Kirby Problem 4.80

Definitions and mathematical statement

Let $X$ be a closed simply connected smooth $4$-manifold. The groups $\pi_0\operatorname{Diff}(X)$ and $\pi_0\operatorname{Homeo}(X)$ consist respectively of smooth and topological isotopy classes of self-maps, with composition as the group operation. Use the $C^\infty$ and compact-open topologies to define their path components. Set \[ K_X=\ker\left(\pi_0\operatorname{Diff}(X) \longrightarrow\pi_0\operatorname{Homeo}(X)\right). \] Its elements are smooth mapping classes represented by diffeomorphisms topologically isotopic to the identity.

Do there exist an integer $n>2$, such a manifold $X$, and an element of order exactly $n$ in $K_X$? In representative terms, there should be a diffeomorphism $f$ topologically isotopic to the identity for which $f^n$ is smoothly isotopic to the identity, while $f^j$ is not for $1\le j<n$. More generally, does $K_X$ contain a finite subgroup of order $n>2$ for some $X$? That subgroup need not be cyclic.

Finite order here belongs to the mapping class, not necessarily to the diffeomorphism as an actual map. Thus the question does not require $f^n=\mathrm{id}$ pointwise or realization of the whole finite subgroup by an action. The existence of infinite-order elements alone would not answer either finite-order request. The order requested is exact, rather than merely a divisor of a chosen integer; the two formulations distinguish cyclic torsion from more general finite subgroups.

Short English statement

Can the kernel from smooth to topological mapping classes of a simply connected closed four-manifold contain torsion of order greater than two, or a finite subgroup of such order?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2956 (KP-4.80), 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 4.80. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.