Universal Lagrangian Bundles

The obstruction to construct a Lagrangian bundle over a fixed integral affine manifold was constructed by Dazord and Delzant in \cite{daz_delz} and shown to be given by `twisted' cup products in \cite{sepe_lag}. This paper uses the topology of universal Lagrangian bundles, which classify Lagrangian bundles topologically (cf. \cite{sepe_topc}), to reinterpret this obstruction as the vanishing of a differential on the second page of a Leray-Serre spectral sequence. Using this interpretation, it is shown that the obstruction of Dazord and Delzant depends on an important cohomological invariant of the integral affine structure on the base space, called the radiance obstruction, which was introduced by Goldman and Hirsch in \cite{goldman}. Some examples, related to non-degenerate singularities of completely integrable Hamiltonian systems, are discussed.