Dehn surgery and Seifert surface system

Dehn surgery and Seifert surface system

Abstract.

For a compact connected 3-submanifold with connected boundary in the 3-sphere, we relate the existence of a Seifert surface system for a surface with a Dehn surgery along a null-homologous link. As its corollary, we obtain a refinement of the Fox’s re-embedding theorem.

Key words and phrases:
link, Dehn surgery, Seifert surface
2010 Mathematics Subject Classification:
57M25
The first author is partially supported by Grant-in-Aid for Scientific Research (C) (No. 23540105 and No. 26400097), The Ministry of Education, Culture, Sports, Science and Technology, Japan
The second author is partially supported by Grant-in-Aid for Scientific Research (C) (No. 25400080), The Ministry of Education, Culture, Sports, Science and Technology, Japan

1. Introduction

Definition 1.1.

Let be a compact connected 3-manifold with connected boundary of genus . A spanning surface system for is a set satisfying the following.

  1. is a set of disjoint orientable surfaces properly embedded in .

  2. is a set of disjoint loops which do not separate .

A spanning surface system for is completely disjoint if is a set of disjoint orientable surfaces.

Remark 1.2.

We remark by [10, Corollary 1.4] that if is a handlebody and is a completely disjoint spanning surface system for , then there exists a meridian disk system for such that for .

By a homological argument, we have the following proposition.

Proposition 1.3.

Any compact connected 3-submanifold with connected boundary in admits a spanning surface system.

However, a compact connected 3-submanifold with connected boundary in does not always admit a completely disjoint spanning surface system.

Remark 1.4.

We remark by [9] that there exists a compact connected 3-submanifold with connected boundary of genus 2 in which does not admit a completely disjoint spanning surface system.

Definition 1.5.

Let be a genus closed surface in , and put . A Seifert surface system for is a pair of sets satisfying the following.

  1. (resp. ) is a spanning surface system for (resp. ).

  2. for , where and

A Seifert surface system for is completely disjoint if and are completely disjoint.

Definition 1.6.

Let be a link in . Following [11], we say that is a reflexive link if the 3-sphere can be obtained by a non-trivial Dehn surgery along . In particular, if the surgery slope for is for some integer , then we call the Dehn surgery a -Dehn surgery.

Suppose that is contained in a compact 3-submanifold in . We say that is null-homologous in if in , and that is completely null-homologous in if in for .

Theorem 1.7.

Let be a compact connected 3-submanifold with connected boundary in . Then the followings hold.

  1. There exists a null-homologous link in , which is reflexive in , such that a handlebody can be obtained from by a -Dehn surgery along .

  2. admits a completely disjoint spanning surface system if and only if there exists a completely null-homologous link in , which is reflexive in , such that a handlebody can be obtained from by a -Dehn surgery along .

We remark that in Theorem 1.7 (2), we can take a completely null-homologous reflexive link so that it is disjoint from the completely disjoint spanning surface system.

Remark 1.8.

Let be a compact connected 3-submanifold of with connected boundary of genus . Let be a map onto a genus handlebody . We say that is a boundary preserving map of onto if is continuous and is a homeomorphism onto . We say that is retractable if can be retracted onto a wedge of simple closed curves. If such a wedge can be chosen to be in , then is called boundary retractable. Set and define , , . Then the following conditions are equivalent.

  1. admits a completely disjoint spanning surface system.

  2. There exists a boundary preserving map from onto a handlebody.

  3. is boundary retractable.

  4. The natural map is an epimorphism.

The equivalence between (1) and (2) was shown in [9, Theorem 2]. The equivalence between (2) and (3) was shown in [7, Theorem 3]. The equivalence between (3) and (4) was shown in [6, Theorem 2, 3].

Let be a compact connected 3-submanifold of . By Proposition 1.3, each component of the exterior of admits a spanning surface system. If we adapt Theorem 1.7 (1) to the exterior of , then we obtain a refinement of the Fox’s re-embedding theorem as the following corollary.

Corollary 1.9 ([3, 16, 13]).

Every compact connected 3-submanifold of can be re-embedded in so that the exterior of the image of is a union of handlebodies.

Remark 1.10.

In relation with Remark 1.8, there is an another equivalence condition. Let be a compact connected 3-submanifold of with connected boundary of genus . By Corollary 1.9, there exists a re-embedding of so that its exterior is a genus handlebody . A handcuff graph shaped spine of is a boundary spine if its constituent link is a boundary link that admits a pair of disjoint Seifert surfaces whose interiors are contained in . A handlebody is -knotted if it does not admit any boundary spine. Then it was shown in [1, Theorem 3.10] that admits a completely disjoint spanning surface system if and only if is not -knotted.

Theorem 1.7 also deduces the following corollary. It follows from (2) of the next corollary that every closed surfaces with completely disjoint Seifert surface systems can be related by -Dehn surgeries along completely disjoint null-homologous reflexive links.

Corollary 1.11.

Let be a closed surface in which separates into 3-submanifolds and Then the followings hold.

  1. There exist null-homologous links and in and , which are reflexive in , such that handlebodies can be obtained from and by -Dehn surgeries along and .

  2. admits a completely disjoint Seifert surface system if and only if there exist completely null-homologous links and in , which are reflexive in , and such that handlebodies can be obtained from and by -Dehn surgeries along and .

By Corollary 1.11 (1), we can obtain a Seifert surface system from a meridian-longitude disk system for the handlebodies by tubing along the null-homologous links.

Corollary 1.12.

Any closed surface in admits a Seifert surface system.

Let be a 3-manifold. Let be a submanifold with or without boundary. When is 1 or 2-dimensional, we write . When is of 3-dimension, we write .

2. Proof

Let be a genus handlebody in , and be a meridian disk system for . Since is a 3-ball, there exists a spine of such that:

  1. consists of loops and arcs connecting to a point .

  2. The point is the center of the 3-ball , and which is homeomorphic to .

  3. Each loop is dual to .

We call this spine a -handcuff graph shaped spine for with respect to .

Figure 1. A -handcuff graph shaped spine for with respect to

Next, let be a set of orientable surfaces with boundary and without closed component. We say that is a Seifert surface system for if .

Lemma 2.1.

Any -handcuff graph shaped spine in admits a Seifert surface system.

Proof.

We take a regular diagram of such that has no crossing. Then we apply the Seifert’s algorithm ([17]) to loops with arbitrary orientations, and obtain a Seifert surfaces for the loops. ∎

The following lemma states that from any meridian disk system for a handlebody, we can obtain a Seifert surface system for the boundary of the bandlebody.

Lemma 2.2.

Let be a genus handlebody in with a meridian disk system . Then there exists a spanning surface system for such that is a Seifert surface system for .

Proof.

Let be a -handcuff graph shaped spine for with respect to . By Lemma 2.1, admits a Seifert surface system . The restriction to gives a spanning surface system, say , for such that is a Seifert surface system for . ∎

Let be a -handcuff graph shaped spine with a Seifert surface system . We call the operation of (1) in Figure 2 a band-crossing change of , and the operation of (2) in Figure 2 a full-twist of . We remark that these operations can be obtained by a -Dehn surgery along trivial links in the complement of .

(1) a band-crossing change (2) a full-twist
Figure 2. A band-crossing change and a full-twist of
Lemma 2.3.

Any -handcuff graph shaped spine with a Seifert surface system can be unknotted by band-crossing changes and full-twists of .

Figure 3. A -handcuff graph shaped spine with a Seifert surface system , which is a “standard planar form”
Proof.

It is observed that with can be transformed to be a “standard planar form” (cf. [8]) by the following operations.

  1. a band-crossing change of

  2. a full-twists of

  3. a crossing change between and

  4. a crossing change among

However, the operations (3) and (4) can be exchanged for the operation (1). If with has a standard planar form, then is unknotted and this completes the proof. We remark that in a standard planar form, is the trivial link. ∎

Lemma 2.4.

Let be a reflexive link in which is contained in a compact 3-submanifold in . Suppose that in null-homologous (resp. completely null-homologous) in . Then the core link in the 3-submanifold obtained by a -Dehn surgery along is also null-homologous (resp. completely null-homologous) in .

Proof.

Suppose that in null-homologous (resp. completely null-homologous) in . Then bounds a Seifert surface (resp. completely disjoint Seifert surface) in . Put . By a -Dehn surgery, the meridian of the core link intersects each component of in one point. This shows that can be extended to a a Seifert surface (resp. completely disjoint Seifert surface) for in . Thus is also null-homologous (resp. completely null-homologous) in . ∎

Lemma 2.5.

Let be a handlebody in . Then the followings hold.

  1. admits a Seifert surface system if and only if there exists a null-homologous link in , which is reflexive in , such that a handlebody can be obtained from by a -Dehn surgery along .

  2. admits a completely disjoint Seifert surface system if and only if there exists a completely null-homologous link in , which is reflexive in , such that a handlebody can be obtained from by a -Dehn surgery along .

Proof.

(1) Suppose that there exists a null-homologous reflexive link in such that a handlebody, say , can be obtained from by a -Dehn surgery along . There exists a meridian disk system for . Since is null-homologous in , by Lemma 2.4, the core link is also null-homologous in . Therefore, we can obtain a Seifert surface system for by tubing along .

Conversely, suppose that admits a Seifert surface system , where and are spanning surface systems for and . By Remark 1.2, we may assume that each is a disk. Then there exists a -handcuff graph shaped spine and can be extended to a Seifert surface system for . By Lemma 2.3, with can be unknotted, hence is a handlebody, by band-crossing changes and full-twists of . This operations can be obtained by a -Dehn surgery along a trivial link in . Since is contained in , is a null-homologous link in . Hence we obtain a null-homologous reflexive link in such that a handlebody can be obtained from by a -Dehn surgery along .

(2) This can be proved by the argument similar to (1). ∎

Lemma 2.6.

Let be a Heegaard surface in which decomposes into two handlebodies and . Let be a meridian disk system for . Then there exist a null-homologous reflexive link in which yields a handlebody by a -Dehn surgery on and a meridian disk system for such that is a completely disjoint Seifert surface system for in .

Proof of Lemma 2.6.

We take a -handcuff graph shaped spine of with respect to . Since can be unknotted by crossing changes, there exists a null-homologous reflexive link in such that after a -Dehn surgery along , every loops of bound mutually disjoint disks. Therefore, a handlebody obtained from by a -Dehn surgery along admits a meridian disk system so that is a completely disjoint Seifert surface system for . ∎

Proof of Theorem 1.7.

(1) We prove by an induction on the genus . Since the 3-sphere does not contain an incompressible closed surface, there exists a compressing disk for in . We divide the proof into two cases.

Case 1:

Case 2:

In Case 1, put . By the assumption of the induction, there exists a null-homologous reflexive link in such that handlebodies can be obtained from by a -Dehn surgery along . This proves the theorem since is obtained by adding 1-handle to .

In Case 2, we take a maximal compression body for in [2]. If is a handlebody (i.e. ), then the theorem follows Lemma 2.2 and Lemma 2.5 (1). Otherwise as , by the assumption of the induction, there exists a null-homologous reflexive link in each component of such that handlebodies can be obtained from the component by a -Dehn surgery along . After these -Dehn surgery, is a handlebody. Therefore, again by Lemma 2.2 and Lemma 2.5 (1), there exists a null-homologous reflexive link in such that a handlebody can be obtained from by a -Dehn surgery along . Finally, we recover the previous -Dehn surgery on each component of to obtain the original .

(2) Suppose that there exists a completely null-homologous reflexive link in such that a handlebody can be obtained from by a -Dehn surgery along . There exists a meridian disk system for the resultant handlebody. Since is completely null-homologous in , we can obtain a completely disjoint spanning surface system for by tubing along .

Conversely, suppose that admits a completely disjoint spanning surface system . In the following 3 steps, we convert and into two handlebodies and so that admits a meridian disk system with .

Step 1: By (1) of this theorem, there exists a null-homologous link reflexive in such that a handlebody can be obtained from by a -Dehn surgery along . Let the resultant handlebody obtained from and note that is again the 3-sphere.

Step 2: We note that there exists a degree one map from to a handlebody which sends each to a meridian disk of and preserves the boundary of (cf. [9, Theorem 2], [4, Theorem 5]). We naturally extend this degree one map to a degree one map as follows.

  1. is contained in by an inclusion.

  2. Each is sent to a meridian disk of the handlebody .

  3. The remnant is sent to the 3-ball .

Since is surjective [5, Lemma 15.12], is homeomorphic to [14, 15, 12].

Step 3: By Lemma 2.6, there exists a null-homologous reflexive link in and a meridian disk system for a handlebody obtained from by a -Dehn surgery along such that is a completely disjoint Seifert surface system for .

Since the degree one map is a boundary preserving map by the condition (1), is a completely disjoint Seifert surface system for . By Lemma 2.5 (2), there exists a completely null-homologous reflexive link in such that a handlebody can be obtained from by a -Dehn surgery along . Moreover, by the proof of Lemma 2.5 (2), we can take so that . Thus the completely disjoint spanning surface system is contained in the resultant handlebody obtained from by a -Dehn surgery along . ∎


References

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  12. J. Morgan and G. Tian, Ricci flow and Poincaré conjecture, Clay Mathematics Monographs Vol.3, 2007.
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