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 $V \mapsto emb (V, N)$ 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 $V \mapsto emb (V, N)$ are explicitly determined. In a sequel to this paper, we introduce the concept of an analytic cofunctor from $'$ 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 $V \mapsto emb (V, N)$ is analytic when $dim (N) - dim (M) \geq 3$ .

Keywords

Embedding, immersion, calculus of functors

Mathematical Subject Classification

Primary: 57R40

Secondary: 57R42

References

Forward citations

Publication

Received: 10 May 1998

Revised: 5 May 1999

Accepted: 13 May 1999

Published: 28 May 1999

Proposed: Ralph Cohen

Seconded: Haynes Miller, Gunnar Carlsson

Correction: 11 March 2011

Authors

Michael Weiss
Department of Mathematics University of Aberdeen Aberdeen AB24 3UE United Kingdom

Volume 3, issue 1 (1999)

Michael Weiss