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



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