We saw in Chapter 4 that a map Vv → Mm in general position is already an embedding if. If this condition fails, we still have effective techniques for constructing embeddings, and will describe some of the main results in this chapter.

For immersions, the results give a complete reduction of the problem to a problem in homotopy theory. The proof of this major result is somewhat technical, and the details will not be required elsewhere.

We will now need to assume rather more familiarity with homotopy theory than in earlier chapters, and refer to Appendix B for a summary of the relevant definitions and results.

The theory of embeddings begins with a technique introduced by Whitney for removing pairs of self-intersections of a smooth n-manifolds in a 2n-manifold (if n ≥ 3). We describe this in some detail in §6.3: it was used as a key tool in §5.5. We then apply it to discuss embeddings of Sn in 2n-manifolds.

The essential idea of this technique was generalised by Haefliger to maps Vv → Mm whenever 2m ≥ 3(v + 1) – a conditionwe call the metastable range. There are several related results giving homotopy theoretic criteria for deforming a map to an immersion, or to an embedding, or for finding a regular homotopy of an immersion to an embedding; each one also has a simplified form when the target is Euclidean space, and also a companion criterion for finding a diffeotopy of the constructed embeddings. We describe these results, but confine ourselves to an outline of the rather involved proof.

Fibration theorems

A map f : E → B is said to be a fibration if given a space K, a map a : K → E and a homotopy b : K × I → B such that, there exists a homotopy such that and. We also say that f has the covering homotopy property (CHP). If this holds for K a finite CW-complex, it follows for any CW-complex; it also follows if (K, L) is a CWpair that c can be chosen to extend a lift already given on L × I. It suffices to require this condition for pairs (K, L) = (Dn, Sn−1).