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The compression theorem III: Applications. (English) Zbl 1032.57029

This is the third in a series of papers about the compression theorem: let \(M\) and \(Q\) be smooth manifolds of dimensions \(m\) and \(q\), and suppose that \(M\) is embedded in \(Q\times {\mathbb R}\) with a normal vector field and that \(q-m\geq 1\); then the vector field can be made parallel to the given \({\mathbb R}\) direction by an isotopy of \(M\) and a normal field in \(Q\times {\mathbb R}\). The first two papers gave proofs of this theorem. In the present paper the authors deal with applications of this theorem and its extension (the multi-compression theorem). In fact four kinds of applications are exhibited and each one is discussed in a separate section.
Sections 2 and 3 give short new and constructive proofs of the immersion theorem of Hirsch and Smale and the loop-suspension theorem of James and its generalization due to May and Segal et al. using configuration space models of multiple-loops-suspension spaces. The proofs given here make the best use of what the compression theorem provides, namely the fact that a given embedding becomes isotopic to an embedding covering an immersion in \(Q\).
In section 4 it is shown that the compression theorem can be used to give a new approach to classifying embeddings and knots of manifolds in codimension one or more, and in section 5 the authors consider the \(C^0\)-singularity problem in the cases of codimension one or more using the compression theorem. Moreover the authors include many developmental comments on further information at the end of every section. Especially section 5 contains five extra comments one of which announces the publication of papers on further applications of the compression theorem. Also in sections 2 and 4 the authors announce the publications of subsequent papers relevant to the subjects here. Anyway one finds that the range of this compression theorem in differential topology is indeed extensive. So it seems that more deep significance lies hidden in this field.

MSC:

57R25 Vector fields, frame fields in differential topology
57R27 Controllability of vector fields on \(C^\infty\) and real-analytic manifolds
57R40 Embeddings in differential topology
57R42 Immersions in differential topology
55P35 Loop spaces
55P40 Suspensions
55P47 Infinite loop spaces
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References:

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