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

TL;DR

This paper applies the $h$-principle to manifold calculus, demonstrating that for symplectic manifolds, the analytic approximation of Lagrangian embeddings aligns with totally real embeddings, extending to other directed embeddings.

Contribution

It establishes the analyticity of the $ ext{Emb}_{ ext{A}}$ functor for subsets where the $h$-principle applies, linking geometric conditions with functor calculus.

Findings

Analytic approximation of Lagrangian embeddings equals totally real embeddings for symplectic manifolds.

Proves analyticity of $ ext{Emb}_{ ext{A}}$ functor under the $h$-principle.

Extends the $h$-principle application to $ ext{A}$-directed embeddings.

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 $\mathrm{Emb}_{\mathcal{A}}(-,N)$.