Kirby Problem 4.131

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.

A Lipschitz structure is an atlas with locally bi-Lipschitz transition maps; compatible smooth structures are those whose smooth charts give this Lipschitz structure. Uniqueness of a smoothing is up to diffeomorphism. For a closed oriented topological four-manifold $X$, put $b_2(X)=\dim_{\mathbb R}H^2(X;\mathbb R)$ and let $\sigma(X)$ be its intersection-form signature. Indefinite means that form has both positive and negative directions. Spin means that the second Stiefel--Whitney class of its oriented topological tangent microbundle vanishes, $w_2(X)=0$. A Lipschitz embedded manifold with boundary in $\mathbb R^4$ is locally bi-Lipschitz equivalent to the standard whole-space or half-space model, as appropriate.

Statement recorded in the supplied source.

Does every Lipschitz four-manifold admit a compatible smooth structure, and is that structure unique up to diffeomorphism? In addition:

1. (a) Is there a closed spin indefinite topological four-manifold with a Lipschitz structure that violates $b_2(X)\geq\tfrac54|\sigma(X)|+2$, or that violates $b_2(X)\geq\tfrac54|\sigma(X)|+4$ when $X$ is not one of $S^4,S^2\times S^2,$ or K3?
2. (b) If $X\subset\mathbb R^4$ is a Lipschitz embedded four-manifold with $\partial X\cong S^3$, does $X$ admit a unique smooth structure?

[S2]

Short English statement

Are Lipschitz four-manifolds smoothable uniquely, and can a Lipschitz spin manifold evade the smooth spin inequalities; what happens for Lipschitz domains with three-sphere boundary?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3007 (KP-4.131). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman (eds.), K3: A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295, American Mathematical Society (2026), Problem 4.131, author preliminary version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.