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.