Ribbonness of a stable-ribbon surface-link, II. General case

TL;DR

This paper proves that all stable-ribbon surface-links are ribbon surface-links and establishes conditions under which connected sums and certain surface-links are ribbon, generalizing previous results in the field.

Contribution

It generalizes the understanding of stable-ribbon surface-links, showing their equivalence to ribbon surface-links and characterizing their behavior under connected sums.

Findings

Stable-ribbon surface-links are all ribbon surface-links.

Connected sum of surface-links is ribbon if and only if both summands are ribbon.

Characterization of when surface-links with ribbon components are ribbon.

Abstract

It is shown that any handle-irreducible summand of every stable-ribbon surface-link is a unique ribbon surface-link up to equivalences, so that every stable-ribbon surface-link is a ribbon surface-link. This is a generalization of a previously observed result for a stably trivial surface-link. Two observations are given. One observation is that a connected sum of two surface-links is a ribbon surface-link if and only if both the connected summands are ribbon surface-links. The other observation is a characterization of when a surface-link consisting of ribbon surface-knot components becomes a ribbon surface-link.

Equations

$$(D\times I)\cap (D^{\prime }\times I)=(\partial D)\times I\cap (\partial D^{\prime })\times I$$

$$F(B)\cong F(D\times I,D^{\prime }\times I)\cong F(D\times I)\cong F(D^{\prime }\times I).$$

$$f(\partial D)=\ell \text{mbox}andf(\partial D^{\prime })={\ell }^{\prime }.$$

$$g(\tilde{\ell },\tilde{\ell })=(\ell ,{\ell }^{\prime }).$$

$$G^t=g^{-1}(\text{mbox}cl(F_E(\dot{h})\setminus \dot{h})\#T)\cup g^{-1}{i}_{t}g(\ddot{h})(t\in [0,1])$$

$$(g^{-1}{i}_{t}g(f(D)\times I,f(D^{\prime })\times I)$$

$$g^{-1}{i}_{0}g(f(D)\times I,f(D^{\prime })\times I)$$

$$g^{-1}{i}_{1}g(f(D)\times I,f(D^{\prime })\times I)$$

$$G^0(f(D)\times I,f(D^{\prime })\times I)$$

$$g{G}^1(d\times I,d^{\prime }\times I)$$

$$(\text{mbox}cl(F_E(\dot{h})\setminus \dot{h})\#T\cup i_{1}g(\ddot{h}))(d\times I,d^{\prime }\times I)$$

$$G^0(f(D)\times I,f(D^{\prime })\times I)\cong g{G}^1(d\times I,d^{\prime }\times I),$$

$$F$$

$$G$$

$$F^{(1)}=F^{\ast }{\#}_2{n}_{2}T{\#}_3\dots {\#}_r{n}_{r}T$$

$$G^{\ast }{\#}_1(n_1^{\prime }-n_1)T{\#}_2{n}_2^{\prime }T{\#}_3\dots {\#}_r{n}_r^{\prime }T,$$

$$G^{(1)}=G^{\ast }{\#}_2{n}_2^{\prime }T{\#}_3\dots {\#}_r{n}_r^{\prime }T.$$

$$[{\ell }^{\prime \prime }]=\sum _{j=1}^g{a}_j[{\ell }_j]+b_j[m_j]$$

$$a_j=n_j{a}_1+r_j$$

$$[{\ell }^{\prime \prime }]=a_1[{\ell }_1]+b_1[m_1]+\sum _{j=2}^g{r}_j[{\ell }_j]+{\tilde{b}}_j[m_j]$$

$$[{\ell }^{\prime \prime }]=a[{\ell }_1]+\sum _{j=1}^g{\tilde{b}}_j[m_j]$$

$$[{\ell }^{\prime \prime \prime }]=\sum _{j=1}^g{b}_j^{\prime }[m_j]$$

Full text

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Keywords: Ribbon,  Stable-ribbon,  Surface-link.

Mathematics Subject Classification 2010: Primary 57Q45; Secondary 57N13

1. Introduction

In this paper, a generalization of the result of the paper [9] on a trivial surface-link to a result on a ribbon surface-link is explained.

A surface-link is a closed oriented (possibly disconnected) surface $F$ embedded in the 4-space ${\mathbf{R}}^{4}$ by a smooth (or a piecewise-linear locally flat) embedding. When $F$ is connected, it is also called a surface-knot. When a (possibly disconnected) closed surface $\mathbf{F}$ is fixed, it is also called an $\mathbf{F}$-link. If $\mathbf{F}$ is the disjoint union of some copies of the 2-sphere $S^{2}$, then it is also called a 2-link. When $\mathbf{F}$ is connected, it is also called a surface-knot, and a 2-knot for ${\mathbf{F}}=S^{2}$. Two surface-links $F$ and $F^{\prime}$ are equivalent by an equivalence $f$ if $F$ is sent to $F^{\prime}$ orientation-preservingly by an orientation-preserving diffeomorphism (or piecewise-linear homeomorphism) $f:{\mathbf{R}}^{4}\to{\mathbf{R}}^{4}$. A *trivial * surface-link is a surface-link $F$ which is the boundary of the union of mutually disjoint handlebodies smoothly embedded in ${\mathbf{R}}^{4}$, where a handlebody is a 3-manifold which is a 3-ball, solid torus or a disk sum of some number of solid tori. A trivial surface-knot is also called an *unknotted * surface-knot. A trivial disconnected surface-link is also called an unknotted-unlinked surface-link. For any given closed oriented (possibly disconnected) surface ${\mathbf{F}}$, a trivial ${\mathbf{F}}$-link exists uniquely up to equivalences (see [3]). A ribbon surface-link is a surface-link $F$ which is obtained from a trivial $nS^{2}$-link $O$ for some $n$ (where $nS^{2}$ denotes the disjoint union of $n$ copies of the 2-sphere $S^{2}$) by the surgery along an embedded 1-handle system (see [4], [11, II]). A stabilization of a surface-link $F$ is a connected sum $\bar{F}=F\#_{k=1}^{s}T_{k}$ of $F$ and a system $T$ of trivial torus-knots $T_{k}\,(k=1,2,\dots,s)$. By granting $s=0$, we understand that a surface-link $F$ itself is a stabilization of $F$. The trivial torus-knot system $T$ is called the stabilizer with stabilizer components $T_{k}\,(k=1,2,...,s)$ on the stabilization $\bar{F}$ of $F$. A stable-ribbon surface-link is a surface-link $F$ such that a stabilization $\bar{F}$ of $F$ is a ribbon surface-link.

For every surface-link $F$, there is a surface-link $F^{*}$ with minimal total genus such that $F$ is equivalent to a stabilization of $F^{*}$. The surface-link $F^{*}$ is called a handle-irreducible summand of $F$.

The following result called Stable-Ribbon Theorem is our main theorem.

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Theorem 1.1. A handle-irreducible summand $F^{*}$ of every stable-ribbon surface-link $F$ is a ribbon surface-link which is determined uniquely from $F$ up to equivalences.

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Since any stabilization of a ribbon surface-link is a ribbon surface-link, Theorem 1.1 implies the following corollary:

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Corollary 1.2. Every stable-ribbon surface-link is a ribbon surface-link.

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The following corollary of a ribbon surface-link is a standard consequence of Corollary 1.2, and contrasts with a behavior of a classical ribbon knot, for every classical knot is a connected summand of a ribbon knot.

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Corollary 1.3. A connected sum $F=F_{1}\#F_{2}$ of surface-links $F_{i}\,(i=1,2)$ is a ribbon surface-link if and only if the surface-links $F_{i}\,(i=1,2)$ are both ribbon surface-links.

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Proof of Corollary 1.3. The ‘if’  part of Corollary 1.3 is seen from the definition of a ribbon surface-link. The proof of the ‘only if’  part of Corollary 1.3 uses an argument of [3] showing the fact that every surface-link is made a trivial surface-knot by the surgery along a finite number of (possibly non-trivial) 1-handles. The connected summand $F_{2}$ is made a trivial surface-knot by the surgery along 1-handles within the 4-ball defining the connected sum, so that the surface-link $F$ changes into a new ribbon surface-link and hence $F_{1}$ is a stable-ribbon surface-link. By Corollary 1.2, $F_{1}$ is a ribbon surface-link. By interchanging the roles of $F_{1}$ and $F_{2}$, $F_{2}$ is also a ribbon surface-link. $\square$

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A stably trivial surface-link is a surface-link $F$ such that a stabilization $\bar{F}$ of $F$ is a trivial surface-link. Since a trivial surface-link is a ribbon surface-link, Theorem 1.1 also implies the following corollary, which is a main result in [9]:

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Corollary 1.4. A handle-irreducible summand of every stably trivial surface-link is a trivial 2-link.

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This corollary implies that every stably trivial surface-link is a trivial surface-link as observed in [9]. See [9, 10] for further results on a trivial surface-link.

The plan for the proof of Theorem 1.1 is to show the following two theorems by an argument based on [9].

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Theorem 1.1.1 Any two handle-irreducible summands of any (not necessarily ribbon) surface-link are equivalent.

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Theorem 1.1.2 Any stable-ribbon surface-link is a ribbon surface-link.

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The proofs of Theorem 1.1.1 and 1.1.2 are given in § 2 and § 3, respectively. The proof of Theorem 1.1 is completed by these theorems as follows:

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Proof of Theorem 1.1. By Theorem 1.1.2, a handle-irreducible summand of every stable-ribbon surface-link is a ribbon surface-link which is unique up to equivalences by Theorem 1.1.2. $\square$

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2. Proof of Theorem 1.1.1

A 2-handle on a surface-link $F$ in ${\mathbf{R}}^{4}$ is an embedded 2-handle $D\times I$ on $F$ with $D$ a chore disk such that $(D\times I)\cap F=(\partial D)\times I$, where $I$ denotes a closed interval containing [math] and $D\times 0$ is identified with $D$. An orthogonal 2-handle pair (or simply, an O2-handle pair) on $F$ is a pair $(D\times I,D^{\prime}\times I)$ of 2-handles $D\times I$, $D^{\prime}\times I)$ on $F$ such that the core disks $D$ and $D^{\prime}$ meet transversely at just one point $p$ in $F$ with

[TABLE]

which is homeomorphic to the square $Q=I\times I$ with $p$ the central point.

Let $(D\times I,D^{\prime}\times I)$ be an O2-handle pair on a surface-link $F$. Let $F(D\times I)$ and $F(D^{\prime}\times I)$ be the surface-links obtained from $F$ by the surgeries along $D\times I$ and $D^{\prime}\times I$, respectively. Let $F(D\times I,D^{\prime}\times I)$ be the surface-link which is the union of the bounded surface $F^{c}_{D}=\mbox{cl}(F\setminus((\partial D)\times I\cup(\partial D^{\prime})\times I))$ and the plumbed disk $\delta_{D}=D\times(\partial I)\cup Q\cup D^{\prime}\times(\partial I)$. A compact once-punctured torus of a torus $T$ is simply called a punctured torus and denoted by $T^{o}$. A punctured torus $T^{o}$ in a 3-ball $B$ is trivial if $T^{o}$ is smoothly and properly embedded in $B$ and there is a solid torus $V$ in $B$ with $\partial V=T^{o}\cup\delta_{B}$ for a disk $\delta_{B}$ in $\partial B$.

A bump of a surface-link $F$ is a 3-ball $B$ in ${\mathbf{R}}^{4}$ with $F\cap B=T^{o}$ a trivial punctured torus in $B$. Let $F(B)$ be a surface-link $F^{c}\cup\delta_{B}$ for the surface $F^{c}_{B}=\mbox{cl}(F\setminus T^{o})$ and a disk $\delta_{B}$ in $\partial B$ with $\partial\delta_{B}=\partial T^{o}$, where note that $F(B)$ is uniquely determined up to cellular moves on $\delta_{B}$ keeping $F^{c}$ fixed. For an O2-handle pair $(D\times I,D^{\prime}\times I)$ on a surface-link $F$, let $\Delta=D\times I\cup D^{\prime}\times I$ is a 3-ball in ${\mathbf{R}}^{4}$ called the 2-handle union. By adding a boundary collar to the 2-handle union $\Delta$, we have a bump $B=B_{D}$ of $F$, which we call the associated bump of the O2-handle pair $(D\times I,D^{\prime}\times I)$ (see [9, Fig. 2]).

An O2-handle pair and a bump on a surface-link are shown to be essentially equivalent notions in [9]. In particular, it is observed in [9] that for any O2-handle pair $(D\times I,D^{\prime}\times I)$ on any surface-link $F$ and the associated bump $B$, there are equivalences

[TABLE]

A punctured torus $T^{o}$ in a 4-ball $A$ is trivial if $T^{o}$ is smoothly and properly embedded in $A$ and there is a solid torus $V$ in $A$ with $\partial V=T^{o}\cup\delta_{A}$ for a disk $\delta_{A}$ in the 3-sphere $\partial A$. A 4D bump of a surface-link $F$ is a 4-ball $A$ in ${\mathbf{R}}^{4}$ with $F\cap A=T^{o}$ a trivial punctured torus in $A$. A 4D bump $A$ is obtained from a bump $B$ of a surface-link $F$ by taking a bi-collar $c(B\times[-1,1])$ of $B$ in ${\mathbf{R}}^{4}$ with $c(B\times 0)=B$. The following lemma is proved by using a 4D bump $A$.

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Lemma 2.1. For an O2-handle pair $(D\times I,D^{\prime}\times I)$ on a surface-link $F$, let $F(D\times I,D^{\prime}\times I)=F^{c}_{D}\cup\delta_{D}$. Then for a trivial torus-knot $T$ with a spin loop basis $(\ell,\ell^{\prime})$, there is an equivalence $f:{\mathbf{R}}^{4}\to{\mathbf{R}}^{4}$ from the surface-link $F$ to a connected sum $F(D\times I,D^{\prime}\times I)\#T$ keeping $F^{c}_{D}$ fixed such that

[TABLE]

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Proof of Lemma 2.1. Let $A$ be a 4D bump associated with the O2-handle pair $(D\times I,D^{\prime}\times I)$ on $F$. Let $\delta_{A}$ be a disk in the 3-sphere $\partial A$ such that the union of $\delta_{D}$ and the trivial punctured torus $F\cap A=P$ bounds a solid torus $V$ in $A$. Then there is an equivalence $f^{\prime}:F\cong F(D\times I,D^{\prime}\times I)\#T$ by deforming $V$ in $A$ so that $P$ is isotopically deformed into the summand $T^{o}$ of a connected sum $\delta_{D}\#T$ in $A$. Then the spin loop pair $(\partial D,\partial D^{\prime})$ on $F_{1}$ is sent to a spin loop basis $(\tilde{\ell},\tilde{\ell}^{\prime})$ of $T^{o}$. By [2] (see [9, (2.4.2)]), there is an orientation-preserving diffeomorphism $g:{\mathbf{R}}^{4}\to{\mathbf{R}}^{4}$ with $g|_{\mbox{cl}({\mathbf{R}}^{4}\setminus A)}=1$ such that

[TABLE]

By the composition $gf^{\prime}$, we have a desired equivalence $f$. $\square$

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A surface-link $F$ has only unique O2-handle pair in the rigid sense if for any O2-handle pairs $(D\times I,D^{\prime}\times I)$ and $(E\times I,E^{\prime}\times I)$ on $F$ with $(\partial D)\times I=(\partial E)\times I$ and $(\partial D^{\prime})\times I=(\partial E^{\prime})\times I$, there is an equivalence $f:{\mathbf{R}}^{4}\to{\mathbf{R}}^{4}$ from $F$ to $F$ such that $f(D\times I)=E\times I$ and $f(D^{\prime}\times I)=E^{\prime}\times I$. It is shown in [9] that every surface-link $F$ has only unique O2-handle pair in the rigid sense with an additional condition that there is an ambient isotopy $f_{t}\,(t\in[0,1])$ with $f_{0}=1$ and $f_{1}=f$ keeping $F^{c}_{D}$ fixed.

A surface-link $F$ has only unique O2-handle pair in the soft sense if for any O2-handle pairs $(D\times I,D^{\prime}\times I)$ and $(E\times I,E^{\prime}\times I)$ on $F$ attached to the same connected component of $F$, there is an equivalence $f:{\mathbf{R}}^{4}\to{\mathbf{R}}^{4}$ from $F$ to $F$ such that $f(D\times I)=E\times I$ and $f(D^{\prime}\times I)=E^{\prime}\times I$.

A surface-link not admitting any O2-handle pair is understood as a surface-link with only unique O2-handle pair in both the rigid and soft senses.

The following lemma shows that the uniqueness of an O2-handle pair in the soft sense is derived from the uniqueness of an O2-handle pair in the rigid sense.

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Lemma 2.2. Every surface-link has only unique O2-handle pair in the soft sense.

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Proof of Lemma 2.2. Let $(D\times I,D^{\prime}\times I)$ and $(E\times I,E^{\prime}\times I)$ be any two O2-handle pairs on a surface-link $F$ attached to the same connected component of $F$.

By Lemma 2.1, there is an equivalence $f:{\mathbf{R}}^{4}\to{\mathbf{R}}^{4}$ from $F$ to to $F(E\times I,E^{\prime}\times I)\#T$ keeping $F^{c}_{E}$ fixed.

Let $F_{E}(\dot{h})$ be a trivial surface-knot obtained from $F_{E}$ by the surgery along a system $h$ of mutually disjoint 1-handles $h_{j}\,(j=1,2,\dots,s)$ on $F_{E}$.

Let $\dot{h}$ be the system of cylinders $\dot{h}_{j}=h_{j}\cap F_{E}(\dot{h})\,(j=1,2,\dots,s)$, and $\ddot{h}$ is the system of two disks $\ddot{h}_{j}=\mbox{cl}(\partial h_{j}\setminus\dot{h}_{j})\,(j=1,2,\dots,s)$.

Let $(d\times I,d^{\prime}\times I)$ be a standard O2-handle pair on $T^{0}$ in the 4-ball defining the connected summand $T^{o}$ in $F_{E}\#T$, and $(e,e^{\prime})=(\partial d,\partial d^{\prime})$ which is a spin loop basis of $T^{o}$. By construction, the system $h$ of 1-handles $h_{j}\,(j=1,2,\dots,s)$ is disjoint from the disk pair $(d,d^{\prime})$. By an isotopic deformation of $f$, we can assume that the system $f^{-1}(h)$ of the 1-handles $f^{-1}(h_{j})\,(j=1,2,\dots,s)$ on $F$ is disjoint from $(D\times I,D^{\prime}\times I)$.

By [2] (see [9, (2.4.2)]), there is an orientation-preserving diffeomorphism $g:{\mathbf{R}}^{4}\to{\mathbf{R}}^{4}$ sending $F_{E}(\dot{h})\#T$ to itself such that the spin loop pair $(gf(\partial D),gf(\partial D^{\prime})=(e,e^{\prime})$ and the restriction of $g$ to the system $\dot{h}$ of the cylinders $\dot{h}_{j}\,(j=1,2,\dots,s)$ is the identity map. This last condition is assumed by a choice of a spin loop basis on $F_{E}(\dot{h})\#T$.

By the uniqueness of an O2-handle pair in the rigid sense given in [9], there is an ambient isotopy $i_{t}:{\mathbf{R}}^{4}\to{\mathbf{R}}^{4}\,(t\in[0,1])$ keeping $(F_{E}(\dot{h})\#T)^{c}$ fixed such that $i_{0}$ is the identity and $i_{1}g(f(D)\times I,f(D^{\prime})\times I)=(d\times I,d^{\prime}\times I)$. Let

[TABLE]

be a surface-link family with $G^{0}=F_{E}\#T$. There is an $O2$-handle pair

[TABLE]

on the surface-link $G^{t}$, where

[TABLE]

Then the surface-link $G^{0}(f(D)\times I,f(D^{\prime})\times I$ is given by

[TABLE]

and the surface-link $gG^{1}(d\times I,d^{\prime}\times I)$ is given by

[TABLE]

where the equivalence

[TABLE]

is obtained from the uniqueness of an O2-handle pair in the rigid sense given in [9]. Since there is an equivalence

[TABLE]

there is an equivalence $f^{\prime}$ from $F_{D}=F_{D}^{c}\cup\delta_{D}$ to $F_{E}=F_{E}^{c}\cup\delta_{E}$ for disks $\delta_{D}$ and $\delta_{E}$. By a disk move, we can assume that $f^{\prime}(\delta_{D})=\delta_{E}$. The map $f^{\prime}$ is isotopic to a diffeomorphism $f^{\prime\prime}:{\mathbf{R}}^{4}\to{\mathbf{R}}^{4}$ sending the associated bump $B_{D}$ of $(D\times I,D^{\prime}\times I)$ to the associated bump $B_{E}$ of $(E\times I,E^{\prime}\times I)$. The diffeomorphism $f^{\prime\prime}:{\mathbf{R}}^{4}\to{\mathbf{R}}^{4}$ is modified into an equivalence $f^{\prime\prime\prime}:{\mathbf{R}}^{4}\to{\mathbf{R}}^{4}$ from $F$ to $F$ such that $f^{\prime\prime\prime}(D\times I)=E\times I$ and $f^{\prime\prime\prime}(D^{\prime}\times I)=E^{\prime}\times I$ because the bumps $B_{D}$ and $B_{E}$ recover the unordered O2-handle pairs $(D\times I,D^{\prime}\times I)$ and $(E\times I,E^{\prime}\times I)$, respectively (cf. [9, Lemma 2.4]). Thus, every surface-link $F$ has only unique O2-handle pair in the soft sense. $\square$

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We use the following corollary to Lemma 2.2.

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Corollary 2.3. Let $F,F^{\prime}$ be surface-links with ordered components $F_{i},F^{\prime}_{i}\,(i=1,2,\dots,r)$, respectively, and $\bar{F}=F\#_{i}T,\bar{F}^{\prime}=F^{\prime}\#_{i}T$ the stabilizations of $F,F^{\prime}$ with induced ordered components obtained by the connected sums $F_{i}\#T,F^{\prime}_{i}\#T$ of the $i$th components $F_{i},F^{\prime}_{i}$ and a trivial torus-knot $T$ for some $i$, respectively. Assume that $\bar{F}$ is equivalent to $\bar{F}^{\prime}$ by a component-order-preserving equivalence. Then $F$ is equivalent to $F^{\prime}$ by a component-order-preserving equivalence.

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Remark 2.4. Corollary 2.3 for ribbon surface-links $F,F^{\prime}$ has a different proof using the result of [8].

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The proof of Theorem 1.1.1 is done as follows.

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Proof of Theorem 1.1.1. A surface-link $F$ with $r$ ordered components is *$k$*th-handle-reducible if $F$ is equivalent to a stabilization $F^{\prime}\#_{k}n_{k}T$ of a surface-link $F^{\prime}$ for a positive integer $n_{k}$, where $\#_{k}n_{k}T$ denotes the stabilizer components $n_{k}T$ attaching to the $k$th component of $F^{\prime}$. Otherwise, the surface-link $F$ is *$k$*th-handle-irreducible. Note that if a $k$th-handle-irreducible surface-link $F$ is component-order-preserving equivalent to a surface-link $G$, then $G$ is also $k$th-handle-irreducible.

Let $F$ and $G$ be ribbon surface-links with components $F_{i}\,(i=1,2,\dots,r)$ and $G_{i}\,(i=1,2,\dots,r)$, respectively. Let $F^{*}$and $G^{*}$ be handle-irreducible summands of $F$ and $G$, respectively.

Assume that there is an equivalence $f$ from $F$ to $G$. Then we show that $F^{*}$ and $G^{*}$ are equivalent. Changing the indexes if necessary, we assume that $f$ sends $F_{i}$ to $G_{i}$ for every $i$. Let

[TABLE]

Taking the inverse equivalence $f^{-1}$ instead of $f$ if necessary, we may assume that $n^{\prime}_{1}\geq n_{1}$. If $n^{\prime}_{1}>n_{1}$, then by (*), there is an equivalence $f^{(1)}$ from the first-handle-irreducible surface-link

[TABLE]

to the first-handle-reducible surface-link

[TABLE]

which has a contradiction. Thus, $n^{\prime}_{1}=n_{1}$ and the first-handle-irreducible surface-link $F^{(1)}$ is equivalent to the first-handle-irreducible ribbon surface-link

[TABLE]

By continuing this process, it is shown that $F^{*}$ is equivalent to $G^{*}$. This completes the proof of Theorem 1.1.1. $\square$

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3. Proof of Theorem 1.1.2

A chord graph is a pair $(o,\alpha)$ of a trivial ink $o$ and an arc system $\alpha$ attaching to $o$ in the 3-space ${\mathbf{R}}^{3}$, where $o$ and $\alpha$ are called a based loop system and a chord system, respectively. A chord diagram is a diagram $C(o,\alpha)$ in the plane ${\mathbf{R}}^{2}$ of a chord graph $(o,\alpha)$ as a spatial graph. Let $D^{+}$ be a proper disk system in the upper half-space ${\mathbf{R}}_{+}^{4}$ obtained from a disk system $d^{+}$ in ${\mathbf{R}}^{3}$ bounded by $o$ by pushing the interior into ${\mathbf{R}}_{+}^{4}$. Similarly, let $D^{-}$ be a proper disk system in the lower half-space ${\mathbf{R}}_{-}^{4}$ obtained from a disk system $d^{-}$ in ${\mathbf{R}}^{3}$ bounded by $o$ by pushing the interior into ${\mathbf{R}}_{-}^{4}$. Let $O$ be the union of $D^{+}$ and $D^{-}$ which is a trivial $nS^{2}$-link in the 4-space ${\mathbf{R}}^{4}$, where $n$ is the number of components of $o$. The union $O\cup\alpha$ is called a chorded sphere system constructed from a chord graph $(o,\alpha)$.

By using the Horibe-Yanagawa lemma in [11, I], the chorded sphere system $O\cup\alpha$ up to orientation-preserving diffeomorphisms of ${\mathbf{R}}^{4}$ is independent of choices of $d^{+}$ and $d^{-}$ and uniquely determined by the chord graph $(o,\alpha)$. A ribbon surface-link $F=F(o,\alpha)$ is uniquely constructed from the chorded sphere system $O\cup\alpha$ so that $F$ is the surgery of $O$ along a 2-handle system $N(\alpha)$ on $O$ with core arc system $\alpha$ (see [5, 6, 7, 8]), where note by [3] that the surface-link $F$ up to equivalences is unaffected by choices of the 2-handle $N(\alpha)$.

A semi-unknotted punctured handlebody system (or simply a SUPH system) for a surface-link $F$ is a punctured handlebody system $V$ in ${\mathbf{R}}^{4}$ such that the boundary $\partial V$ of $V$ is a union $F\cup O$ of $F$ and a trivial $S^{2}$-link $O$ with $F\cap O=\emptyset$. The following lemma is a characterization of a ribbon surface-link (cf. [11, II], Yanagawa [12]).

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Lemma 3.1. A surface-link F is a ribbon surface-link if and only if there is a punctured SUPH system $V$ for $F$.

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Proof of Lemma 3.1. Given a ribbon surface-link, a SUPH system $V$ is constructed by a thickening $O\times I$ of $O$ in ${\mathbf{R}}^{4}$ by attaching a 1-handle system. Conversely, given a SUPH system $V$ in ${\mathbf{R}}^{4}$ such that $\partial V=F\cup O$ for a trivial $S^{2}$-link $O$ with $F\cap O=\emptyset$, there is a chord system $\alpha$ in $V$ attaching to $O$ such that the frontier of the regular neighborhood of $O\cup\alpha$ in $V$ is parallel to $F$, showing that $F$ is a ribbon surface-link. $\square$

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The following lemma is basic to the proof of Theorem 1.1.2.

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Lemma 3.2. The following (1) and (2) hold.

(1) For a surface-link $F$ and a trivial torus-knot $T$, if a connected sum $F\#T$ is a ribbon surface-link, then $F$ is a ribbon surface-link.

(2) If $F$ is a ribbon surface-link and $(D\times I,D^{\prime}\times I)$ is an O2-handle pair on $F$, then $F(D\times I,D^{\prime}\times I)$ is a ribbon surface-link.

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Theorem 1.1.2 is a consequence of Lemma 3.2 as follows:

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Proof of Theorem 1.1.2. If a stabilization $\bar{F}$ of a surface-link $F$ is a ribbon surface-link, then $F$ is a ribbon surface-link by an inductive use of Lemma 3.2 (1). $\square$

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We are in a position to show Lemma 3.2.

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Proof of Lemma 3.2. The assertion (1) $\Rightarrow$ (2) holds. In fact, by Lemma 2.1, there is a connected sum splitting $F\cong F(D\times I,D^{\prime}\times I)\#T$ for a trivial torus-knot $T$. Thus, if $F$ is a ribbon surface-link, then $F(D\times I,D^{\prime}\times I)$ is a ribbon surface-link by (1).

We show (1). Let $F\#T=F_{1}\#T\cup F_{2}\cup\dots\cup F_{r}$ be a ribbon surface-link for a trivial torus-knot $T$. The following claim (3.2.1) is shown later.

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(3.2.1) There is a stabilization $\bar{F}=\bar{F}_{1}\cup F_{2}\cup\dots\cup F_{r}$ of $F\#T$ with $\bar{F}_{1}=F_{1}\#T\#_{i=1}^{2m}T_{i}$ such that the following conditions (i) and (ii) hold:

(i) There is an O2-handle pair $(D\times I,D^{\prime}\times I)$ on $\bar{F}$ attached to $\bar{F}_{1}$ such that the surface-link $\bar{F}(D\times I)$ is a ribbon surface-link admitting a SUPH system with the 1-handles $h^{\prime}_{i}\,(i=1,2,\dots,2m)$ trivially attached.

(ii) There is an O2-handle pair $(E\times I,E^{\prime}\times I)$ on $\bar{F}$ attached to $\bar{F}_{1}$ such that the surface-link $\bar{F}(E\times I)$ is the surface-link $F$ with the 1-handles $h^{\prime\prime}_{i}\,(i=1,2,\dots,2m)$ trivially attached.

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By assuming (3.2.1), the proof of Lemma 3.2 is completed as follows.

By (i), the surface-link $F^{\prime\prime}=\bar{F}(D\times I,D^{\prime}\times I)\cong\bar{F}(D\times I)$ is a ribbon surface-link and further the surface-link $F^{*}$ obtained from $F^{\prime\prime}$ by the surgery on O2-handle pairs of all the trivial 1-handles $h^{\prime}_{i}\,(i=1,2,\dots,2m)$ is also a ribbon surface-link. By (ii), the surface-link $\bar{F}(E\times I,E^{\prime}\times I)\cong\bar{F}(E\times I)$ is the surface-link $F$ with the 1-handles $h^{\prime\prime}_{i}\,(i=1,2,\dots,2m)$ trivially attached. By an inductive use of Lemma 2.2 (or Theorem 1.1.1), the surface-link $F$ is equivalent to the ribbon surface-link $F^{*}$. Hence $F$ is a ribbon surface-link, obtaining (3). Thus, the proof of Lemma 3.2 is completed except for the proof of (3.2.1). $\square$

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We are in a position to prove the claim (3.2.1).

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Proof of (3.2.1). Let $V$ be a SUPH system for $F\#T$ by Lemma 3.1. Let the component of the SUPH system $V$ containing $F_{1}\#T$ be a disk sum $U\#_{\partial}W$ for a punctured 3-ball $U$ and a handlebody $W$. Let $A$ be a 4D bump defining the connected sum $F\#T$ with $(F\#T)\cap A=T^{o}$.

We proceed the proof by assuming the following claim (3.2.2) shown later.

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(3.2.2) There are a spin loop basis $(\ell,\ell^{\prime})$ for $T^{o}$ and a spin simple loop $\tilde{\ell}^{\prime}$ in $F\#T$ such that $\mbox{Int}(\ell,\tilde{\ell}^{\prime})=1$ and $\tilde{\ell}^{\prime}$ bounds a disk $D^{\prime}$ in $W$.

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By assuming (3.2.2), the proof of (3.2.1) is completed as follows.

Let $p_{i}\,(i=0,1,\dots,2m)$ be the intersection points of $\ell$ and $\tilde{\ell}^{\prime}$. For every $i>0$, let $\alpha_{i}$ be an arc neighborhood of $p_{i}$ in $\ell$, and $h_{i}$ a 1-handle on $F\#T$ with a core arc $\hat{\alpha}_{i}$ obtained by pushing the interior of $\alpha_{i}$ into ${\mathbf{R}}^{4}\setminus V$. Let $\tilde{\alpha}_{i}$ be a proper arc in $\dot{\partial}h_{i}=\mbox{cl}(\partial h_{i}\setminus h_{i}\cap F\#T)$ parallel to $\hat{\alpha}_{i}$ in $h_{i}$ with $\partial\tilde{\alpha}_{i}=\partial\alpha_{i}$.

Let $\bar{F}=F\#T\#_{i=1}^{2m}T_{i}$ be a stabilization of $F$ associated with the system of mutually disjoint trivial 1-handles $h_{i}\,(i=1,2,\dots,2m)$.

Let $\tilde{\ell}$ be a simple loop obtained from $\ell$ by replacing $\alpha_{i}$ with $\tilde{\alpha}_{i}$ for every $i>0$. The loop $\tilde{\ell}$ is taken to be a spin loop in $\bar{F}$ meeting $\ell^{\prime}$ transversely in just one point.

Let $A$ be a 4D bump of the associated bump $B$ of an O2-handle pair $(E\times I,E^{\prime}\times I)$ on $F\#T$ in ${\mathbf{R}}^{4}$ attached to $T^{o}$ with $(\ell,\ell^{\prime})=(\partial E,\partial E^{\prime})$. Then the loop $\tilde{\ell}$ and the trivial 1-handles $h_{i}\,(i=1,2,\dots,2m)$ are taken in $A$.

Let $W^{+}(D^{\prime})$ be the handlebody obtained from the handlebody $W^{+}=W\cup_{i=1}^{2m}h_{i}$ by splitting along a thickened disk $D^{\prime}\times I$ of $D^{\prime}$. Then the manifold $V^{+}(D^{\prime})$ obtained from $V^{+}=V\cup_{i=1}^{2m}h_{i}$ by replacing $W^{+}$ with $W^{+}(D^{\prime})$ is a SUPH system.

The SUPH system $V^{+}$ is ambient isotopic in ${\mathbf{R}}^{4}$ to a SUPH system $\tilde{V}^{+}$ which is the union of $V^{+}(D^{\prime})$ and a solid torus $W_{1}$ in $A$ connected by a 1-handle $h_{W}$ in $A$, where the solid torus $W_{1}$ has a deformed disk $\tilde{D}^{\prime}$ of $D^{\prime}$ as a meridian disk and the loop $\tilde{\ell}$ as a longitude. Since the trivial 1-handles $h_{i}\,(i=1,2,\dots,2m)$ are taken in the bump $B$, the solid torus $W_{1}$ is moved into a 4-ball disjoint from $T^{o}\cup_{i=1}^{2m}h_{i}$ and hence the loop $\tilde{\ell}$ bounds a disk $\tilde{D}$ in $A$ not meeting $T^{o}$, $h_{i}$ for all $i>0$ and $h_{W}$. By putting back the ambient isotopy from the SUPH system $\tilde{V}^{+}$ to the SUPH system $V^{+}$, we see that there is an O2-handle pair $(D\times I,D^{\prime}\times I)$ on the surface-link $\bar{F}$ such that $\bar{F}(D\times I)$ is a ribbon surface-link admitting trivial 1-handles $h^{\prime}_{i}\,(i=1,2,\dots,2m)$. This shows (i).

On the other hand, the 1-handles $h_{i}\,(i=1,2,\dots,2m)$ on $F\#T$ are isotopically deformed in $A$ into 1-handles $h^{\prime\prime}_{i}\,(i=1,2,\dots,2m)$ on $F\#T$ disjoint from the disk pair $(E,E^{\prime})$ such that the core arcs of the 1-handles $h_{i}\,(i=1,2,\dots,2m)$ are deformed into simple arcs in $F\#T$ away from the disk pair $(E,E^{\prime})$ in $A$. Hence the surface-link $\bar{F}(E\times I,E^{\prime}\times I)$ is the surface-link $F$ with the trivial 1-handles $h^{\prime\prime}_{i}\,(i=1,2,\dots,2m)$ attached. This shows (ii). Thus, the proof of (3.2.1) is completed except for the proof of (3.2.2). $\square$

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The proof of (3.2.2) is given as follows:

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Proof of (3.2.2) Consider a disk sum decomposition of the handlebody $W$ into a 3-ball $B_{0}$ and solid tori $V_{j}=S^{1}\times D^{2}_{j}\,(j=1,2,\dots,g)$ pasting along mutually disjoint disks in $\partial B_{0}$. Let $(\ell_{j},m_{j})$ be a longitude-meridian pair of the solid torus $V_{j}$ for all $j$. By [1] (see [9, (2.4.1)]), the loop basis $(\ell_{j},m_{j})$ for $\partial V_{j}$ is taken as a spin loop basis in ${\mathbf{R}}~{}4$ for all $j$. The homology $H_{1}(\partial W;{\mathbb{Z}})$ has the basis $[\ell_{j}],[m_{j}],\,(j=1,2,\dots,g)$.

For a loop basis $(\ell,\ell^{\prime})$ of $T^{o}$ with the intersection number $\mbox{Int}(\ell,\ell^{\prime})=1$ in $T^{o}$, the image $I(T^{o})$ and the kernel $K(T^{o})$ of the natural homomorphism $\iota_{*}:H_{1}(T^{o};{\mathbb{Z}})\to H_{1}(W;{\mathbb{Z}})$ are infinite cyclic groups. Let $x$ be an element of $H_{1}(T^{o};{\mathbb{Z}})$ such that the image $\iota_{*}(x)$ is a generator of $I(T^{o})$, and $x^{\prime}$ a generator of $K(T^{o})$. By noting that the intersection number Int(x,x’)=1 in $T^{o}$, let $x=a[\ell]+b[\ell^{\prime}]$ and $x^{\prime}=a^{\prime}[\ell]+b^{\prime}[\ell^{\prime}]$ for coprime integral pairs $(a,b)$ and $(a^{\prime},b^{\prime})$ with $ab^{\prime}-a^{\prime}b=1$. Let $(\ell^{\prime\prime},\ell^{\prime\prime\prime})$ be a loop basis for $T^{o}$ such that $[\ell^{\prime\prime}]=x$ and $[\ell^{\prime\prime\prime}]=x^{\prime}$.

The homology class $[\ell^{\prime\prime}]\in H_{1}(\partial W;{\mathbb{Z}})$ is written as the sum

[TABLE]

for an integral system $a_{j},b_{j}\,(j=1,2,\dots,g)$. Since $\iota_{*}([\ell^{\prime\prime}])\neq 0$, there is a non-zero integer in the integers $a_{j}\,(j=1,2,\dots,g)$. By changing the orientations of $\ell_{j}$ and the indexes of the solid tori $V_{j}$ if necessary, assume that $a_{j}\geq 0$ for all $j$ and $a_{1}$ is the smallest non-zero integer in the integral system $a_{j}\,(j=1,2,\dots,g)$. For $j\geq 2$, let

[TABLE]

for an integer $r_{j}$ with $0\leq r_{j}<a_{1}$. By handle slides of $W$, we have a new disk sum decomposition of $W$ into a 3-ball $B_{0}$ and solid tori $V_{j}=S^{1}\times D^{2}_{j}\,(j=1,2,\dots,g)$ such that

[TABLE]

for some integers $\tilde{b}_{j}\,(j=2,3,\dots,g)$ By repeating this process, we have a disk sum decomposition of $W$ into a 3-ball $B_{0}$ and solid tori $V_{j}=S^{1}\times D^{2}_{j}\,(j=1,2,\dots,g)$ such that

[TABLE]

for some integers $\tilde{b}_{j}\,(j=1,2,\dots,g)$, where $a$ is the greatest common divisor of the integers $a_{j}\,(j=1,2,\dots,g)$.

Let

[TABLE]

for an integral system $b^{\prime}_{j}\,(j=1,2,\dots,g)$. Since the intersection number $\mbox{Int}(\ell^{\prime\prime},\ell^{\prime\prime\prime})=1$ in $\partial W$, we have $ab^{\prime}_{1}=1$ and hence $a=1$. By [1] (see [9, (2.4.1)]), the loop basis $(\ell^{\prime\prime},\ell^{\prime\prime\prime})$ of $T^{o}$ is taken spin if we consider $x+x^{\prime}$ instead of $x$ if necessary since $\ell^{\prime\prime\prime}$ is a spin loop. Since the intersection number $\mbox{Int}(\ell^{\prime\prime},m_{1})=1$ in $F\#T$, we can take $(\ell^{\prime\prime},\ell^{\prime\prime\prime})$, $m_{1}$ and a meridian disk of $m_{1}$ in $V_{1}$ as $(\ell,\ell^{\prime})$, $\tilde{\ell}^{\prime}$ and $D^{\prime}$ in (3.2.2), respectively. Thus, the proof of (3.2.2) is completed. $\square$

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This completes the proof of Lemma 3.2. $\square$

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Acknowledgements. This work was in part supported by Osaka City University Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics).

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