Local path-connectedness of Ham_c(M, ω) in the C0 topology

Ascertain whether the group of compactly supported Hamiltonian diffeomorphisms Ham_c(M, ω) is locally path-connected in the C0 topology for symplectic manifolds beyond the known cases of closed surfaces and the standard symplectic 2n-ball.

Background

The authors highlight a broader context linking their divisorial approach to a classical problem: the local path-connectedness of the Hamiltonian group in the C0 topology. While this property is established for closed surfaces and the standard ball, it is unknown in general.

A positive resolution would have implications for the topology of symplectomorphism groups and could be approached via divisorial decompositions or Floer-theoretic methods, as suggested by the paper.

References

This in turn is related to another long-standing open problem concerning the local path-connectedness of the Hamiltonian group [Ba78]. To our knowledge, local path-connectedness remains wide open outside these cases.

— $C^0$-rigidity of the Hamiltonian diffeomorphism group of symplectic rational surfaces (2508.20285 - Atallah et al., 27 Aug 2025) in Introduction, Relations to other works