Kirby Problem 4.33

Definitions and mathematical statement

Let $S_1,S_2$ be smoothly embedded closed surfaces in a closed oriented smooth four-manifold $X$. Assume they are topologically ambiently isotopic: a path of self-homeomorphisms beginning at the identity carries $S_1$ to $S_2$. Smooth ambient isotopy requires such a path through diffeomorphisms.

An external stabilization replaces each pair $(X,S_i)$ by $(X,S_i)\#(S^2\times S^2,\varnothing)$, taking the connected sum away from the surface. A standard internal stabilization replaces it by $(X,S_i)\#(S^4,T)$, where $T$ is the standard unknotted torus. It adds a trivial local tube to the surface without changing $X$. A general internal stabilization adds an embedded tube along an arc joining two disks on the surface; the arc need not lie in a small ball. Make compatible choices so that the resulting surfaces have the same abstract type.

Under what hypotheses do $S_1$ and $S_2$ become smoothly isotopic after one external stabilization, after one standard internal stabilization, or after one general internal stabilization? More generally, for each of these operations, is there a finite upper bound depending only on $X$ for the number needed for arbitrary such pairs?

The ambient manifold stays fixed in the internal versions and changes in the external version. A uniform bound for fixed $X$ must be independent of the surfaces and their complexity. Where even eventual isotopy is not known, existence of a finite number of stabilizations is itself part of the problem.

Short English statement

When does stabilization remove exotic smooth knotting of topologically isotopic surfaces, and can the number depend only on the ambient manifold?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2909 (KP-4.33), 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 4.33. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.