Almost Lagrangian Obstruction

The aim of this paper is to describe the obstruction for an almost Lagrangian fibration to be Lagrangian, a problem which is central to the classification of Lagrangian fibrations and, more generally, to understanding the obstructions to carry out surgery of integrable systems, an idea introduced by Zung. It is shown that this obstruction (namely, the homomorphism of Dazord and Delzant) is related to the cup product in cohomology with local coefficients on the base space B of the fibration. The map is described explicitly and some examples are calculated, thus providing the first examples of non-trivial Lagrangian obstructions.