Calculus of Embeddings

Let $M$ and $N$ be smooth manifolds, where $M\subset N$ and $\dim (N)-\dim (M)\ge 3$. A disjunction lemma for embeddings proved recently by Goodwillie leads to a calculation up to extension problems of the base point component of the space of smooth embeddings of $M$ in $N$. This is mostly in terms of ${\text{\bf imm}}(M,N)$, the space of smooth immersions, which is well understood, and embedding spaces $\text{\bf emb}(S,N)$ for finite subsets $S$ of $M$ with few elements. The meaning of few depends on the precision desired.