Existence of symplectic structures on torus bundles over surfaces (Q2576749)

The author shows that a total space of a locally trivial torus bundle over a surface \(\Sigma_g\) of genus \(g > 1\) carries a symplectic structure if and only if the homology class of the fiber \([T^2]\) is nonzero in \(H_2(E, \mathbb R)\). He proves this result using Seiberg-Witten theory and spectral sequences.