Kirby Problem 4.81

Definitions and mathematical statement

For a closed simply connected smooth $4$-manifold $X$, give $\operatorname{Diff}(X)$ the $C^\infty$ topology and $\operatorname{Homeo}(X)$ the compact-open topology. Composition makes each set a topological group. Their component groups are the corresponding mapping class groups, and forgetting smoothness induces \[ q_X:\pi_0\operatorname{Diff}(X)\longrightarrow \pi_0\operatorname{Homeo}(X). \] Write $K_X=\ker q_X$. A class belongs to $K_X$ precisely when its representative diffeomorphism is topologically isotopic to the identity, although it may fail to be smoothly isotopic to it.

Does there exist such an $X$ for which $K_X$ is finitely generated? Equivalently, can one find a finite list of smooth mapping classes $[f_1],\ldots,[f_r]\in K_X$ so that every class in $K_X$ is a finite product of these classes and their inverses? The empty list is allowed if the kernel is trivial.

The question concerns the whole kernel as an abstract group, not merely its abelianization, a subgroup of detected classes, or the number of distinct numerical invariants. The manifold may be chosen, but must be closed and simply connected. No bound on $r$ is prescribed. The conclusion does not require actual finite generation of the subgroup of all topologically trivial diffeomorphisms before quotienting by smooth isotopy.

Short English statement

Can the full group of topologically trivial smooth mapping classes of a closed simply connected four-manifold be finitely generated?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2957 (KP-4.81), 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 4.81. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.