Ribbonness of a Stable-Ribbon Surface-Link, II: General Case

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.

1. Introduction

This paper generalizes the previous result that a stably trivial surface-link is a trivial surface-link to the result that a stable-ribbon surface-link is a ribbon surface-link [1,2]. A surface-link is a closed oriented (possibly disconnected) surface F which is embedded in the 4-space ${\mathbf{R}}^4$ by a smooth embedding. When F is connected, it is also called a surface-knot. When a fixed (possibly disconnected) closed surface $\mathbf{F}$ is smoothly embedded into ${\mathbf{R}}^4$, 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 an $S^2$-link. When $\mathbf{F}$ is connected, it is also called an $\mathbf{F}$-knot, and an $S^2$-knot for $\mathbf{F}=S^2$. Two surface-links F and $F^{\prime }$ are equivalent by an equivalence f if f is an orientation-preserving diffeomorphism $f:{\mathbf{R}}^4\to {\mathbf{R}}^4$ sending F to $F^{\prime }$ that preserves orientation. A trivial surface-link is a surface-link F which bounds 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 $n{S}^2$-link O for some n (where $n{S}^2$ denotes the disjoint union of n copies of the 2-sphere $S^2$) by surgery along an embedded 1-handle system [4,5,6,7]. This object is an old concept in surface-knot theory, but in recent years, it has been considered as a chord diagram, which is a relaxed version of a virtual graph including a virtual knotoid in a plane diagram [8,9,10,11]. A stabilization of a surface-link F is a connected sum $\overline{F}=F{\#}_{k=1}^s{T}_k$ of F and a system of trivial torus-knots $T_k(k=1,2,\cdots ,s)$. By granting $s=0$, a surface-link F itself is regarded as a stabilization of F. The trivial torus-knot system T is called the stabilizer with stabilizer components $T_k(k=1,2,\cdots ,s)$ on the stabilization $\overline{F}$ of F. A stable-ribbon surface-link is a surface-link F such that a stabilization $\overline{F}$ of F is a ribbon surface-link. Every surface-link F is equivalent to a stabilization of a surface-link $F_{\ast }$ with minimal total genus. This surface-link $F_{\ast }$ is called a handle-irreducible summand of F. The following result called Stable-Ribbon Theorem is our main theorem.

Theorem 1. 

Any handle-irreducible summand $F_{\ast }$ of every stable-ribbon surface-link F is a ribbon surface-link that is determined uniquely from F up to equivalences and stabilizations.

Any stabilization of a ribbon surface-link is a ribbon surface-link. So, the following corollary is obtained from Theorem 1.

Corollary 1. 

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

A stably trivial surface-link is a surface-link F such that a stabilization $\overline{F}$ of F is a trivial surface-link. Since a trivial surface-link is a ribbon surface-link, Theorem 1 also implies the following corollary, which is used to prove smooth unknotting conjecture for a surface-link. This result leads to 4D smooth and then classical Poincaré conjectures [12,13,14,15].

Corollary 2. 

Any handle-irreducible summand of every stably trivial surface-link is a trivial $S^2$-link, so that every stably trivial surface-link is a trivial surface-link.

The plan for the Proof of Theorem 1 is to show the following two lemmas by using the previous techniques [1,2].

Lemma 1. 

Any handle-irreducible summand of any surface-link is unique up to equivalences and stabilizations.

Lemma 2. 

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

The Proof of Theorem 1 is completed by these lemmas as follows:

Proof of Theorem 1 assuming Lemmas 1 and 2. 

By Lemma 2, any handle-irreducible summand of every stable-ribbon surface-link is a ribbon surface-link, which is unique up to equivalences and stabilizations by Lemma 1. This completes the Proof of Theorem 1. □

An idea of the Proof of Lemma 1 is to generalize the unique result of an O2-handle pair on a surface-link earlier established to the case where the restriction on the attaching part is relaxed (see Theorem 4). An idea of the Proof of Lemma 2 is to consider a semi-unknotted multi-punctured handlebody system, simply called a SUPH system, of a ribbon surface-link. Two applications of Theorem 1 are made. One observation on Theorem 1 is the following theorem.

Theorem 2. 

A connected sum $F=F_1\#F_2$ of surface-links $F_i(i=1,2)$ in $S^4$ is a ribbon surface-link if and only if both the surface-links $F_i(i=1,2)$ are ribbon surface-links.

This theorem contrasts with the behavior of classical ribbon knot, because every classical knot is a connected summand of a connected sum ribbon knot. In fact, for every knot k and the inversed mirror image $-k^{\ast }$ of k in the 3-sphere $S^3$, the connected sum $k\#(-k^{\ast })$ is a ribbon knot in $S^3$ [16,17,18,19]. A natural presentation of $k\#(-k^{\ast })$ is seen in a chord diagram of the spun $S^2$-knot of k as a ribbon $S^2$-knot [9].

A surface-knot $F^{\prime }$ in $S^4$ is obtained from a surface-link F of r components in $S^4$ by fusion if $F^{\prime }$ is obtained from F by surgery along $r-1$ disjointedly embedded 1-handles on F in $S^4$. In an earlier preprint of this paper, it is claimed that every $S^2$-link L in $S^4$ consisting of trivial components in $S^4$ is a ribbon $S^2$-link without restrictions. However, the proof contains an error. In fact, there was a non-ribbon $S^2$-link L consisting of two trivial components such that the $S^2$-link obtained from L by a fusion is a non-ribbon $S^2$-knot [20]. As a revised content, the following theorem is shown, giving a characterization of when a surface-link consisting of ribbon surface-knot components is a ribbon surface-link. Since a trivial surface-knot is a ribbon surface-knot, this theorem also gives a characterization of when a surface-link consisting of trivial components is a ribbon surface-link.

Theorem 3. 

The following statements (1)–(3) on a surface-link F consisting of ribbon surface-knot components in $S^4$ are mutually equivalent:

(1) 

F is a ribbon surface-link.

(2) 

The surface-knot obtained from F by every fusion is a ribbon surface-knot.

(3) 

The surface-knot obtained from F by a fusion is a ribbon surface-knot.

There are a lot of classical non-ribbon links consisting of trivial components producing a ribbon-knot by a fusion such as Hopf link, the split sum of Whitehead link and its mirror image, and the split sum of Borromean rings and its mirror image, etc., [16,21]. Thus, this theorem also contrasts with the behavior of a classical ribbon link.

The Proofs of Lemmas 1 and 2 are given in Section 2 and Section 3, respectively. In Section 4, the Proofs of Theorems 2 and 3 are given.

2. Proof of Lemma 1

A 2-handle on a surface-link F in ${\mathbf{R}}^4$ is a 2-handle $D\times I$ on F with D a core disk embedded in ${\mathbf{R}}^4$ such that $D\times I\cap F=\partial D\times I$, where I denotes a closed interval containing 0 and $D\times 0$ is identified with D. Two 2-handles $D\times I$ and $E\times I$ on F are equivalent if there is an equivalence $f:{\mathbf{R}}^4\to {\mathbf{R}}^4$ from F to itself such that the restriction ${f|}_F:F\to F$ is the identity map and $f(D\times I)=E\times I$.

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

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

and $\partial D\times I$ and $\partial D^{\prime }\times I$ meet orthogonally on F, that is, the boundary circles $\partial D$ and $\partial D^{\prime }$ meet transversely at one point p so that the intersection $\partial D\times I\cap \partial D^{\prime }\times I$ is homeomorphic to the square $Q=p\times I\times I$. 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 $\delta \cup F_{\delta }^c$ of the plumbed disk

$$\delta ={\delta }_{D\times I,D^{\prime }\times I}=D\times \partial I\cup Q\cup D^{\prime }\times \partial I$$

and the surface

$$F_{\delta }^c=\mathrm{cl}(F\setminus (\partial D\times I\cup \partial D^{\prime }\times I)).$$

A once-punctured torus $T^o$ in a 3-ball B is trivial if $T^o$ is smoothly and properly embedded in B which splits B into two solid tori. A bump of a surface-link F is a 3-ball B in ${\mathbf{R}}^4$ with $F\cap B=T^o$ a trivial once-punctured torus in B. Let $F\left(B\right)$ be a surface-link $F_B^c\cup {\delta }_B$ which is the union of the surface $F_B^c=\mathrm{cl}(F\setminus T^o)$ and a disk ${\delta }_B$ in the 2-sphere $\partial B$ with $\partial {\delta }_B=\partial T^o$. A cellular move of a compact (possibly, bounded) surface P in ${\mathbf{R}}^4$ is a compact surface $\tilde{P}$ such that the intersection $P^o=P\cap \tilde{P}$ is a once-punctured compact surface of P and $\tilde{P}$ with $d=\mathrm{cl}(P\setminus P^o)$ and $\tilde{d}=(\tilde{P}\setminus P^o)$ disks in the interiors of P and $\tilde{P}$, respectively such that the union $d\cup \tilde{d}$ is a 2-sphere bounding a 3-ball smoothly embedded in ${\mathbf{R}}^4$ and not meeting the interior of $P^o$. Note that $F\left(B\right)$ is uniquely determined up to cellular moves on the disk ${\delta }_B$ keeping $F_B^c$ 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. Consider the 3-ball $\Delta$ as a Seifert hypersurface of the trivial $S^2$-knot $K=\partial \Delta$ in ${\mathbf{R}}^4$ to construct a 3-ball $B_{\Delta }$ obtained from $\Delta$ by adding an outer boundary collar. This 3-ball $B_{\Delta }$ is a bump of F, which we call the associated bump of the O2-handle pair $(D\times I,D^{\prime }\times I)$. When the union of the 3-ball $\Delta$ and a boundary collar of $F_{\delta }^c$ are deformed into the 3-space ${\mathbf{R}}^3\subset {\mathbf{R}}^4$, this associated bump $B_{\Delta }$ is also considered as a regular neighborhood of $\Delta$ in ${\mathbf{R}}^3$. It is observed that an O2-handle unordered pair $(D\times I,D^{\prime }\times I)$ on a surface-link F is constructed uniquely from any given bump B of F in ${\mathbf{R}}^4$ with $F(D\times I,D^{\prime }\times I)\cong F\left(B\right)$ [1]. Further, for any O2-handle pair $(D\times I,D^{\prime }\times I)$ on any surface-link F and the associated bump B, there are identifications

$$F\left(B\right)=F(D\times I,D^{\prime }\times I)=F(D\times I)=F(D^{\prime }\times I)$$

by equivalences which are attained by cellular moves on the disk $\delta ={\delta }_{D\times I,D^{\prime }\times I}$ keeping $F_{\delta }^c$ fixed [1]. A once-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 once-punctured torus in A. A 4D bump A is obtained from a bump B of a surface-link F by taking a bi-collar $c(B\times [-1,1\left]\right)$ of B in ${\mathbf{R}}^4$ with $c(B\times 0)=B$. The following lemma is proved by using a 4D bump A.

Lemma 3. 

Let $(D\times I,D^{\prime }\times I)$ be any O2-handle pair on any surface-link F in ${\mathbf{R}}^4$ and T a trivial torus-knot in ${\mathbf{R}}^4$ with any given spin loop basis $(e,e^{\prime })$. Then, 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 $F_{\delta }^c$ fixed such that $f(\partial D)=e$ and $f(\partial D^{\prime })=e^{\prime }$.

Proof of Lemma 3. 

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 there is a solid torus V in A whose boundary is the union of the trivial once-punctured torus $P=F\cap A$ and the disk ${\delta }_A$. This solid torus V induces an equivalence $f^{\prime }:({\mathbf{R}}^4,F)\to ({\mathbf{R}}^4,F(D\times I,D^{\prime }\times I)\#T)$ sending P to the connected summand $T^o$ of a connected sum $F(D\times I,D^{\prime }\times I)\#T$ in A. Let $(\tilde{e},{\tilde{e}}^{\prime })$ be the spin loop basis of $T^o$ which is the image of the spin loop pair $(\partial D,\partial D^{\prime })$ on F under $f^{\prime }$. There is an orientation-preserving diffeomorphism $g:{\mathbf{R}}^4\to {\mathbf{R}}^4$ with ${g|}_{\mathrm{cl}({\mathbf{R}}^4\setminus A)}=1$ such that $g(\tilde{e},{\tilde{e}}^{\prime })=(e,e^{\prime })$ by the previous techniques [1,22]. The composition $f=g{f}^{\prime }$ is a desired equivalence. This completes the Proof of Lemma 3. □

A surface-link F only has a 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 itself keeping $F_{\delta }^c$ fixed such that $f(D\times I)=E\times I$ and $f(D^{\prime }\times I)=E^{\prime }\times I$. It is proved that every surface-link F has only unique O2-handle pair in the rigid sense [1,2]. 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, say $F_1$ of F, there is an equivalence $f:{\mathbf{R}}^4\to {\mathbf{R}}^4$ from F to itself keeping $F^{\left(1\right)}=F\setminus F_1$ fixed 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 a unique O2-handle pair in both the rigid and soft senses. The following uniqueness of an O2-handle pair in the soft sense is essentially a consequence of the uniqueness of an O2-handle pair in the rigid sense.

Theorem 4 (Uniqueness of an O2-handle pair in the soft sense).

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

Proof of Theorem 4. 

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 $F_1$ of F. Let $F^{\left(1\right)}=F\setminus F_1$. By Lemma 3, there is an equivalence $f:{\mathbf{R}}^4\to {\mathbf{R}}^4$ from F to the connected sum

$$F(D\times I,D^{\prime }\times I)\#T=F^{\left(1\right)}\cup {\tilde{F}}_1\#T$$

keeping $F^{\left(1\right)}$ fixed and sending $F_1$ to ${\tilde{F}}_1\#T$, where ${\tilde{F}}_1=F_1(D\times I,D^{\prime }\times I)$ and T is a trivial torus-knot in ${\mathbf{R}}^4$. Similarly, there is an equivalence $f^{\prime }:{\mathbf{R}}^4\to {\mathbf{R}}^4$ from F to the connected sum

$$F(E\times I,E^{\prime }\times I)\#T^{\prime }=F^{\left(1\right)}\cup {\tilde{F}}_1^{\prime }\#T^{\prime }$$

keeping $F^{\left(1\right)}$ fixed and sending $F_1$ to ${\tilde{F}}_1^{\prime }\#T^{\prime }$, where ${\tilde{F}}_1^{\prime }=F_1(E\times I,E^{\prime }\times I)$ and $T^{\prime }$ is a trivial torus-knot in ${\mathbf{R}}^4$. The diffeomorphism $g=f^{\prime }{f}^{-1}:{\mathbf{R}}^4\to {\mathbf{R}}^4$ is an equivalence from $F^{\left(1\right)}\cup {\tilde{F}}_1\#T$ to $F^{\left(1\right)}\cup {\tilde{F}}_1^{\prime }\#T^{\prime }$ keeping $F^{\left(1\right)}$ fixed. The connected sum ${\tilde{F}}_1\#T$ is obtained from the split union ${\tilde{F}}_1+T$ in ${\mathbf{R}}^4$ by surgery along an embedded 1-handle h connecting a disk $d_1\subset {\tilde{F}}_1$ and a disk $d\subset T$, and the connected sum ${\tilde{F}}_1^{\prime }\#T^{\prime }$ is obtained from the split union ${\tilde{F}}_1^{\prime }+T^{\prime }$ in ${\mathbf{R}}^4$ by surgery along an embedded 1-handle $h^{\prime }$ connecting a disk $d_1^{\prime }\subset {\tilde{F}}_1^{\prime }$ and a disk $d^{\prime }\subset \tilde{T}$. Then there is a 4-ball A in ${\mathbf{R}}^4$ such that $T^o=A\cap (F^{\left(1\right)}\cup {\tilde{F}}_1\#T)$ is a trivial once-punctured torus of T in A with $d_1$ a disk bounded by the trivial knot $\partial T^o$ in the 3-sphere $\partial A$. Similarly, there is a 4-ball $A^{\prime }$ in ${\mathbf{R}}^4$ such that ${\left(T^{\prime }\right)}^o=A^{\prime }\cap (F^{\left(1\right)}\cup {\tilde{F}}_1^{\prime }\#T^{\prime })$ is a trivial once-punctured torus of $T^{\prime }$ in $A^{\prime }$ with $d_1^{\prime }$ a disk bounded by the trivial knot $\partial {\left(T^{\prime }\right)}^o$ in the 3-sphere $\partial A^{\prime }$. It may be assumed that $g(\partial d_1)=\partial d_1^{\prime }$ by sliding the attaching loop $g(\partial d_1)$ in $g({\tilde{F}}_1\#T)$ and/or the attaching loop $\partial d_1^{\prime }$ in ${\tilde{F}}_1^{\prime }\#T^{\prime }$. Then, it is assumed that $g\left(T^o\right)={\left(T^{\prime }\right)}^o$ (For a special case that $g\left(T^0\right)=\mathrm{cl}({\tilde{F}}_1^{\prime }\#T^{\prime }\setminus {\left(T^{\prime }\right)}^o)$, there is a deformation from $g\left(A\right)$ into $A^{\prime }$ to obtain $g\left(T^0\right)={\left(T^{\prime }\right)}^o$). Further, by Lemma 3, it is assumed that $g(f(\partial D),f(\partial D^{\prime }))=(f^{\prime }(\partial E),f^{\prime }(\partial E^{\prime }))$. Then

$$(f^{\prime }(\partial D),f^{\prime }(\partial D^{\prime }))=g(f(\partial D),f(\partial D^{\prime }))=(f^{\prime }(\partial E),f^{\prime }(\partial E^{\prime })).$$

Since every surface-link only has a unique O2-handle pair in the rigid sense, there is an equivalence $g^{\prime }:{\mathbf{R}}^4\to {\mathbf{R}}^4$ from $F^{\left(1\right)}\cup {\tilde{F}}_1^{\prime }\#T^{\prime }$ to itself keeping $F^{\left(1\right)}$ fixed such that $g^{\prime }(f^{\prime }\left(D\right)\times I,f^{\prime }\left(D^{\prime }\right)\times I)=(f^{\prime }\left(E\right)\times I,f^{\prime }\left(E^{\prime }\right)\times I)$. The composite equivalence $g^{\ast }={\left(f^{\prime }\right)}^{-1}{g}^{\prime }gf:{\mathbf{R}}^4\to {\mathbf{R}}^4$ is an equivalence from F to itself keeping $F^{\left(1\right)}$ fixed and sending $(D\times I,D^{\prime }\times I)$ to $(E\times I,E^{\prime }\times I)$. Thus, every surface-link F only has a unique O2-handle pair in the soft sense. This completes the Proof of Theorem 4. □

The following corollary is obtained from the Proof of Theorem 4.

Corollary 3. 

Let F and $F^{\prime }$ be surface-links with ordered components $F_i(i=1,2,\cdots ,r)$ and $F_i^{\prime }(i=1,2,\cdots ,r)$, respectively. Assume that the stabilizations $\overline{F}=F{\#}_{i}T,{\overline{F}}^{\prime }=F^{\prime }{\#}_{i}T$ of $F,F^{\prime }$ with induced ordered components obtained by the connected sums $F_i\#T,F_i^{\prime }\#T$ of the ith components $F_i,F_i^{\prime }$ and a trivial torus-knot T, respectively, are equivalent by a component-order-preserving equivalence ${\mathbf{R}}^4\to {\mathbf{R}}^4$. Then, F is equivalent to $F^{\prime }$ by a component-order-preserving equivalence ${\mathbf{R}}^4\to {\mathbf{R}}^4$.

Remark 1. 

For the case of ribbon surface-links F and $F^{\prime }$, Corollary 3 has a different proof [9,23].

The Proof of Lemma 1 is performed as follows.

Proof of Lemma 1. 

A surface-link F with r ordered components is kth handle-reducible if F is equivalent to a stabilization $F^{\prime }{\#}_k{n}_{k}T$ of a surface-link $F^{\prime }$ for an integer $n_k>0$, where ${\#}_k{n}_{k}T$ denotes the stabilizer components $n_{k}T$ attaching to the kth component of $F^{\prime }$. Otherwise, the surface-link F is said to be kth handle-irreducible. Note that if a surface-link G is equivalent to a kth handle-irreducible surface-link F by component-order-preserving equivalence, then G is also kth handle-irreducible. Let F and G be ribbon surface-links with components $F_i(i=1,2,\cdots ,r)$ and $G_i(i=1,2,\cdots ,r)$, respectively. Let $F_{\ast }$ and $G_{\ast }$ be handle-irreducible summands of F and G, respectively. Assume that there is an equivalence f from F to G. Then, it is shown that $F_{\ast }$ and $G_{\ast }$ are equivalent, as follows. Changing the indexes if necessary, assume that f sends $F_i$ to $G_i$ for every i. □

Let

$$\begin{array}{ccc}\hfill F& =& F_{\ast }{\#}_1{n}_{1}T{\#}_2{n}_{2}T{\#}_3\cdots {\#}_r{n}_{r}T,\hfill \\ \hfill G& =& G_{\ast }{\#}_1{n}_1^{\prime }T{\#}_2{n}_2^{\prime }T{\#}_3\cdots {\#}_r{n}_r^{\prime }T.\hfill \end{array}$$

If necessary, by taking the inverse equivalence $f^{-1}$ instead of f, assume that $n_1^{\prime }\ge n_1$. If $n_1^{\prime }>n_1$, then there is a component-order-preserving equivalence $f^{\left(1\right)}$ from the first-handle-irreducible surface-link

$$F_{\left(1\right)}=F_{\ast }{\#}_2{n}_{2}T{\#}_3\cdots {\#}_r{n}_{r}T$$

to the first-handle-reducible surface-link

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

by Corollary 3, which contradicts the first handle-irreducibility. Thus, $n_1^{\prime }=n_1$, and the first handle-irreducible surface-link $F_{\left(1\right)}$ is equivalent to the first-handle-irreducible ribbon surface-link

$$G_{\left(1\right)}=G_{\ast }{\#}_2{n}_2^{\prime }T{\#}_3\cdots {\#}_r{n}_r^{\prime }T.$$

By continuing this process, it is shown that $F_{\ast }$ is equivalent to $G_{\ast }$. This completes the Proof of Lemma 1.

3. Proof of Lemma 2

A chorded loop system is a pair $(o,\alpha )$ of a trivial link 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 chorded loop diagram or simply a chord diagram is a diagram $C(o,\alpha )$ in the plane ${\mathbf{R}}^2$ of the spatial graph $o\cup \alpha$. 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 $n{S}^2$-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 chorded loop system $(o,\alpha )$. 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 chorded loop system $(o,\alpha )$ by the Horibe–Yanagawa lemma [19]. Thus, every ribbon surface-link F is uniquely constructed from a chorded loop system $(o,\alpha )$ via the chorded sphere system $O\cup \alpha$ so that $F=F(o,\alpha )$ is obtained from O by surgery along a 1-handle system $N\left(\alpha \right)$ on O with core arc system $\alpha$, where note that the surface-link F up to equivalences is unaffected by choices of any 1-handle system $N\left(\alpha \right)$ fixing $\alpha$ [3]. The moves on a chorded loop system $(o,\alpha )$ giving the same ribbon surface-link up to equivalences are determined [9]. A semi-unknotted multi-punctured handlebody system or simply a SUPH system for a surface-link F in ${\mathbf{R}}^4$ is a multi-punctured handlebody system V (smoothly embedded) in ${\mathbf{R}}^4$ such that $\partial V=F\cup O$ for a trivial $S^2$-link O in ${\mathbf{R}}^4$. Note that the numbers of connected components in F and V are equal. The following lemma makes a characterization of a ribbon surface-link [4,7].

Lemma 4. 

A surface-link F is a ribbon surface-link if and only if F admits a SUPH system V in ${\mathbf{R}}^4$.

Proof of Lemma 4. 

A SUPH system V for a ribbon surface-link F is constructed from a chorded sphere system $O\cup \alpha$ by taking the union of a thickening $O\times [0,1]$ of O in ${\mathbf{R}}^4$ and the 1-handle system $N\left(\alpha \right)$ attaching only to $O\times 0$. Conversely, given a SUPH system V in ${\mathbf{R}}^4$ with $\partial V=F\cup O$ for a trivial $S^2$-link O, then take a chord system $\alpha$ in V attaching to O so that the frontier of the regular neighborhood of $O\cup \alpha$ in V is parallel to F in V. The chorded sphere system $O\cup \alpha$ shows that F is a ribbon surface-link. This completes the Proof of Lemma 4. □

Let F be a surface-link of components $F_i(i=1,2,\cdots ,r)$ in ${\mathbf{R}}^4$. Let $F\#T$ be the connected sum of F and a trivial torus-knot T in ${\mathbf{R}}^4$ consisting of the components $F_1\#T,F_i(i=2,3,\cdots ,r)$. Assume that $F\#T$ is a ribbon surface-link. By Lemma 4, let V be a SUPH system for $F\#T$ in ${\mathbf{R}}^4$. Let $V_1$ be the component of V for $F_1\#T$ and write $V_1=U{\#}_{\partial }W$, a disk sum for a multi-punctured 3-ball U and a handlebody W. The following lemma is needed to prove Lemma 2.

Lemma 5. 

For a suitable spin loop basis $(\ell ,{\ell }^{\prime })$ for $T^o$, there is a spin simple loop ${\tilde{\ell }}^{\prime }$ in the ribbon-surface-link $F_1\#T$ with intersection number $Int(\ell ,{\tilde{\ell }}^{\prime })\ne 0$ in $F_1\#T$ such that the loop ${\tilde{\ell }}^{\prime }$ bounds a disk $D^{\prime }$ in the handlebody W.

Proof of Lemma 5. 

Consider a disk sum decomposition of the handlebody W into solid tori $S^1\times D_j^2(j=1,2,\cdots ,g)$ pasting along mutually disjoint disks. Let $({\ell }_j,m_j)$ be a longitude-meridian pair of the solid torus $S^1\times D_j^2$ for all j. The loop basis $({\ell }_j,m_j)$ for $S^1\times D_j^2$ is chosen to be a spin loop basis in ${\mathbf{R}}^4$ for all j [24]. By a choice of a spin loop basis $(\ell ,{\ell }^{\prime })$ for $T^o$, the loop ℓ meets a meridian loop $m_j$ with a non-zero intersection number in $\partial W$. The loop $m_j$ is taken to be a loop ${\tilde{\ell }}^{\prime }$ in $F_1\#T$ bounding a disk $D^{\prime }$ in W with intersection number $\mathrm{Int}(\ell ,{\tilde{\ell }}^{\prime })\ne 0$ since $m_j$ bounds a meridian disk $1\times D_j^2$ of the solid torus $S^1\times D_j^2\subset W$. This completes the Proof of Lemma 5. □

The following lemma is obtained by using Lemma 5.

Lemma 6. 

There is a stabilization $\overline{F}$ of the ribbon surface-link $F\#T$ in ${\mathbf{R}}^4$ consisting of the components ${\overline{F}}_1,F_i(i=2,3,\cdots ,r)$ where ${\overline{F}}_1$ is the connected sum of $F_1\#T$ and trivial torus-knots $T_i(i=1,2,\cdots ,m)$ for some $m\ge 0$ such that the surface-link $\overline{F}$ has the following conditions (i) and (ii).

(i)

There is an O2-handle pair $(D\times I,D^{\prime }\times I)$ on $\overline{F}$ attached to ${\overline{F}}_1$ such that the surface-link $\overline{F}(D^{\prime }\times I)$ is a ribbon surface-link with trivial 1-handles $h_i^{\prime }(i=1,2,\cdots ,m)$ attached.

(ii)

There is an O2-handle pair $(E\times I,E^{\prime }\times I)$ on $\overline{F}$ attached to ${\overline{F}}_1$ such that the surface-link $\overline{F}(E^{\prime }\times I)$ is F with trivial 1-handles $h_i^{″}(i=1,2,\cdots ,m)$ attached.

Proof of Lemma 6. 

Let $p_i(i=0,1,\cdots ,m)$ be the intersection points of transversely meeting simple loops ℓ and ${\tilde{\ell }}^{\prime }$ in $F_1\#T$ given by Lemma 5. For every $i>0$, let ${\alpha }_i$ be an arc neighborhood of $p_i$ in ℓ, and $h_i$ a 1-handle on $F\#T$ with a core arc ${\widehat{\alpha }}_i$ obtained by pushing the interior of ${\alpha }_i$ outside the SUPH system V. Let $\overline{F}=F\#T{\#}_{i=1}^m{T}_i$ be a stabilization of $F\#T$ with the component ${\overline{F}}_1=F_1\#T{\#}_{i=1}^m{T}_i$ obtained from $F_1\#T$ by surgery along the disjoint trivial 1-handle system $h_i(i=1,2,\cdots ,m)$. Let ${\alpha }_i^{+}={\alpha }_i\cup (h_i\cap \ell )$ be the arc in ℓ extending ${\alpha }_i$. Let ${\tilde{\alpha }}_i$ be a proper arc in the annulus $\mathrm{cl}(\partial h_i\setminus h_i\cap F\#T)$, which is parallel to the core arc ${\widehat{\alpha }}_i$ of $h_j$ in $h_i$ with $\partial {\tilde{\alpha }}_i=\partial {\alpha }_i^{+}$. Let $\tilde{\ell }$ be a simple spin loop in $\overline{F}$ obtained from ℓ by replacing ${\alpha }_i^{+}$ with ${\tilde{\alpha }}_i$ for every $i>0$, which meets ${\tilde{\ell }}^{\prime }$ transversely in just one point. Let $W^{+}\left(D^{\prime }\right)$ be the handlebody obtained from the handlebody $W^{+}=W{\cup }_{i=1}^m{h}_i$ by removing a thickened disk $D^{\prime }\times I$ of $D^{\prime }$. The manifold $V^{+}\left(D^{\prime }\right)$ obtained from the SUPH system $V^{+}=V{\cup }_{i=1}^m{h}_i$ for the ribbon surface-link $\overline{F}$ by replacing $W^{+}$ with $W^{+}\left(D^{\prime }\right)$ is a SUPH system for a surface-link ${\overline{F}}^{\prime }$ in ${\mathbf{R}}^4$ consisting of $F_i(i=2,3,\cdots ,r)$ and a component ${\overline{F}}_1^{\prime }$ with genus reduced by 1 from ${\overline{F}}_1$. By Lemma 4, ${\overline{F}}^{\prime }$ is a ribbon-surface-link in ${\mathbf{R}}^4$. The SUPH system $V^{+}$ for $\overline{F}$ is a disk sum of $V^{+}\left(D^{\prime }\right)$ and a solid torus $W_1$ with the disk $D^{\prime }$ as a meridian disk and the loop $\tilde{\ell }$ as a longitude. Let $d_W=V^{+}\left(D^{\prime }\right)\cap W_1$ be the pasting disk between $V^{+}\left(D^{\prime }\right)$ and $W_1$, which is regarded as a 1-handle $h_W$ joining $V^{+}\left(D^{\prime }\right)$ and $W_1$. Let $(E\times I,E^{\prime }\times I)$ be an O2-handle pair on $F\#T$ in ${\mathbf{R}}^4$ attached to $T^o$ with $(\partial E,\partial E^{\prime })=(\ell ,{\ell }^{\prime })$ in Lemma 5. Let A be a 4D bump of the associated bump B of $(E\times I,E^{\prime }\times I)$. In the case of (i), since there is no need to worry about the intersection of A with $E\cup E^{\prime }$, the 4D ball A is deformed so that $V^{+}\cap A=W_1\cup h_W$ by observing that $V\setminus V_1$ and U are disjoint from A by construction of A and by taking spine graphs of $W^{+}\left(D^{\prime }\right)$, $W_1$ and $h_W$. Then, the loop $\tilde{\ell }$ bounds a disk D in A not meeting the interior of $W_1$ and $h_W$. This means that there is an O2-handle pair $(D\times I,D^{\prime }\times I)$ on the surface-link $\overline{F}$ such that $\overline{F}(D^{\prime }\times I)$ is a ribbon surface-link with trivial 1-handles $h_i^{\prime }(i=1,2,\cdots ,m)$ attached, showing (i). For the case of (ii), note that the 1-handles $h_i(i=1,2,\cdots ,m)$ on $F\#T$ are deformed isotopically in A into 1-handles $h_i^{″}(i=1,2,\cdots ,m)$ on $F\#T$ disjoint from the disk pair $(E,E^{\prime })$ because the core arcs of the 1-handles $h_i(i=1,2,\cdots ,m)$ are deformed to be disjoint from the disk pair $(E,E^{\prime })$ in A. The surface-link $\overline{F}(E^{\prime }\times I)$ that is equivalent to $\overline{F}(E\times I,E^{\prime }\times I)$ is the surface-link F with the trivial 1-handles $h_i^{″}(i=1,2,\cdots ,m)$ attached, showing (ii). Thus, the Proof of Lemma 6 is completed. □

The following lemma is a combination of Lemma 6 and the uniqueness of an O2-handle pair in the soft sense (Theorem 4).

Lemma 7. 

If a connected sum $F\#T$ of a surface-link F and a trivial torus-knot T in ${\mathbf{R}}^4$ is a ribbon surface-link, then F is a ribbon surface-link.

Proof of Lemma 7. 

Let $F\#T=F_1\#T\cup F_2\cup \cdots \cup F_r$ be a ribbon surface-link for a trivial torus-knot T. By Lemma 6 (i), the surface-link $F^{″}=\overline{F}(D\times I,D^{\prime }\times I)$ equivalent to $\overline{F}(D\times I)$ is a ribbon surface-link and further the surface-link $F^{\ast }$ obtained from $F^{″}$ by the surgery on O2-handle pairs of all the trivial 1-handles $h_i^{\prime }(i=1,2,\cdots ,m)$ is a ribbon surface-link. By Lemma 6 (ii), the surface-link $\overline{F}(E\times I,E^{\prime }\times I)$ equivalent to $\overline{F}(E\times I)$ is the surface-link F with the 1-handles $h_i^{″}(i=1,2,\cdots ,m)$ trivially attached. By an inductive use of Theorem 4 (uniqueness of an O2-handle pair in the soft sense), the surface-link F is equivalent to the ribbon surface-link $F^{\ast }$. Thus, F is a ribbon surface-link, and the Proof of Lemma 7 is completed. □

Lemma 2 is a direct consequence of Lemma 7 as follows.

Proof of Lemma 2. 

If a stabilization $\overline{F}$ of a surface-link F is a ribbon surface-link, then F is a ribbon surface-link by an inductive use of Lemma 7. This completes the Proof of Lemma 2. □

4. Proofs of Theorems 2 and 3

The Proof of Theorem 2 is performed as follows.

Proof of Theorem 2. 

The ‘if’ part of Theorem 2 is seen from the definition of a ribbon surface-link. The Proof of the ‘only if’ part of Theorem 2 uses the fact that every surface-link is made a trivial surface-knot by surgery along a finite number of possibly non-trivial 1-handles [3]. The connected summand $F_2$ of $F_1\#F_2$ is made a trivial surface-knot by surgery along 1-handles within the 4-ball defining the connected summand $F_2$ 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 $F_2$, the connected summand $F_2$ is also a ribbon surface-link. This completes the Proof of Theorem 2 □

The following lemma is not used in the present version of Theorem 3, but this lemma remains here since it is an interesting property.

Lemma 8. 

Let K be an $S^2$-knot in $S^4$ obtained from a trivial surface-knot F of genus n in $S^4$ by surgery along disjoint 2-handles $D_i\times I(i=1,2,\cdots ,n)$. Then, there is a disjoint O2-handle pair system $(D_i\times I,D_i^{\prime }\times I)(i=1,2,\cdots ,n)$ on F in $S^4$ if and only if K is a trivial $S^2$-knot in $S^4$.

Proof of Lemma 8. 

If there is a disjoint O2-handle pair system $(D_i\times I,D_i^{\prime }\times I)(i=1,2,\cdots ,n)$ on F in $S^4$, then K is a trivial $S^2$-knot by Corollary 1.3. Note that the 2-handle system $D_i\times I(i=1,2,\cdots ,n)$ on F is a 1-handle system on K. If K is a trivial $S^2$-knot, then the 1-handle system $D_i\times I(i=1,2,\cdots ,n)$ on the trivial $S^2$-knot K is always a trivial 1-handle system on K [3]. Hence, there is a disjoint O2-handle pair system $(D_i\times I,D_i^{\prime }\times I)(i=1,2,\cdots ,n)$ on F in $S^4$. This completes the Proof of Lemma 8. □

The Proof of Theorem 3 is performed as follows.

Proof of Theorem 3. 

The assertions (1) → (2) and (2) → (3) are obvious by definitions. The assertion (3) → (1) is shown as follows: When F is a ribbon surface-knot, there is nothing to prove. By inductive assumption, the result for F of $r-1$ components is assumed. Let F be a surface-link of r ribbon surface-knot components $F_i,(i=1,2,\cdots ,r)$. Since a fusion of F makes a ribbon surface-knot by assumption, let K be a ribbon surface-knot obtained form F by a fusion along a disjoint 1-handle system h on F with only one 1-handle $h_r$ connecting to the component $F_r$. Let $K^{\prime }$ be a surface-knot obtained from the sublink $F\setminus F_r$ by the fusion along the 1-handle system $h\setminus h_r$. Since $F_r$ is a ribbon surface-knot, let $(O_r,{\alpha }_r)$ be a chorded sphere system for the ribbon surface-knot $F_r$. Since $\left[K^{\prime }\right]=0$ in $H_2(S^4\setminus O_r;Z)=0$, there is a compact connected oriented 3-manifold $V^{\prime }$ in $S^4$ with $\partial V^{\prime }=K^{\prime }$ and $V^{\prime }\cap O_r=\varnothing$. Since the arc system ${\alpha }_r$ transversely meets with the interior of $V^{\prime }$, a multi-punctured manifold ${\left(V^{\prime }\right)}^{\left(0\right)}$ of $V^{\prime }$ does not meet the chorded sphere system $O_r\cup {\alpha }_r$ so that $W_r\cap {\left(V^{\prime }\right)}^{\left(0\right)}=\varnothing$ for a SUPH system $W_r$ for $F_r$ with $\partial W_r=F_r\cup O_r$, constructed from $O_r\cup {\alpha }_r$. Let g be a disjoint 1-handle system on $F^{\prime }$ embedded in ${\left(V^{\prime }\right)}^{\left(0\right)}$ such that the closed complement $H^{\left(0\right)}=\mathrm{cl}({(V^{\prime })}^{\left(0\right)}\setminus g)$ is a multi-punctured handlebody of a genus, say n. Let $\partial H^{\left(0\right)}={\left(K^{\prime }\right)}^{+}\cup O^H$ where ${\left(K^{\prime }\right)}^{+}$ is the surface-knot obtained from $K^{\prime }$ by surgery along the 1-handle system g. The union $W=H^{\left(0\right)}\cup W_r$ is a SUPH system for the surface-link ${\left(K^{\prime }\right)}^{+}\cup F_r$ in $S^4$, so that ${\left(K^{\prime }\right)}^{+}\cup F_r$ is a ribbon surface-link in $S^4$ by Lemma 4. By replacing W with a multi-puncture manifold of W, the union $W^{+}=W\cup h_r$ is a SUPH system for the ribbon surface-knot $K^{+}$ in $S^4$ obtained from the surface-link ${\left(K^{\prime }\right)}^{+}\cup F_r$ by fusion along $h_r$. Note that the ribbon surface-knot $K^{+}$ is also obtained from the ribbon surface-knot K by surgery along g. Let $W_K$ be a SUPH system for the ribbon surface-knot K in $S^4$. By replacing $W_K$ with a multi-punctured manifold of $W_K$, the union $W_K^{+}=W_K\cup g$ is a SUPH system for the ribbon surface-knot $K^{+}$ in $S^4$. Equivalent ribbon surface-links are faithfully equivalent and they are moved into each other by the moves M0, M1, M2, [23]. This means that there is an orientation-preserving diffeomorphism f of $S^4$ sending a multi-punctured manifold of the SUPH system ${\left(W^{+}\right)}^{\left(0\right)}$ of $W^{+}$ to a multi-punctured manifold ${\left(W_K^{+}\right)}^{\left(0\right)}$ of the SUPH system $W_K^{+}$ and keeping $K^{+}$ set-wise fixed. Let $D\left(h_r\right)$ and $D\left(g\right)$ be a proper disk and a proper disk system in $h_r$ and g parallel to the attaching disks with one disk for every 1-handle, respectively. The proper disk $f\left(D\left(h_r\right)\right)$ and the proper disk system $D\left(g\right)$ may meet transversely with simple loops in the interior of the punctured handlebody ${\left(W_K^{+}\right)}^{\left(0\right)}$. Let $f{\left(D\left(h_r\right)\right)}^{\prime }$ be a proper disk disjoint from $D\left(g\right)$ with $\partial f{\left(D\left(h_r\right)\right)}^{\prime }=\partial f\left(D\left(h_r\right)\right)$ in ${\left(W_K^{+}\right)}^{\left(0\right)}$ obtained from $f\left(D\left(h_r\right)\right)$ by using a cutting technique along an innermost loop of $f\left(D\left(h_r\right)\right)\cap D\left(g\right)$ in $D\left(g\right)$ inductively. By splitting ${\left(W^{+}\right)}^{\left(0\right)}$ along the disk system $f{\left(D\left(h_r\right)\right)}^{\prime }\cup D\left(g\right)$, a SUPH system $W\left(K^{\prime }\right)\cup W\left(F_r\right)$ for a ribbon surface-link with $K^{\prime }\cup F_r$ as a fusion with a trivial $S^2$-link is obtained, meaning that the surface-link $K^{\prime }\cup F_r$ is a ribbon surface-link. In particular, $K^{\prime }$ is a ribbon surface-knot in $S^4$. By inductive assumption, $F\setminus F_r$ is a ribbon surface-link. Let $W(F\setminus F_r)$ be a SUPH system for the ribbon surface-link $F\setminus F_r$ in $S^4$. A multi-punctured manifold of the SUPH system $W(F\setminus F_r)\cup (h\setminus h_r)$ for the ribbon surface-knot $K^{\prime }$ is sent to a multi-punctured manifold of the SUPH system $W\left(K^{\prime }\right)$ for $K^{\prime }$ by an orientation-preserving diffeomorphism $f^{\prime }$ of $S^4$. After replacing $W(F\setminus F_r)\cup (h\setminus h_r)$ and $W\left(K^{\prime }\right)\cup W\left(F_r\right)$ with multi-punctured manifolds, respectively, the preimage ${\left(f^{\prime }\right)}^{-1}(W\left(K^{\prime }\right)\cup W\left(F_r\right))$ is the union of $W(F\setminus F_r)\cup (h\setminus h_r)$ and ${\left(f^{\prime }\right)}^{-1}\left(W\left(F_r\right)\right)$, showing that $W(F\setminus F_r)\cup {\left(f^{\prime }\right)}^{-1}\left(W\left(F_r\right)\right)$ is a SUPH system for the surface-link F. Thus, F is a ribbon surface-link. By induction on r, (3) → (1) is obtained and the Proof of Theorem 3 is completed. □

5. Conclusions

The ribbonness of a stable-ribbon surface-link shown in Theorem 1 is applied to determine the Ribbonness of some classes of surface-links. Theorem 3 appears unrelated to Theorem 1, but the result that equivalent ribbon surface-links are faithfully equivalent used in the Proof of Theorem 3 comes from a prior result of Theorem 1 [23]. The ribbonness of a surface-link relates not only to smooth unknotting conjecture for a surface-link leading to classical and 4D smooth Poincaré conjectures but also to J. H. C. Whitehead asphericity conjecture for aspherical 2-complex [25,26,27,28] as well as Kervaire conjecture on group weight [29,30]. In another direction, it may be an interesting problem to investigate a canonical relationship between a chorded loop diagram of a ribbon surface-knot and a knot diagram. In conclusion, ribbon surface-knot theory will be a tool for studies of low-dimensional topology.