Geometry & Topology, Vol. 3 (1999) Paper no. 3, pages 67--101.

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

Michael Weiss

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 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) > 2.

Keywords. Embedding, immersion, calculus of functors

AMS subject classification. Primary: 57R40. Secondary: 57R42.

DOI: 10.2140/gt.1999.3.67

E-print: arXiv:math.GT/9905202

Submitted to GT on 10 May 1998. (Revised 5 May 1999.) Paper accepted 13 May 1999. Paper published 28 May 1999.

Notes on file formats

Michael Weiss
Department of Mathematics, University of Aberdeen
Aberdeen, AB24 3UE, UK

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