Kirby Problem 4.5

Definitions and mathematical statement

Here an open $4$-manifold $M$ is a topological manifold without boundary which is noncompact; assume it is connected, Hausdorff and second countable. A smoothing is an atlas of local coordinates in $\mathbb R^4$ with smooth transition maps inducing its given topology. Two smoothings are considered the same if there is some diffeomorphism between the resulting smooth manifolds, regardless of whether that map is isotopic to the identity on the underlying topological manifold.

Does every such $M$ support uncountably many pairwise nondiffeomorphic smoothings? Explicitly, the desired conclusion is an uncountable index set $A$ and smooth manifolds $M_\alpha$, each homeomorphic to $M$, with $M_\alpha\not\cong_{\mathrm{diff}}M_\beta$ whenever $\alpha\ne\beta$.

The topology and end structure of $M$ remain fixed throughout the family. No finite-generation condition on homology or fundamental group, no finiteness of the number of ends, and no orientability hypothesis is part of the question. The smooth structures may behave differently near infinity, and a distinguishing diffeomorphism is allowed to move points arbitrarily far out the ends.

The equivalence relation is essential: uncountably many marked atlases, or even many smoothings inequivalent under homeomorphisms isotopic to the identity, need not give uncountably many diffeomorphism types. The question asks for the latter, for every connected open manifold, not only for Euclidean space or punctured closed manifolds.

Short English statement

Does every connected noncompact boundaryless four-manifold have uncountably many smoothings up to diffeomorphism?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2881 (KP-4.5), 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 4.5. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.