Kirby Problem 4.85

Definitions and mathematical statement

Let $X$ be a closed orientable smooth $4$-manifold. Let $D=\operatorname{Diff}_0(X)$ be the identity component of its diffeomorphism group in the $C^\infty$ topology; its elements are diffeomorphisms smoothly isotopic to the identity. Use the commutator convention $[a,b]=aba^{-1}b^{-1}$. For $f$ in the commutator subgroup $[D,D]$, define its commutator length by \[ \operatorname{cl}_D(f)=\min\{r\ge0: f=[a_1,b_1]\cdots[a_r,b_r],\ a_i,b_i\in D\}. \] The empty product represents the identity.

A group is perfect if it equals its commutator subgroup. It is uniformly perfect if it is perfect and some finite integer $C$ bounds the commutator length of every element. In the present setting $D$ is perfect, so the question is whether its commutator length can be unbounded.

Does there exist such a manifold $X$ for which $D$ is not uniformly perfect? Equivalently, for each integer $N\ge0$, there must be $f_N\in D$ which is not a product of $N$ commutators of elements of $D$.

The commutator factors are arbitrary identity-isotopic diffeomorphisms; no common support, isotopy length, derivative bound, or fragmentation restriction is imposed. The desired unboundedness occurs in one fixed manifold's group. Allowing the manifold itself to vary with $N$ would not prove this existence assertion.

Short English statement

Can commutator length be unbounded in the identity component of the diffeomorphism group of a closed orientable smooth four-manifold?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2961 (KP-4.85), 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 4.85. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.