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Discreteness of the flux group Γ_ω for open symplectic manifolds

Determine whether the flux group Γ_ω appearing in the exact sequence 1 → Ham_c(M, ω) → Symp_{c,0}(M, ω) → H^1_c(M; R)/Γ_ω → 0 induced by the flux homomorphism is discrete for arbitrary connected open symplectic manifolds (M, ω).

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Background

The flux homomorphism gives rise to an exact sequence relating Ham_c(M, ω), Symp_{c,0}(M, ω), and H1_c(M; R)/Γω. Discreteness of Γω is crucial for establishing local contractibility and certain topological properties of Ham_c(M, ω). While Γ_ω vanishes for exact symplectic manifolds, its discreteness is not known in general for open symplectic manifolds.

References

The reason is that we do not know whether the flux group \Gamma_\omega showing up in the exact sequence induced by the flux homomorphism is discrete for a general open symplectic manifold (M,\omega).

— Smooth perfectness of Hamiltonian diffeomorphism groups (2509.16327 - Edtmair, 19 Sep 2025) in Section 2.1 (Hamiltonian diffeomorphism groups and their topology)