This the first of a set of three papers about the Compression Theorem: if M is embedded in Q ×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 [5; 2.4.5… (More)