Kirby Problem 4.76

Definitions and mathematical statement

Let $X$ be a closed oriented simply connected smooth $4$-manifold and fix an integer $k>0$. For $n\ge0$, write $X_n=X\#_{n}(S^2\times S^2)$, with $X_0=X$. Connected sum removes an interior smooth $4$-ball from each oriented summand and identifies their boundary $3$-spheres by an orientation-reversing diffeomorphism.

Let $\operatorname{Diff}^+(X_n)$ have its $C^\infty$ topology and $\operatorname{Homeo}^+(X_n)$ its compact-open topology. Both consist of orientation-preserving self-maps of the indicated kind. Inclusion gives a homomorphism on homotopy groups, based at the identity: \[ i_{n,*}:\pi_k(\operatorname{Diff}^+(X_n)) \longrightarrow\pi_k(\operatorname{Homeo}^+(X_n)). \] Does there exist an integer $N=N(X,k)\ge0$ such that $i_{n,*}$ is surjective for every $n\ge N$?

A positive answer means that each based continuous $S^k$-family of orientation-preserving homeomorphisms of $X_n$ is based-homotopic, through homeomorphism families, to one consisting of diffeomorphisms and continuous in the smooth topology. The number of stabilizations may depend on $X$ and $k$, but must work for every class at each of the indicated later stages.

This is eventual surjectivity at each stabilized manifold, a stronger requirement than smoothing one chosen family after a family-dependent stabilization. No injectivity assertion is made, and no map between diffeomorphism groups of different closed connected sums is needed to state the question.

Short English statement

After sufficiently many S²×S² stabilizations, is every positive-degree family of homeomorphisms represented up to homotopy by a family of diffeomorphisms?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2952 (KP-4.76), 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 4.76. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.