Kirby Problem 4.77

Definitions and mathematical statement

For a closed smooth four-manifold $X$, remove an open ball to obtain $X^\circ$. Let $D=\operatorname{Diff}_\partial(X^\circ)$ be the topological group of diffeomorphisms equal to the identity on a fixed boundary collar, with the $C^\infty$ topology. Let $H=\operatorname{Homeo}_\partial(X^\circ)$ have the corresponding compact-open topology and boundary convention. Inclusion induces $i_*:\pi_k(D)\to\pi_k(H)$, based at the identity.

Attach copies of $S^2\times S^2$ in an added boundary collar and extend all maps by the identity. These choices define group stabilization maps and a homomorphism \[ s_*:\pi_k(D)\longrightarrow \operatorname*{colim}_{N\to\infty} \pi_k\operatorname{Diff}_\partial \left(X^\circ\#_N(S^2\times S^2)\right). \] For which $X$ and integers $k\ge0$ is $\ker i_*=\ker s_*$?

For $k=0$, the homotopy groups are groups of path components, so the first kernel consists of smooth mapping classes topologically isotopic to the identity relative to the boundary. The second consists of classes smoothly isotopic to the identity after some finite stabilization. For $k>0$, the same question concerns based sphere-parametrized families and their null-homotopies.

Vanishing in the direct limit requires vanishing at a finite stage. The statement does not claim that one stabilization suffices or that one stage works simultaneously for every class. No simple-connectivity assumption is imposed on the general classification of cases where these kernels agree.

Short English statement

When are topological triviality and smooth triviality after finite stabilization equivalent for families of four-manifold diffeomorphisms?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2953 (KP-4.77), 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 4.77. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.