Embeddings from the point of view of immersion theory : Part I
نویسنده
چکیده
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 7→ 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 7→ 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 7→ emb(V,N) is analytic when dim(N)− dim(M) ≥ 3. AMS Classification numbers Primary: 57R40 Secondary: 57R42
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