The compression theorem III: applications

Rourke, Colin¹ ; Sanderson, Brian¹

¹Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK

Algebraic and Geometric Topology, Tome 3 (2003) no. 2, pp. 857-872

Résumé

This is the third of three papers about the Compression Theorem: if Mm is embedded in Qq × ℝ with a normal vector field and if q − m ≥ 1, 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 × 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 C0–small isotopy and give a theoretical solution in the codimension ≥ 1 case.

Détail

DOI : 10.2140/agt.2003.3.857

Keywords: compression, embedding, isotopy, immersion, singularities, vector field, loops–suspension, knot, configuration space

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