C\({}^ 1\) local parametrized Morse theory
The author starts from the following unpublished result of H. Chaltin: any family of parametrised smooth functions \(f_ t: M^ m\to {\mathbb{R}}\) \((t\in P^ p)\), which is nonsingular on \(\partial M\), can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.
scientific article; zbMATH DE number 4192676