It is known (since Maslov observations) that the analogues of Bohr–Sommerfeld conditions in the asymptotic quantisation (know also as the Maslov’s canonical operator method) have a topological nature. This condition has a form of annihilation of some cohomology classes (the Maslov–Arnold classes). The Maslov–Arnold classes are, in fact, the examples of characteristic classes, which are completely defined by a universal construction of a ‘classifying map’ into a ‘classifying space’ phase space T∗M. The topology of this Lagrangian Grassmannian and its Z2−cohomology ring H∗ (LG,Z2) are well known (A. Borel, D. Fuchs). The classes of Maslov–Arnold contain an important information about singularities for the Lagrangian projections. In this paper, we review and describe one of the generalizations of Maslov–Arnold classes associated with a topological study of Monge–Ampère equations and their solutions. This important tool for studies of Monge–Ampère solution singularities