2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/100862Making use of the extended flux homomorphism on the group of symplectomorphisms of a closed oriented surface of genus at least 2, we introduce new characteristic classes of foliated surface bundles with symplectic, equivalently area-preserving, total holonomy. These characteristic classes are stable with respect to the genus and we show that they are highly non-trivial. We also prove that the second homology of the group of Hamiltonian symplectomorphisms equipped with the discrete topology is very large.minor improvements to the exposition; to appear in Journal of Differential GeometrySymplectic GeometryAlgebraic TopologyGeometric Topology57R17; 57R50; 57M99text