Kirby Problem 4.34

Definitions and mathematical statement

Let $\Sigma$ be a closed surface smoothly embedded in $S^4$. A torus surgery in its complement chooses an embedded torus $T\subset S^4\setminus\Sigma$ with trivial normal bundle, removes $T^2\times\operatorname{int}(D^2)$, and glues back $T^2\times D^2$ by a boundary diffeomorphism. The surgery torus is disjoint from the surface, so the operation carries $\Sigma$ unchanged into the new ambient manifold.

Can every such surface be unknotted by finitely many torus surgeries for which the ambient manifold after every individual step is again diffeomorphic to $S^4$? Identifications with $S^4$ are allowed between steps. In the locally flat topological version, replace smooth embeddings and diffeomorphisms by locally flat embeddings and homeomorphisms, with product tubular neighborhoods for the surgery tori.

For an orientable connected surface, unknotted means isotopic to the standard genus-$g$ surface contained in an equatorial $S^3$, bounding a standard handlebody there. For nonorientable components use the standard connected sums of projective planes with normal Euler numbers $+2$ or $-2$, matching the surface's normal data; standard separated models apply to multiple components.

The genus and abstract surface type are unchanged by these complementary surgeries. Allowing arbitrary changes of the ambient four-manifold gives a different problem. Nor is it enough for only the final ambient manifold to be $S^4$: preservation at every intermediate stage is the stated requirement.

Short English statement

Can every knotted surface be unknotted by complementary torus surgeries that keep the ambient manifold a four-sphere at every stage?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2910 (KP-4.34), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 4.34. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.