Classes caractéristiques isotropes. (Isotropic characteristic classes) (Q1113486)

Let \(E\to B\) be a symplectic fiber bundle and let I,C\(\subset E\) be a pair consisting of a real isotropic subbundle of complementary dimensions. We associate to the triple (I,C,E) a classifying map with values in a stable isotropic Grassmannian \(\alpha\) : \(B\to {\mathfrak I}G(k)\). The pull-back by \(\alpha\) of the generators of the cohomology ring \(H^*({\mathfrak I}G(k);A)\) are the isotropic characteristic classes of the triple (I,C,E). We show that these classes are fine obstructions for any homomorphic deformation of the pair I, C into a pair \(I'\), \(C'\) everywhere transversal and we describe the geometric applications of this situation. Then we formulate the h-principle for isotropic immersions of a manifold into a symplectic manifold, from which we infer some consequences when the symplectic manifold is \({\mathbb{C}}^ n\) with the standard form. Finally, we compute the stable and unstable cohomologies of isotropic Grassmannians.