The compression theorem III: applications

Abstract

This is the third of three papers about the Compression Theorem: if $M^m$ is embedded in $Q^q\times ℝ$ with a normal vector field and if $q-m\ge 1$, then the given vector field can be straightened (ie, made parallel to the given $ℝ$ direction) by an isotopy of $M$ and normal field in $Q\times ℝ$.

The theorem can be deduced from Gromov’s theorem on directed embeddings [Partial differential relations, Springer–Verlag (1986); 2.4.5 C’] and the first 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 controlling the singularities of a smooth projection up to $C^0$–small isotopy and give a theoretical solution in the codimension $\ge 1$ case.