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

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

“…A stably trivial surface-link is a surface-link F such that a stabilization 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]: Corollary 1.4. A handle-irreducible summand of every stably trivial surface-link is a trivial 2-link.…”

“…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.…”

Smooth homotopy 4-sphere

“…and (∂D) × I and (∂D ′ ) × I meet orthogonally on F , that is, ∂D and ∂D ′ meet transversely at one point p and the intersection (∂D) × I ∩ (∂D ′ ) × I is diffeomorphic to the square Q = p × I × I (see [7]).…”

“…Proof of (2. In a process of this deformation, every disk in the neighborhoods meets F in two points apart from the part of 2-handle attachments, but in the end of this deformation the interior of the 2-handle core image u ′ (D ′ 1 ) no longer meets F (see the proof of [7,Lemma 3.2]). This deformation does not touch the 2-handles D 1 × I and D ′ 1 × I.…”

Classical Poincar{é} conjecture via 4D topology

“…The positive proof of this conjecture for any surface-knot is claimed in [7], where a revised proof of the uniquness of an O2-handle pair is in preparation. (See also [8] for a generalization to a surfece-link).…”