×

Deformations of maps on complete intersections, Damon’s \({\mathcal K}_ V\)-equivalence and bifurcations. (English) Zbl 0847.58007

Brasselet, Jean-Paul (ed.), Singularities. Papers of the international congress ‘Singularities in geometry and topology’, held in Lille (France), 3-8 June, 1991. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 201, 263-284 (1994).
The authors prove a generalization of J. Damon’s result [Lect. Notes Math. 1462, 93-121 (1991; Zbl 0822.32005)], relating the \(A_e\)-versal unfoldings of a map-germ \(f\) with the \(K_{D(G)}\)-versal unfoldings of an associated map germ which induces \(f\) from a stable map \(G\), to the case where the source of \(f\) is a complete intersection with an isolated singularity. Further, as an application, the authors give a generalization of J. Damon and D. Mond’s results [Invent. Math. 106, No. 2, 217-242 (1991; Zbl 0772.32022)], which relate the topology of a stabilization of an unstable map-germ \(f\), to the \(A_e\)-codimension of \(f\). Also a relation to the bifurcation problem is discussed.
One of main results of this paper is as follows: Let \(X\) be an isolated complete intersection singularity, and \(f : X \to P\) be of finite singularity type. If \(f\) is induced from a stable map \(G : N \to Q\) by the base change \(\gamma : P \to Q\), then a \(K_{D(G),e}\)-versal deformation \(\Gamma_0 : P \times \mathbb{C}^d\), \(0 \to Q\) of \(\gamma\) induces a versal unfolding \(F_0\) of \(f\).
For the entire collection see [Zbl 0796.00018].

MSC:

58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
32S05 Local complex singularities
32S55 Milnor fibration; relations with knot theory
PDFBibTeX XMLCite