On extensions of a symplectic class

Abstract

Let F be a fibration on a simply-connected base with symplectic fiber ( M , ω ) . Assume that the fiber is nilpotent and T 2 k - separable for some integer k or a nilmanifold. Then our main theorem, Theorem 1.8, gives a necessary and sufficient condition for the cohomology class [ ω ] to extend to a cohomology class of the total space of F . This allows us to describe Thurstonʼs criterion for a symplectic fibration to admit a compatible symplectic form in terms of the classifying map for the underlying fibration. The obstruction due to Lalond and McDuff for a symplectic bundle to be Hamiltonian is also rephrased in the same vein. Furthermore, with the aid of the main theorem, we discuss a global nature of the set of the homotopy equivalence classes of fibrations with symplectic fiber in which the class [ ω ] is extendable.