Kirby Problem 4.71

Definitions and mathematical statement

Let $X$ be a closed orientable smooth $4$-manifold. Its full diffeomorphism group $D=\operatorname{Diff}(X)$ is given the $C^\infty$ topology. For $k\ge1$, take $\pi_k(D)$ based at the identity; for $k=0$, $\pi_0(D)$ is the group of smooth isotopy classes under composition. A group is finitely generated if some finite set and its inverses generate every element.

Part (a) asks whether there exist such an $X$ and an integer $k\ge0$ for which $\pi_k(D)$ is finitely generated. The group may be trivial; no nonzero-rank requirement is imposed.

Let $BD$ denote a classifying space for the topological group $D$, for example the base of its universal principal bundle $ED\to BD$ with contractible total space. Part (b) asks whether there exist such an $X$ and an integer $k>0$ for which the integral singular homology group $H_k(BD;\mathbb Z)$ is finitely generated as an abelian group.

The manifold and degree may be chosen independently in the two parts. Neither asks for finite generation in every degree. The second part uses positive degree to exclude the automatic degree-zero homology answer. The full diffeomorphism group includes any orientation-reversing diffeomorphisms; orientability of $X$ does not silently replace it by $\operatorname{Diff}^+(X)$. Finally, the topological group is not being treated as a discrete group when forming its classifying space.

Short English statement

Can a closed orientable smooth four-manifold have a finitely generated homotopy group of its diffeomorphism group, or a positive-degree integral homology group of its classifying space?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2947 (KP-4.71), 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 4.71. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.