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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
The compression theorem III: applications

Colin Rourke and Brian Sanderson

Algebraic & Geometric Topology 3 (2003) 857–872

arXiv: math.GT/0301356


This is the third of three papers about the Compression Theorem: if Mm is embedded in Qq × with a normal vector field and if q m 1, then the given vector field can be straightened (ie, made parallel to the given direction) by an isotopy of M and normal field in Q × .

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 C0–small isotopy and give a theoretical solution in the codimension 1 case.

compression, embedding, isotopy, immersion, singularities, vector field, loops–suspension, knot, configuration space
Mathematical Subject Classification 2000
Primary: 57R25, 57R27, 57R40, 57R42, 57R52
Secondary: 57R20, 57R45, 55P35, 55P40, 55P47
Forward citations
Received: 31 January 2003
Revised: 16 September 2003
Accepted: 24 September 2003
Published: 25 September 2003
Colin Rourke
Mathematics Institute
University of Warwick
Coventry CV4 7AL
Brian Sanderson
Mathematics Institute
University of Warwick
Coventry CV4 7AL