The compression theorem III : applications ( preliminary

This is the third of three papers about the Compression Theorem: if M m is embedded in Q q R with a normal vector eld and if q ? m 1, then the given vector eld can be straightened (ie, made parallel to the given R direction) by an isotopy of M and normal eld in Q R. The theorem can be deduced from Gromov's theorem on directed em-beddings 3; 2.4.5 (C 0)] and the rst two parts gave proofs. Here we are concerned with applications. We give short new (and constructive) proofs for immersion theory and for the loops{suspension theorem of James et al and a new approach to classifying embeddings of manifolds in codimension one or more, which leads to theoretical solutions. We also consider the general problem of simplifying (or specifying) the singularities of a smooth projection up to C 0 {small isotopy and give a theoretical solution in the codimension 1 case.