Kirby Problem 4.70

Definitions and mathematical statement

For a closed smooth $4$-manifold $X$, let $\operatorname{Diff}(X)$ be its full diffeomorphism group with the $C^\infty$ topology, and let $\operatorname{Homeo}(X)$ be its full homeomorphism group with the compact-open topology. Forgetting differentiability defines a continuous homomorphism $i:\operatorname{Diff}(X)\hookrightarrow \operatorname{Homeo}(X)$.

For $k\ge1$, use the identity map as basepoint when taking $\pi_k$; a representative is a continuous based map from $S^k$ into the relevant transformation group, modulo based homotopy. For $k=0$, $\pi_0$ is the group of path components, with multiplication induced by composition. Thus these degree-zero groups are smooth and topological mapping class groups respectively.

Do there exist a closed smooth $4$-manifold $X$ and an integer $k\ge0$ for which \[ i_*:\pi_k(\operatorname{Diff}(X)) \longrightarrow\pi_k(\operatorname{Homeo}(X)) \] is an isomorphism? Both the manifold and the single degree may be chosen. It is not required that the map be an isomorphism in all degrees or that the inclusion be a homotopy equivalence.

Injectivity would mean that a family which becomes nullhomotopic topologically was already nullhomotopic smoothly; surjectivity would mean every topological family has a smooth representative up to homotopy. Both conditions are required in the chosen degree. The source does not restrict $X$ to be orientable or simply connected.

Short English statement

Is the inclusion of diffeomorphisms into homeomorphisms an isomorphism on some homotopy group for some closed smooth four-manifold?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2946 (KP-4.70), 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 4.70. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.