Ribbonness on boundary surface-link, revised

A revised proof of the author’s earlier result is given. It is shown that a boundary surface-link in the 4-sphere is a ribbon surface-link if the surface-link obtained from it by surgery along a pairwise nontrivial fusion 1-handle system is a ribbon surface-link. As a corollary, the surface knot obtained from the anti-parallel surface-link of a non-ribbon surface-knot by surgery along a nontrivial or trivial fusion 1-handle is a non-ribbon or trivial surface-knot, respectively. This result answers Cochran’s conjecture on non-ribbon sphere-knots in the affirmative.

💡 Research Summary

The paper by Akio Kawauchi revisits the relationship between two fundamental classes of surface‑links in the 4‑sphere S⁴: boundary surface‑links and ribbon surface‑links. A boundary surface‑link is a collection of oriented closed surfaces F = F₁ ∪ … ∪ F_r each of which bounds a compact oriented 3‑manifold (a Seifert hypersurface) V_i embedded in S⁴. A ribbon surface‑link, on the other hand, is obtained from a trivial S²‑link O by attaching a system of 1‑handles h; the resulting surface‑link O(h) is called ribbon. The central question is whether a boundary surface‑link that becomes ribbon after a certain surgery must already be ribbon.

The main result, Theorem 1.1, answers this affirmatively under a precise hypothesis: let F be a boundary surface‑link with at least two components (r ≥ 2) and let h be a pairwise non‑trivial fusion 1‑handle system on F. “Fusion” means that performing surgery along h reduces the number of components from r to r − s (s ≥ 1). “Pairwise non‑trivial” requires that each 1‑handle’s core arc is non‑trivial in the complement of the other handles and cannot be isotoped away from the 3‑ball neighborhoods used in the construction. If the surface‑link F(h) obtained by surgery along h is a ribbon 2‑surface‑link, then the original F is already a ribbon surface‑link.

To prove this, Kawauchi introduces a new technical device called a SUPH system (short for “2‑handle–1‑handle move”). A SUPH system is a compact, possibly multi‑punctured 3‑manifold W embedded in S⁴ whose boundary consists of the surface‑link under study together with a trivial S²‑link O. The key property is that if a surface‑link admits a SUPH system, then it is ribbon; conversely, any ribbon surface‑link can be described by a SUPH system. Lemma 2.1 shows that changing a 2‑handle core disk by an elementary SUPH move preserves ribbonness. Lemma 2.2 and Corollary 2.3 give a method for replacing a spanning arc (or equivalently a 1‑handle) by a band‑sum with a meridian loop system; this allows one to “thicken” a 1‑handle and insert a trivial S²‑link around it.

The proof of Theorem 1.1 proceeds by decomposing the boundary surface‑link F into Seifert hypersurfaces V_i and disjoint 3‑balls B_i that together form a local model of a trivial S²‑link L = ∂B₁ ∪ ∂B₂. The fusion 1‑handle h on F is localized to a fusion 1‑handle h_L on L, which is non‑trivial by hypothesis. The surface‑link L(h_L) is a ribbon 2‑sphere link, and a SUPH system W(L; h) for it is constructed. By attaching the remaining pieces of the Seifert hypersurfaces (the V_i’s) to this SUPH system, a new SUPH system W′ for F(h) is obtained. Since F(h) is assumed ribbon, it already possesses a SUPH system; by a careful isotopy and a diffeomorphism of S⁴ that matches the two SUPH systems, one shows that the original surface‑link F also admits a SUPH system, hence is ribbon.

Theorem 1.2 applies Theorem 1.1 to the anti‑parallel surface‑link P(F) of a surface‑knot F. P(F) consists of two parallel copies of F with opposite orientations, glued along a product neighbourhood. P(F) is always a boundary surface‑link because each copy bounds a copy of a Seifert hypersurface. Consequently, if F is a non‑ribbon 2‑knot and h is a fusion 1‑handle on P(F), then:

• if h is non‑trivial, the resulting surface‑knot P(F)(h) is non‑ribbon;
• if h is trivial, P(F)(h) is a trivial (unknotted) surface‑knot. Thus the paper resolves Cochran’s conjecture (1990s) which asserted that any sufficiently complicated fusion 1‑handle attached to a non‑ribbon sphere‑knot yields a non‑ribbon sphere‑knot. The corollary (Corollary 1.3) further refines this by showing that for any non‑trivial fusion 1‑handle h there exists a trivial S²‑link O_h⁰ surrounding it such that P(F)(h) can be expressed as a connected sum of P(F)(h₀) (with a trivial handle h₀) and O_h⁰, linked by another fusion 1‑handle system. This “handle exchange” picture gives a flexible way to move between ribbon and non‑ribbon configurations.

The paper also corrects an earlier erroneous theorem (Theorem 1.4 in