Kirby Problem 4.86

Definitions and mathematical statement

Let $X$ be a closed connected topological four-manifold admitting at least one smooth structure. A finite group action is a homomorphism $\rho:G\to\operatorname{Homeo}(X)$. It is locally linear if, near every point $x$, there is a chart to a neighborhood of zero in $\mathbb R^4$ equivariant for the stabilizer $G_x$, acting there by a linear representation.

Given a smoothing $\mathcal S$ of $X$, the action is smoothable if some self-homeomorphism $h:X\to X$ conjugates it into $\operatorname{Diff}(X,\mathcal S)$: \[ h\rho(g)h^{-1}\in\operatorname{Diff}(X,\mathcal S) \quad\text{for every }g\in G. \] Does every such $X$ admit a locally linear finite group action which is not smoothable for any smoothing $\mathcal S$? In full, choose $G$ and $\rho$ first, then require failure of the displayed smoothness condition for every smoothing and every possible conjugating homeomorphism.

The group and action may depend on the underlying topological manifold. An ineffective kernel can be removed, so it is enough to seek effective actions; the trivial action cannot be an example. No orientation-preserving condition is imposed unless a restricted version is explicitly considered.

Failure to be smooth in one chosen atlas is much weaker than the target: changing coordinates by a topological conjugacy or changing the smooth structure altogether is allowed. The underlying manifold is required to be smoothable, so its lack of any smoothing cannot serve as the obstruction.

Short English statement

Does every closed smoothable four-manifold carry a locally linear finite action that cannot be smoothed in any smooth structure?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2962 (KP-4.86), 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 4.86. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.