#### Volume 5, issue 1 (2001)

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The compression theorem I

### Colin Rourke and Brian Sanderson

Geometry & Topology 5 (2001) 399–429
 arXiv: math.GT/9712235
##### Abstract

This the first of a set of three papers about the Compression Theorem: if ${M}^{m}$ is embedded in ${Q}^{q}×ℝ$ 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 $Q×ℝ$.

The theorem can be deduced from Gromov’s theorem on directed embeddings and is implicit in the preceeding discussion. Here we give a direct proof that leads to an explicit description of the finishing embedding.

In the second paper in the series we give a proof in the spirit of Gromov’s proof and in the third part we give applications.

##### Keywords
compression, embedding, isotopy, immersion, straightening, vector field
##### Mathematical Subject Classification 2000
Primary: 57R25
Secondary: 57R27, 57R40, 57R42, 57R52