Kirby Problem 4.2

Definitions and mathematical statement

A closed smooth $4$-manifold is compact and has empty boundary. A smooth manifold $X'$ is an exotic copy of a given smooth manifold $X$ if there exists a homeomorphism $X'\to X$ but no diffeomorphism $X'\to X$. Thus the topological manifold is unchanged while the smooth structures are inequivalent up to diffeomorphism. A homeomorphism and a diffeomorphism are both bijections with continuous inverse; the latter and its inverse must additionally be smooth in the specified atlases.

Does every closed smooth $4$-manifold $X$ have an exotic copy? More strongly, for every such $X$, does there exist a sequence $(X_j)_{j\ge1}$ of smooth manifolds, each homeomorphic to $X$, such that the $X_j$ are pairwise nondiffeomorphic and none is diffeomorphic to $X$?

The first assertion requests at least one different diffeomorphism type for each underlying topological manifold; the second requests infinitely many. An infinite collection of atlases related by self-diffeomorphisms would not suffice. The comparison does not mark points or a preferred identification with $X$, and no relative-boundary condition occurs because the manifolds are closed. No simple-connectivity, orientability, symplectic structure, or restriction on the intersection form is imposed in the stated question. One may first study connected manifolds; treating arbitrary finite disjoint unions does not replace the universal connected cases. Any proposed family must establish both the common homeomorphism type and its pairwise smooth distinctions.

Short English statement

Does every closed smooth four-manifold admit another smooth structure, and in fact infinitely many, up to diffeomorphism?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2878 (KP-4.2), 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 4.2. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.