Kirby Problem 4.82

Definitions and mathematical statement

Let $X$ be a closed simply connected smooth $4$-manifold and $\phi:X\to X$ a self-diffeomorphism. For embedded closed smooth surfaces $S_0,S_1\subset X$, smooth isotopy means an ambient isotopy: a smooth family $F_t\in\operatorname{Diff}(X)$ with $F_0=\mathrm{id}$ and $F_1(S_0)=S_1$. The surfaces are treated as embedded images, without a specified parameterization.

Suppose that for every smoothly embedded closed surface $\Sigma\subset X$, the two images $\Sigma$ and $\phi(\Sigma)$ are smoothly isotopic. The witnessing isotopy may depend on $\Sigma$; no one family working simultaneously for every surface is assumed.

Part (a) asks whether $\phi$ must be smoothly isotopic to the identity on $X$. Part (b) asks the weaker conclusion that $\phi$ is smoothly isotopic to a diffeomorphism $\psi$ supported inside some smoothly embedded $4$-ball in $X$. Support is the closure of $\{x:\psi(x)\ne x\}$; ball-supported here means this closed set lies in the interior of such a ball, so the map is the identity outside it and near its boundary.

The hypothesis is stronger than preservation of homology classes of surfaces, but weaker than preserving every parameterized surface embedding up to isotopy. These distinctions are retained explicitly. The supporting ball in (b), if it exists, may depend on $\phi$, and (b) does not assert that the remaining diffeomorphism inside that ball is itself isotopically trivial.

Short English statement

If a diffeomorphism preserves the smooth isotopy class of every embedded surface, must it be isotopic to the identity, or at least to a map supported in a ball?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2958 (KP-4.82), 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 4.82. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.