A Haefliger style description of the embedding calculus tower

Let M and N be smooth manifolds. The calculus of embeddings produces, for every k⩾1, a best degree ⩽k polynomial approximation to the cofunctor taking an open V⊂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⩾2. The description is new only for k⩾3. In the case k=2 we recover Haefligerʹs approximation and the known result that it is the best degree ⩽2 approximation.