Floer-theoretic detection of non-trivial symplectic mapping classes on positive rational surfaces

Determine whether every non-trivial mapping class in the symplectic mapping class group π0(Symp(X, ω)) of a positive symplectic rational surface (including type D cases) acts non-trivially on some Floer-theoretic invariant, such as Hamiltonian Floer cohomology, Lagrangian Floer cohomology, or the Fukaya category.

Background

The authors discuss a potential Floer-theoretic route to address C0-closure questions: if non-trivial symplectic mapping classes act non-trivially on C0-robust Floer invariants, they could not be approximated by Hamiltonian isotopies, yielding positive closure results.

Existing results (e.g., Jannaud) show that iterations of Lagrangian Dehn twists are not in the C0-closure of Symp0 under certain assumptions, but a general statement for arbitrary compositions or all mapping classes on positive rational surfaces is not yet established.

References

However, it is an open question whether every non-trivial mapping class of a positive symplectic rational surface acts non-trivially on some Floer theoretic invariants even for type $D$, which explains the advantage of our current approach.

— $C^0$-rigidity of the Hamiltonian diffeomorphism group of symplectic rational surfaces (2508.20285 - Atallah et al., 27 Aug 2025) in Further discussion, Subsection 'Floer theoretic approach'