A Haefliger Style Description of the Embedding Calculus Tower

Let M and N be smooth manifolds. The calculus of embeddings produces, for every k greater than or equal to 1, a best degree less than or equal to k polynomial approximation to the cofunctor taking an open V C M to the space of embeddings from V to N. In this paper, a description of these polynomial approximations in terms of equivariant mapping spaces is given, for k greater than or equal to 2. The description is new only for k greater than or equal to 3. In the case k = 2 we recover Haefliger's approximation and the known result that it is the best degree less than or equal to 2 approximation. (C) 2002 Published by Elsevier Science Ltd.