Kirby Problem 4.72

Definitions and mathematical statement

Let $X$ be a closed connected orientable topological $4$-manifold whose fundamental group is finite. No smoothing is assumed. Give $H=\operatorname{Homeo}(X)$, the full homeomorphism group, the compact-open topology. For $k\ge1$, let $\pi_k(H)$ be its homotopy group based at the identity. For $k=0$, use the group of path components, so $\pi_0(H)$ is the topological mapping class group under composition.

Part (a) asks whether $\pi_k(H)$ is finitely generated for every integer $k\ge0$, for every $X$ satisfying these hypotheses. Finite generation in degree zero has the ordinary possibly nonabelian group meaning; in positive degrees it is finite generation of the corresponding homotopy group.

Let $BH$ be a classifying space for the topological group $H$, constructed as the quotient of a contractible free $H$-space in the usual universal-bundle model. Part (b) asks whether $H_k(BH;\mathbb Z)$ is a finitely generated abelian group for every $k\ge0$ and every such $X$.

The group $H$ retains its topology when $BH$ is formed; discrete group cohomology would be a different invariant. No uniform bound on the number of generators as $k$ or $X$ varies is requested. The hypothesis permits any finite fundamental group, not only the trivial group. Both assertions are universal finiteness statements, rather than requests for just one favorable example.

Short English statement

For a closed orientable topological four-manifold with finite fundamental group, are all homotopy groups of its homeomorphism group and all integral homology groups of its classifying space finitely generated?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2948 (KP-4.72), 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 4.72. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.