#### Geometry & Topology, Vol. 5 (2001)
Paper no. 14, pages 399--429.

## The compression theorem I

### Colin Rourke, Brian Sanderson

**Abstract**.
This the first of a set of three papers about the Compression Theorem:
if M^m is embedded in Q^q X R with a normal vector field and if q-m >
0, 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 X R.

The theorem can be deduced from Gromov's theorem on directed
embeddings [M Gromov, Partial differential relations, Springer-Verlag
(1986); 2.4.5 C'] 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

**AMS subject classification**.
Primary: 57R25.
Secondary: 57R27, 57R40, 57R42, 57R52.

**DOI:** 10.2140/gt.2001.5.399

**E-print:** `arXiv:math.GT/9712235`

Submitted to GT on 25 January 2001.
(Revised 2 April 2001.)
Paper accepted 23 April 2001.
Paper published 24 April 2001.

Notes on file formats
Colin Rourke, Brian Sanderson

Mathematics Institute, University of Warwick

Coventry, CV4 7AL, UK

Email: cpr@maths.warwick.ac.uk, bjs@maths.warwick.ac.uk

URL: http://www.maths.warwick.ac.uk/~cpr/ and ~bjs/

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