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Existence of a closed irreducible 4–manifold with an exotic diffeomorphism

Determine whether there exists a closed irreducible smooth 4–manifold that admits an exotic diffeomorphism (i.e., a diffeomorphism topologically isotopic to the identity but not smoothly isotopic to the identity).

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Background

The authors define exotic diffeomorphisms and survey known constructions producing exotic diffeomorphisms on closed 4–manifolds, typically using comparisons of manifolds rather than compactly supported deformations.

They present Theorem \ref{thm: closed} to systematically produce infinite-order exotic diffeomorphisms in certain closed 4–manifolds, but emphasize that the irreducibility of those manifolds remains unknown, leaving the existence question for closed irreducible examples open.

References

Another motivation for Theorem \ref{thm: closed} comes from the following major open problem: does there exist a closed irreducible 4-manifold that admits an exotic diffeomorphism?

— On four-dimensional Dehn twists and Milnor fibrations (2409.11961 - Konno et al., 18 Sep 2024) in Introduction, Subsection “Exotic Dehn twists on open, contractible, and closed manifolds”