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.

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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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