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Donaldson’s question on Torelli symplectomorphisms and Dehn–Seidel factorizations

Ascertain whether every Torelli symplectomorphism of a closed simply–connected symplectic 4–manifold (M, ω) is symplectically isotopic to a product of squared Dehn–Seidel twists.

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Background

This question, attributed to Donaldson, asks whether all Torelli symplectomorphisms on closed simply-connected symplectic 4-manifolds factor into squared Dehn–Seidel twists. The authors note affirmative results for positive rational surfaces but that the general case remains open.

They also discuss a related broader question about factoring arbitrary symplectomorphisms into Dehn–Seidel twists, mentioning known counterexamples when simple connectivity is dropped.

References

The following question is attributed to Donaldson in the literature : for a closed simply--connected symplectic $4$-manifold $(M, \omega )$, is every Torelli symplectomorphism of $(M, \omega )$ symplectically isotopic to a product of squared Dehn--Seidel twists? The answer is known to be affirmative for positive rational surfaces , but otherwise remains widely open.

— On four-dimensional Dehn twists and Milnor fibrations (2409.11961 - Konno et al., 18 Sep 2024) in Introduction, Subsection “Robust symplectomorphisms and Donaldson’s question”