Kirby Problem 4.75

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $X$ be a connected smooth four-manifold with nonempty boundary and finite fundamental group. Let $G_0=\operatorname{Diff}_{\partial}(X)$ be the group of diffeomorphisms equal to the identity near the boundary, with the smooth topology. For $N\in\{0,1,2,\ldots\}$, stabilize by forming the interior connected sum $X_N=X\#N(S^2\times S^2)$. Fix this construction in an attached boundary collar: at each step use \[ X_{N+1}=X_N\cup_{\partial X_N}\left((\partial X_N\times[0,1])\#(S^2\times S^2)\right). \] Extending by the identity on the added piece gives $s_N:G_N\to G_{N+1}$, where $G_N=\operatorname{Diff}_{\partial}(X_N)$. Homotopy groups are based at the identity; $\pi_0(G_N)$ means the group of path components. The direct limit is taken using $(s_N)_*$, and $s_*$ denotes the map from the unstabilized group to that direct limit.

Statement recorded in the supplied source.

For every such $X$ and every integer $k\ge0$, is \[ \operatorname{im}\left(s_*:\pi_k(G_0)\longrightarrow\underset{N\to\infty}{\operatorname{colim}}\,\pi_k(G_N)\right) \] a finitely generated group? Ask the same question with each $G_N$ replaced by $\operatorname{Homeo}_{\partial}(X_N)$, the compact-open topological group of homeomorphisms equal to the identity near the boundary, and stabilization again defined by extension by the identity. Only the image from the original manifold is required to be finitely generated, not the entire direct limit. The source also asks whether the smooth assertion has a counterexample when $\pi_1(X)$ is infinite: is there a connected smooth four-manifold with nonempty boundary and infinite fundamental group, and some $k\ge0$, for which the same stabilization-image group is not finitely generated? [S2]

Short English statement

For finite fundamental group, ask whether smooth and topological stabilization images are finitely generated in every homotopy degree. Also ask whether an infinite-fundamental-group smooth counterexample exists.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2951 (KP-4.75). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, K3—A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295 (2026), author version, Problem 4.75 and Remarks, PDF page 252. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.