loops on based at as in Figure 3. We identify with the free group of rank by the generators. Since acts on , we have a homomorphism . Griffiths [5] showed that is surjective. Denote by the kernel of . Luft [9] proved that is generated by disk twists. Then we have the exact sequence

Figure 3. Generators for .

The IA-subgroup of is the kernel of the homomorphism induced by the abelianization of . Remark that the image of is included in . We define an element of by and for . Magnus [10] proved the following theorem (see also [4]).

Theorem 2.1 ([10]).

For , is normally generated in by .

Since and is surjective, we have . Denote by the kernel of the homomorphism . is called the Luft-Torelli group in [13]. Then we have the exact sequence

(2.1)

A BP (resp. genus- BP) on is a contractible bounding pair (CBP) (resp. genus- contractible bounding pair (genus- CBP)) if each bounds a disk in . For example, is a genus-1 CBP on , where is a simple closed curve on as in Figure 4. For a CBP (resp. genus- CBP) on , we call a CBP-map (genus- CBP-map). CBP-maps are elements of . Pitsch [13] proved the following theorem.

Figure 4. Simple closed curve on .
Theorem 2.2 ([13]).

For , is generated by CBP-maps.

By Johnson’s argument [7], this theorem is improved as follows.

Proposition 2.3.

For , is normally generated in by a genus-1 CBP-map.

Proof.

Let be a genus- CBP on . Without loss of generality, we can assume that each doesn’t intersect with . Take proper disks and in such that for . By cutting along , we obtain a handlebody of genus which doesn’t include . Then there exist proper disjoint disks in such that the result of cutting along is a disjoint union of handlebodyies of genus , and lie on a boundary of the same component for , and and don’t lie on the same component for (see Figure 5). Then we have

Since each is a genus- CBP-map, is a product of genus- CBP-maps. We get Proposition 2.3. ∎

Figure 5. Disks on .
Proof of Theorem 1.2.

By the exact sequence (2.1) and Proposition 2.3, is normally generated in by and . Hence it is enough for the proof of Theorem 1.2 to show that is a product of conjugations of in . Since , we have

We have completed the proof of Theorem 1.2. ∎

Remark 2.4.

The last relation

in the proof of Theorem 1.2 has the following geometric meaning. Let be a separating disk in as in Figure 6. Then we can regard , and as pushing maps of along simple loops on the boundary of the closure of the complement of the first 1-handle. is obtained from the pushing map along and is obtained from the pushing map along as in Figure 6. The above relation means a product of pushing maps along simple loops which intersect transversely once is equal to the pushing map along the product of these loops.

Figure 6. Disk in and loops and .

2.2. A Condition for conjugations of genus-1 HBP-maps in the handlebody group

In this section, we give a necessary and sufficient condition that a genus-1 HBP-map is conjugate to in . For proper disks in , the pair is a meridian disk system if each is non-separating and we obtain a 3-ball by cutting along . For example, is a meridian disk system, where are disks in whose boundary components are as in Figure 1, respectively. Then we have the following proposition.

Proposition 2.5.

Let be a genus-1 HBP on . Then is conjugate to in if and only if there exist a properly embedded annulus in whose boundary is and a meridian disk system such that are disjoint from and the intersection of and is an arc which doesn’t separate .

Proof.

We suppose that is conjugate to in . Then there exists a diffeomorphism such that the restriction is identity map on and . By Figure 1, there exists a properly embedded annulus in whose boundary is such that the intersection of and is an arc which doesn’t separate and are disjoint from . Thus , , , , satisfy the condition above. We have proved the “only if” part of the proposition.

We suppose that there exist a properly embedded annulus in whose boundary is and a meridian disk system such that are disjoint from and the intersection of and is an arc which doesn’t separate . Note that the arc separates into two disks and in . Let be a 3-ball which is obtained by cutting along . Since doesn’t separate , the image of in is a proper disk in . Hence separates into 3-balls and . Without loss of generality, we can assume that the copies and of and copies and of are included in for some , and the copies and of and the copies and of are included in for any since is a genus-1 HBP on . Denote by and the images of in and , respectively.

Let and be the handlebodies which obtained by cutting along such that is diffeomorphic to and is diffeomorphic to and let , and be the 3-balls which obtained by cutting , and along , respectively. Denote by and the disks on and which are obtained from , by and are copies of disk on for and by and the disks on and which are obtained from by cutting along for , respectively. For and , we regard as a disk in . Since the isotopy classes of and in and (resp. and in and ) fixed , and (, ) (resp. , and (, )) depend on the isotopy classes of arcs which obtained from the center line of (resp. ), there exist orientation preserving diffoemorphisms and such that , , the restriction is the identity map and and are compatible with regluing of along , , , , and for , and . Such diffeomorphisms induce the diffeomorphism such that and . Thus is conjugate to in and we have completed the proof of this proposition. ∎

Let and be simple closed curves on as in Figure 7 for . Since the union bounds an annulus in which intersects with at proper arcs in as in Figure 7, is a genus-1 HBP on . Note that such an annulus is unique up to isotopy by the irreducibility of . Then we show that is not conjugate to in by Proposition 2.5 and the next proposition.

Proposition 2.6.

For , there does not exist a proper disk in which transversely intersects with at a proper arc in and separates into a disk.

Proof.

Suppose that there exists a proper disk in which transversely intersects with at a proper arc in and separates into a disk. Denote by the proper arc in . For proper disks and in whose intersection is disjoint union of proper arcs in , we obtain disks in from the disk by cutting along . Then there exist disks in such that are proper disks in and each is isotopic to a proper disk in which doesn’t intersect with and the other . We call the operation which gives disjoint disks from the disk the surgery on along .

By the irreducibility of , we can assume that the intersection of and is a disjoint union of proper arcs in . Let be proper disks in which are obtained from by the surgery on along and let be the solid torus which obtained from by cutting along . Since , , and don’t intersect , we regard , , and as proper disks, a proper annulus and a proper arc in . Note that the intersection of and in is not a single arc up to ambient isotopy of (see Figure 8). Then there exists such that the proper disk in intersects with at the arc . Since transversely intersects with each at one point, is a non-separating disk in . Hence is isotopic to in by forgetting the copies of throughout the isotopy. This is a contradiction to the fact that the intersection of and in is not a single arc. We have completed the proof of this proposition. ∎

Figure 7. Simple closed curves and in which bound an annulus in .
Figure 8. Annulus and the disk in the solid torus . We express the copies of by the holes on .

3. Applications

In this section, we prove Theorem 1.3.

3.1. Proof of Theorem 1.3

Take a symplectic basis for as in Figure 9. The symplectic group is , where and is the identity matrix of rank . We define

The notation was introduced by Hirose [6]. The last equation and the next lemma is obtained from an argument in Section 2 of [2]. Recall the homomorphism induced by the action of on .

Lemma 3.1 ([2]).

.

Figure 9. Basis for the first homology group of .

We review the next well-known lemma.

Lemma 3.2.

Let , and be groups and let and be homomorphisms. We take a generating set for and a lift of with respect to . Then is generated by and .

Let be the homomorphism induced by the natural projection for . Then we define

For distinct