Ribbonness of a stable-ribbon surface-link, I. A stably trivial surface-link

Abstract

There is a question asking whether a handle-irreducible summand of every stable-ribbon surface-link is a unique ribbon surface-link. This question for the case of a trivial surface-link is affirmatively answered. That is, a handle-irreducible summand of every stably trivial surface-link is only a trivial 2-link. By combining this result with an old result of F. Hosowaka and the author that every surface-knot with infinite cyclic fundamental group is a stably trivial surface-knot, it is concluded that every surface-knot with infinite cyclic fundamental group is a trivial (i.e., an unknotted) surface-knot.

Introduction

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 a (possibly disconnected) closed surface F is fixed, it is also called an F-link. If F is the disjoint union of some copies of the 2-sphere $S^2$, then it is also called a 2-link. When 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) $f:{\mathbf{R}}^4\to {\mathbf{R}}^4$. The notation $F\cong F^{\prime }$ is used for equivalent surface-links F, $F^{\prime }$. 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, a solid torus or a boundary-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 and unlinked surface-link. For any given closed oriented (possibly disconnected) surface F, a trivial F-link exists uniquely up to equivalences (see [6]). A ribbon surface-link is a surface-link F which is obtained from a trivial 2-link O by the surgery along an embedded 1-handle system (see [10], [11], [12], [13], [16, II]). A stabilization of a surface-link F is a connected sum $F^{\mathrm{\#}sT}=F{\mathrm{\#}}_{k=1}^s{T}_k$ of F and a system 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 $T_k(k=1,2,...,s)$ is called the stabilizer on the stabilization $F^{\mathrm{\#}sT}$ of F.

A stable-ribbon surface-link is a surface-link F such that a stabilization $F^{\mathrm{\#}sT}$ 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 question is a central question.

Question 1.0

A handle-irreducible summand of every stable-ribbon surface-link is a ribbon surface-link which is unique up to equivalences?

A stably trivial surface-link is a surface-link F such that a stabilization of F is a trivial surface-link.

In this paper, the following theorem is shown answering affirmatively this question for the case of a stably trivial surface-link. This question in the general case will be answered affirmatively in [15].

Theorem 1.1

Any handle-irreducible summand of every stably trivial surface-link is a trivial 2-link.

The following corollary is directly obtained from Theorem 1.1:

Corollary 1.2

Every stably trivial surface-link is a trivial surface-link.

If a surface-knot F has an infinite cyclic fundamental group, then F is a TOP-trivial surface-knot, which was shown by Freedman for a 2-knot and by [3], [9] for a higher genus surface-knot. In the case of a piecewise linear surface-knot (equivalent to a smooth surface-knot), it is known by [6] that a stabilization of the surface-knot F is a trivial surface-knot, namely the surface-knot F is a stably trivial surface-knot. Hence the following corollary is directly obtained from Corollary 1.2 answering the problem [17, Problem 1.55(A)] on unknotting of a 2-knot positively (see [14] for the surface-link version):

Corollary 1.3

A surface-knot F is a trivial surface-knot if the fundamental group ${\pi }_1({\mathbf{R}}^4\setminus F)$ is an infinite cyclic group.

The exterior of a surface-knot F is the 4-manifold $E=\text{cl}({\mathbf{R}}^4\setminus N(F))$ for a tubular neighborhood $N(F)$ of F in ${\mathbf{R}}^4$. Then the boundary ∂E is a trivial circle bundle over F. A surface-knot F is of Dehn's type if there is a section $F^{\prime }$ of F in the bundle ∂E such that the inclusion $F^{\prime }\to E$ is homotopic to a constant map. By [3, Corollary 4.2], the fundamental group ${\pi }_1({\mathbf{R}}^4\setminus F)$ of a surface-knot F of Dehn's type is an infinite cyclic group. Thus, we have the following corollary (answering the problem [17, Problem 1.51] on unknotting of a 2-knot of Dehn's type positively):

Corollary 1.4

A surface-knot of Dehn's type is a trivial surface-knot.

Unknotting Conjecture asks whether an n-knot $K^n$ (i.e., a smooth embedding image of the n-sphere $S^n$ in the (n+2)-sphere $S^{n+2}$) is unknotted (i.e., bounds a smooth $(n+1)$-ball in $S^{n+2}$) if and only if the complement $S^{n+2}\backslash K^n$ is homotopy equivalent to $S^1$ (see [8] for example). This conjecture was previously known to be true for $n>3$ by [18], for $n=3$ by [20] and for $n=1$ by [5], [19]. The conjecture for $n=2$ was known only in the TOP category by [1] (see also [2]). Corollary 1.3 answers this finally remained smooth unknotting conjecture affirmatively and hence Unknotting Conjecture can be changed into the following:

Unknotting Theorem

A smooth $S^n$-knot $K^n$ in $S^{n+2}$ is unknotted if and only if the complement $S^{n+2}\backslash K^n$ is homotopy equivalent to $S^1$ for every $n\ge 1$.

A main idea in our argument is to use the surgery of a surface-link on an orthogonal 2-handle pair, which is much different from the surgery of a surface-link on a single 2-handle. It is known that every surface-link F in ${\mathbf{R}}^4$ is obtained from a higher genus trivial surface-knot $F^{\prime }$ by the surgery of $F^{\prime }$ on a system of mutually disjoint 2-handles, because a handlebody in ${\mathbf{R}}^4$ is obtained from a connected Seifert hypersurface of F by removing mutually disjoint 1-handles (see [6]). Thus, for example, every 2-twist spun 2-bridge knot in [21] is obtained from a trivial torus-knot T in ${\mathbf{R}}^4$ by the surgery of T on a single 2-handle, because it bounds a once-punctured lens space as a Seifert hypersurface.

In Section 2, it is shown that every stably trivial surface-link is a trivial surface-link if and only if the uniqueness of an orthogonal 2-handle pair on every trivial surface-link holds. In Section 3, the uniqueness of every orthogonal 2-handle pair on every surface-link is shown, by which Theorem 1.1 is obtained.