Dehn surgery and Seifert surface system
Abstract.
For a compact connected 3submanifold with connected boundary in the 3sphere, we relate the existence of a Seifert surface system for a surface with a Dehn surgery along a nullhomologous link. As its corollary, we obtain a refinement of the Fox’s reembedding theorem.
Key words and phrases:
link, Dehn surgery, Seifert surface2010 Mathematics Subject Classification:
57M251. Introduction
Definition 1.1.
Let be a compact connected 3manifold with connected boundary of genus . A spanning surface system for is a set satisfying the following.

is a set of disjoint orientable surfaces properly embedded in .

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 3submanifold with connected boundary in admits a spanning surface system.
However, a compact connected 3submanifold 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 3submanifold 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.

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

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 3sphere can be obtained by a nontrivial 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 3submanifold in . We say that is nullhomologous in if in , and that is completely nullhomologous in if in for .
Theorem 1.7.
Let be a compact connected 3submanifold with connected boundary in . Then the followings hold.

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

admits a completely disjoint spanning surface system if and only if there exists a completely nullhomologous 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 nullhomologous reflexive link so that it is disjoint from the completely disjoint spanning surface system.
Remark 1.8.
Let be a compact connected 3submanifold 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.

admits a completely disjoint spanning surface system.

There exists a boundary preserving map from onto a handlebody.

is boundary retractable.

The natural map is an epimorphism.
Let be a compact connected 3submanifold 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 reembedding theorem as the following corollary.
Corollary 1.9 ([3, 16, 13]).
Every compact connected 3submanifold of can be reembedded 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 3submanifold of with connected boundary of genus . By Corollary 1.9, there exists a reembedding 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 nullhomologous reflexive links.
Corollary 1.11.
Let be a closed surface in which separates into 3submanifolds and Then the followings hold.

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

admits a completely disjoint Seifert surface system if and only if there exist completely nullhomologous 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 meridianlongitude disk system for the handlebodies by tubing along the nullhomologous links.
Corollary 1.12.
Any closed surface in admits a Seifert surface system.
Let be a 3manifold. Let be a submanifold with or without boundary. When is 1 or 2dimensional, we write . When is of 3dimension, we write .
2. Proof
Let be a genus handlebody in , and be a meridian disk system for . Since is a 3ball, there exists a spine of such that:

consists of loops and arcs connecting to a point .

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

Each loop is dual to .
We call this spine 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 bandcrossing change of , and the operation of (2) in Figure 2 a fulltwist of . We remark that these operations can be obtained by a Dehn surgery along trivial links in the complement of .
(1) a bandcrossing change  (2) a fulltwist 
Lemma 2.3.
Any handcuff graph shaped spine with a Seifert surface system can be unknotted by bandcrossing changes and fulltwists of .
Proof.
It is observed that with can be transformed to be a “standard planar form” (cf. [8]) by the following operations.

a bandcrossing change of

a fulltwists of

a crossing change between and

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 3submanifold in . Suppose that in nullhomologous (resp. completely nullhomologous) in . Then the core link in the 3submanifold obtained by a Dehn surgery along is also nullhomologous (resp. completely nullhomologous) in .
Proof.
Suppose that in nullhomologous (resp. completely nullhomologous) 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 nullhomologous (resp. completely nullhomologous) in . ∎
Lemma 2.5.
Let be a handlebody in . Then the followings hold.

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

admits a completely disjoint Seifert surface system if and only if there exists a completely nullhomologous 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 nullhomologous 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 nullhomologous in , by Lemma 2.4, the core link is also nullhomologous 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 bandcrossing changes and fulltwists of . This operations can be obtained by a Dehn surgery along a trivial link in . Since is contained in , is a nullhomologous link in . Hence we obtain a nullhomologous 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 nullhomologous 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 nullhomologous 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 3sphere 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 nullhomologous reflexive link in such that handlebodies can be obtained from by a Dehn surgery along . This proves the theorem since is obtained by adding 1handle 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 nullhomologous 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 nullhomologous 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 nullhomologous 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 nullhomologous 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 nullhomologous 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 3sphere.
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.

is contained in by an inclusion.

Each is sent to a meridian disk of the handlebody .

The remnant is sent to the 3ball .
Since is surjective [5, Lemma 15.12], is homeomorphic to [14, 15, 12].
Step 3: By Lemma 2.6, there exists a nullhomologous 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 nullhomologous 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 . ∎
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