Kirby Problem 4.4

Definitions and mathematical statement

For $g\ge0$, let $\Sigma_g$ be the closed connected orientable surface of genus $g$, with its standard smooth structure; $\Sigma_0=S^2$ and $\Sigma_1=T^2$. Let $Y$ be a closed connected smooth $3$-manifold. The products $S^1\times Y$ and $\Sigma_g\times\Sigma_h$ carry their product smooth structures. An exotic smoothing of one of these products means a smooth $4$-manifold homeomorphic to that product but not diffeomorphic to it with the product smoothing.

Do exotic smoothings exist on any products in these two families? In each family, does every product admit an exotic smoothing? The parameters $g,h$ range over all nonnegative integers, and $Y$ ranges over closed $3$-manifolds. The source does not impose a special geometry or a restriction on the fundamental group of $Y$.

The homeomorphism type must be that of the actual product. A manifold which is only homotopy equivalent to the product is insufficient unless a homeomorphism is also established. Conversely, an exotic smoothing need not itself exhibit any smooth product decomposition or retain the product projections as smooth maps.

The two levels of quantification are separate: one verified exotic example answers an existence question but not the every-product question. The comparison allows arbitrary diffeomorphisms, including ones that do not preserve factors; failure of a factor-preserving smooth identification alone does not prove exoticness.

Short English statement

Which products of a circle with a closed three-manifold, or of two closed orientable surfaces, admit exotic smooth structures?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2880 (KP-4.4), 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 4.4. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.