Symplectic torus bundles and group extensions

Abstract

Symplectic torus bundles $\xi:T^{2}\to E\to B$ are classified by the second cohomology group of $B$ with local coefficients $H_{1}(T^{2})$. For $B$ a compact, orientable surface, the main theorem of this paper gives a necessary and sufficient condition on the cohomology class corresponding to $\xi$ for $E$ to admit a symplectic structure compatible with the symplectic bundle structure of $\xi$ : namely, that it be a torsion class. The proof is based on a group-extension-theoretic construction of J. Huebschmann (Sur les premieres differentielles de la suite spectrale cohomologique d'une extension de groupes, C.R. Acad. Sc. Paris, Serie A, tome 285, 28 novembre 1977, 929-931). A key ingredient is the notion of fibrewise-localization.