An application of the h-principle to manifold calculus

Abstract

Manifold calculus is a form of functor calculus that analyzes contravariant functors from some categories of manifolds to topological spaces by providing analytic approximations to them. In this paper, using the technique of the h-principle, we show that for a symplectic manifold N, the analytic approximation to the Lagrangian embeddings functor $$\mathrm {Emb}_{\mathrm {Lag}}(-,N)$$ is the totally real embeddings functor $$\mathrm {Emb}_{\mathrm {TR}}(-,N)$$. More generally, for subsets $${\mathcal {A}}$$ of the m-plane Grassmannian bundle $${{\,\mathrm{{Gr}}\,}}(m,TN)$$ for which the h-principle holds for $${\mathcal {A}}$$-directed embeddings, we prove the analyticity of the $${\mathcal {A}}$$-directed embeddings functor $${{\,\mathrm{Emb}\,}}_{{\mathcal {A}}}(-,N)$$.