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

Let M and N be smooth manifolds without boundary. Immersion theory suggests that an understanding of the space of smooth embeddings emb(M, N ) should come from an analysis of the cofunctor V emb(V, N ) from the poset 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 emb(V, N ) are explicitly determined. In a sequel to this paper, we introduce the concept of an analytic cofunctor from O 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 emb(V, N ) is analytic when dim(N )dim(M ) 3.