#### Algebraic and Geometric Topology 3 (2003),
paper no. 28, pages 857-872.

## The compression theorem III: applications

### Colin Rourke, Brian Sanderson

**Abstract**.
This is the third 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 [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 C^0-small isotopy and give a theoretical
solution in the codimension >0 case.
**Keywords**. Compression, embedding, isotopy,
immersion, singularities, vector field, loops-suspension, knot,
configuration space

**AMS subject classification**.
Primary: 57R25, 57R27, 57R40, 57R42, 57R52.
Secondary: 57R20, 57R45, 55P35, 55P40, 55P47.

**DOI:** 10.2140/agt.2003.3.857

**E-print:** `arXiv:math.GT/0301356`

Submitted: 31 January 2003.
(Revised: 16 September 2003.)
Accepted: 24 September 2003.
Published: 25 September 2003.

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,
http://www.maths.warwick.ac.uk/~bjs/

AGT home page

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