Embeddings from the point of view of immersion theory : Part I

Abstract

Let $M$ and $N$ be smooth manifolds without boundary. Immersion theory suggests that an understanding of the space of smooth embeddings $emb\left(M,N\right)$ should come from an analysis of the cofunctor $V\mapsto emb\left(V,N\right)$ from the poset $\mathcal{O}$ of open subsets of $M$ to spaces. We therefore abstract some of the properties of this cofunctor, and develop a suitable calculus of such cofunctors, Goodwillie style, with Taylor series and so on. The terms of the Taylor series for the cofunctor $V\mapsto emb\left(V,N\right)$ are explicitly determined. In a sequel to this paper, we introduce the concept of an analytic cofunctor from $\prime$ to spaces, and show that the Taylor series of an analytic cofunctor $F$ converges to $F$. Deep excision theorems due to Goodwillie and Goodwillie–Klein imply that the cofunctor $V\mapsto emb\left(V,N\right)$ is analytic when $dim\left(N\right)-dim\left(M\right)\ge 3$.