On extensions of a symplectic class

TL;DR

This paper establishes a necessary and sufficient condition for extending a symplectic class in certain fibrations, linking it to Thurston's criterion and the Hamiltonian obstruction, thus advancing understanding of symplectic bundle structures.

Contribution

It provides a new cohomological criterion for extending symplectic classes in fibrations with specific fibers, connecting to Thurston's and Lalond-McDuff's criteria.

Findings

Characterizes when the symplectic class extends to the total space.

Rephrases Thurston's criterion in terms of classifying maps.

Reinterprets Lalond and McDuff's obstruction for Hamiltonian structures.

Abstract

Let F be a fibration on a simply-connected base with symplectic fibre (M, \omega). Assume that the fibre is nilpotent and T^{2k}-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 [\omega] to extend to a cohomology class of the total space of F. This allows us to describe Thurston's 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 fibre in which the class [\omega] is extendable.