An Application of the $h$-principle to Manifold Calculus

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 ${\mathrm{E}\mathrm{m}\mathrm{b}}_{\mathrm{L}\mathrm{a}\mathrm{g}}(-,N)$ is the totally real embeddings functor ${\mathrm{E}\mathrm{m}\mathrm{b}}_{\mathrm{T}\mathrm{R}}(-,N)$. More generally, for subsets $\mathcal{A}$ of the $m$-plane Grassmannian bundle $\mathrm{G}\mathrm{r}(m,TN)$ for which the $h$-principle holds for $\mathcal{A}$-directed embeddings, we prove the analyticity of the $\mathcal{A}$-directed embeddings functor ${\mathrm{E}\mathrm{m}\mathrm{b}}_{\mathcal{A}}(-,N)$.