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

Michael Weiss

Geometry & Topology 3 (1999) 67–101

Erratum: Geometry & Topology 15 (2011) 407–409

arXiv: math.GT/9905202

Abstract

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 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.

Keywords
Embedding, immersion, calculus of functors
Mathematical Subject Classification
Primary: 57R40
Secondary: 57R42
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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
Authors
Michael Weiss
Department of Mathematics
University of Aberdeen
Aberdeen
AB24 3UE
United Kingdom