Calculus of Embeddings

Let M and N be smooth manifolds, where M ⊂ N and dim(N)− dim(M) ≥ 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 imm(M,N), the space of smooth immersions, which is well understood, and embedding spaces emb(S,N) for finite subsets S of M with few elements. The meaning of few depends on the precision desired. 0. Introduction: Immersions vs. embeddings Let M and N be smooth manifolds without boundary. Suppose that m ≤ n, and ifm = n, suppose also thatM has no compact component. Let mono(TM, TN) be the space of vector bundle monomorphisms TM → TN . Such a vector bundle monomorphism consists of a continuous map f : M → N and, for each x ∈ M , a linear monomorphism TxM → Tf(x)N which depends continuously on x. For example, an immersion f : M → N has a differential df : TM → TN which belongs to mono(TM, TN). In this way, (*) imm(M,N) ⊂mono(TM, TN) where imm(M,N) is the space of smooth immersions from M to N . The main theorem of immersion theory states that the inclusion (*) is a homotopy equivalence. The clearest references for this are perhaps [HaePo], for the PL analog, and [Hae1], but the theorem goes back to Smale [Sm] and Hirsch [Hi]. Smale’s stunning discovery that the immersions S → R given by x 7→ x and x 7→ −x respectively are regularly homotopic (homotopic through immersions) is a direct consequence of the main theorem. There is reformulation of the theorem which can only be decoded with a little homotopy theory. It goes like this. The rule V 7→ imm(V,N) is a contravariant functor (cofunctor for short) from the poset O of open subsets V ⊂M to spaces. That 1991 Mathematics Subject Classification. Primary 57R40, 57R42.