This is the third of three papers about
the Compression Theorem: if Mm is embedded in
Qq×R with a normal vector field and if q-m≥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×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 C0–small
isotopy and give a theoretical solution in the codimension ≥1 case.