C\({}^ 1\) local parametrized Morse theory
type: Item
instance of: scholarly article
title: C\({}^ 1\) local parametrized Morse theory
zbMATH Open document ID: 0723.57025
DOI: 10.1016/0166-8641(90)90046-5
author: Kiyoshi Igusa
published in: Topology and its Applications
publication date: 1990-01-01T00:00:00Z
review text: The author starts from the following unpublished result of H. Chaltin: any family of parametrised smooth functions \(f_ t: M^ m\to {\mathbb{R}}\) \((t\in P^ p)\), which is nonsingular on \(\partial M\), can be \(C^ 0\)- approximated modulo \(\partial M\) by a family having only non-degenerate \((A_ 1)\) and birth-death \((A_ 2)\) singularities. It is possible to generalise this result in two ways: (i) replace \(C^ 0\) by \(C^ 1\) (but \underbar{not} by \(C^ r\), \(r\geq 2)\) and (ii) argue without deforming the family \(f_ t\) in a region where the singular set is already ``good''. The results obtained are very technical and are either stable (p\(\leq m)\) or unstable \((p>m)\). Their flavour is given by the special case of a foliation. Thus (1) Let \({\mathcal F}\) be a smooth foliation of \(M^ m\) with codimension \(q\leq m/2\), and let f: \(M^ m\to {\mathbb{R}}\) be smooth. Then f can be approximated by \(f'\) having only \(A_ 1\) and \(A_ 2\) singularities when restricted to each leaf of \({\mathcal F}\). (2) With \(q>m/2\) and otherwise unchanged data, the deformed function \(f'\) has only singularities of codimension \(\leq 2q-m+1\) in the sense of Arnol'd. In the stable case proof is by explicit deformation; the unstable is more subtle, and the author introduces the notion of ``total codimension'', which may well find other applications.
Mathematics Subject Classification ID: 57R45
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 4192676
zbMATH Keywords: smooth manifolds
zbMATH Keywords: jets
zbMATH Keywords: singularities
zbMATH Keywords: foliation
zbMATH Keywords: stable
zbMATH Keywords: unstable
zbMATH Keywords: total codimension
reviewed by: Charles B. Thomas
MaRDI profile type: Publication
full work available at URL: 0166-8641(90)90046-5
OpenAlex ID: W2057543837
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cites work: Stability of \(C^ \infty\) mappings. III: Finitely determined map germs
cites work: Singular Points of Complex Hypersurfaces. (AM-61)
description: scientific article; zbMATH DE number 4192676