Abstract
Symplectic torus bundles
ξ:
T
2
→E→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
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 (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.