Geometry & Topology, Vol. 3 (1999)
Paper no. 3, pages 67--101.
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) > 2.
Embedding, immersion, calculus of functors
AMS subject classification.
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
Department of Mathematics, University of Aberdeen
Aberdeen, AB24 3UE, UK
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