Kirby Problem 4.39

Definitions and mathematical statement

Let $K$ be a closed smooth surface embedded by $i:K\hookrightarrow M$ in a smooth $4$-manifold. Use the full mapping class group $\operatorname{Mod}(K)=\pi_0\operatorname{Diff}(K)$, whose elements are smooth-isotopy classes of self-diffeomorphisms, with composition as multiplication. Orientability of $K$ is not required. If oriented versions are considered, all maps can instead be restricted consistently to preserve the specified orientations.

Define the extendable subgroup \[ E(M,i)=\{[h]\in\operatorname{Mod}(K): \exists F\in\operatorname{Diff}(M),\ F\circ i=i\circ h\}. \] Here one representative $h$ of the mapping class must admit the extension. Isotopy extension ensures that the property does not depend on that choice. The ambient diffeomorphism preserves the embedded surface setwise but need not be isotopic to the identity.

Describe $E(M,i)$ in terms of the embedding and the topology of the ambient manifold. Conversely, characterize the subgroups $H\le\operatorname{Mod}(K)$ for which there exist some smooth $4$-manifold $M$ and embedding $i$ with $E(M,i)=H$.

The realization question asks for equality with the entire extendable subgroup. Realizing each element of $H$ separately without excluding other extendable classes is insufficient. Likewise extensions need not form a chosen homomorphic section into $\operatorname{Diff}(M)$; the requirement is individual extendability. No compactness or simple-connectivity condition on $M$ is added to the source statement.

Short English statement

Which mapping classes of an embedded closed surface extend over its ambient four-manifold, and which subgroups can arise this way?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2915 (KP-4.39), 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 4.39. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.