Kirby Problem 4.1

Definitions and mathematical statement

Let $S^4=\{x\in\mathbb R^5:\|x\|=1\}$ carry its usual smooth structure. A smooth structure on a topological $4$-manifold is a maximal atlas of charts to open subsets of $\mathbb R^4$ whose transition maps are smooth. Two smooth structures are equivalent here if the resulting smooth manifolds are diffeomorphic: there must be a bijection that is smooth with smooth inverse. The diffeomorphism need not be the identity on the underlying point set.

Is every smooth closed manifold $X$ homeomorphic to $S^4$ diffeomorphic to the standard $S^4$? Equivalently, does the topological $4$-sphere have exactly one diffeomorphism class of smooth structures? A smooth manifold homeomorphic but not diffeomorphic to the standard sphere would be an exotic smooth $4$-sphere and would give a negative answer.

A closely related formulation uses a smooth homotopy $4$-sphere: a closed smooth manifold homotopy equivalent to $S^4$. The topological four-dimensional Poincaré theorem identifies its underlying topological manifold with $S^4$, so the smooth question asks whether every such homotopy sphere is standard as a smooth manifold. The required comparison is a diffeomorphism, not merely a homeomorphism, a homotopy equivalence, or an agreement of homology groups. No particular atlas, handle decomposition, embedding, or candidate construction is part of the input; the assertion quantifies over all smoothings.

Short English statement

Is every smooth manifold homeomorphic to the four-sphere diffeomorphic to the standard four-sphere?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2877 (KP-4.1), 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 4.1. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.