The compression theorem III: applications

Details The compression theorem III: applications The compression theorem III: applications

2003-01-30

arxiv.org/abs/math/0301356v2

Algebr. Geom. Topol. 3 (2003) 857-872

Mathematics

Geometric Topology

Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-28.abs.html

Scientific paper

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 [M Gromov, Partial differential relations, Springer--Verlag (1986) 2.4.5 C'] and the first two parts (math.GT/9712235 and math.GT/0003026) 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.

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