Defining Lagrangian immersions by phase functions

In order to analyze the singularities of the solutions of certain partial differential equations, Hörmander, in his paper on Fourier integral operators, extends the method of stationary phase by introducing the class of nondegenerate phase functions. Each phase function, in turn, defines a lagrangian submanifold of the cotangent bundle of the manifold which is the domain of the corresponding differential operator. Given a lagrangian submanifold of a cotangent bundle, when is it globally defined by a nondegenerate phase function? A necessary and sufficient condition is here found to be the vanishing of two topological obstructions; one in the cohomology and the other in the k -theory of the given lagrangian submanifold.