Kirby Problem 4.68

Definitions and mathematical statement

For a closed smooth $4$-manifold $X$, a self-diffeomorphism $f:X\to X$ is exotic if it is topologically isotopic to the identity but not smoothly isotopic to it. The topological condition means a continuous family of homeomorphisms $H_t$, with $H_0=\mathrm{id}$ and $H_1=f$. A smooth isotopy would be a smooth family of diffeomorphisms with those endpoints. Both are isotopies of the whole manifold, without a boundary condition.

Does every closed smooth $4$-manifold admit an exotic diffeomorphism? Two highlighted cases ask: (a) does any definite closed smooth $4$-manifold admit one; and (b) does the standard $S^2\times S^2$ or the standard K3 surface admit one? For an oriented manifold, definite means that its real intersection form on $H_2(X;\mathbb R)$ is positive definite or negative definite. The standard K3 smooth manifold may be represented by a nonsingular quartic complex surface in $\mathbb{CP}^3$.

An exotic diffeomorphism is a nontrivial smooth mapping class which becomes trivial in the topological mapping class group. Failure to be equal to the identity map is far weaker; even a nontrivial rotation can be smoothly isotopic to it. Likewise a homotopy of maps is weaker than the required topological isotopy. The universal question and each named special case concern the specified smooth manifold itself, without first changing its smoothing or stabilizing it by connected sum.

Short English statement

Does every closed smooth four-manifold admit a diffeomorphism topologically but not smoothly isotopic to the identity, including definite examples, S²×S² and K3?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2944 (KP-4.68), 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 4.68. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.