Symplectic embeddings in infinite codimension

Abstract

Let X be a union of a sequence of symplectic manifolds of increasing dimension and let M be a manifold with a closed 2-form \(\omega \). We use Tischler’s elementary method for constructing symplectic embeddings in complex projective space to show that the map from the space of embeddings of M in X to the cohomology class of \(\omega \) given by pulling back the limiting symplectic form on X is a weak Serre fibration. Using the same technique we prove that, if \(b_2(M)<\infty \), any compact family of closed 2-forms on M can be obtained by restricting a standard family of forms on a product of complex projective spaces along a family of embeddings.