Symplectic torus bundles and group extensions

Symplectic torus bundles ξ : T 2 → E → B are classified by the second cohomology group of B with local coefficients H1(T ). For B a compact, orientable surface, the main theorem of this paper gives a necessary and sufficient condition on the cohomology class corresponding to ξ for E to admit a symplectic structure compatible with the symplectic bundle structure of ξ : namely, that it be a torsion class. The proof is based on a group-extension-theoretic construction of J. Huebschmann [5]. A key ingredient is the notion of fibrewise-localization.