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Topological triviality of a family of zero-sets. (English) Zbl 0658.58011
As stated in the authors’ abstract, the paper gives conditions on a mapping F: \(U\times R^ k\to K^ p\) \((U\subset K^ n\) open, \(K={\mathbb{R}}\) or \({\mathbb{C}})\) such that the family \(F_ t^{-1}(0)\) is a topologically trivial family. The authors consider ‘admissible’ families, which are deformations of homogeneous polynomial maps, and show that they are locally topological trivial at the origin.
Using their result they are able to simplify the proof of a counterexample to a conjecture of Thom concerning the number of topologically different realizations of a given jet, contained in a paper of S. Koike and W. Kucharz [C. R. Acad. Sci., Paris, Sér. A 288, 457-459 (1979; Zbl 0404.58002)].
Reviewer: J.Timourian

MSC:
58C25 Differentiable maps on manifolds
58A20 Jets in global analysis
57R45 Singularities of differentiable mappings in differential topology
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[1] Jacek Bochnak and Tzee-char Kuo, Different realizations of a non sufficient jet, Nederl. Akad. Wetensch. Proc. Ser. A 75=Indag. Math. 34 (1972), 24 – 31. · Zbl 0225.58001
[2] Michael Buchner, Jerrold Marsden, and Stephen Schecter, Applications of the blowing-up construction and algebraic geometry to bifurcation problems, J. Differential Equations 48 (1983), no. 3, 404 – 433. · Zbl 0464.34030 · doi:10.1016/0022-0396(83)90102-X · doi.org
[3] James Damon, Finite determinacy and topological triviality. I, Invent. Math. 62 (1980/81), no. 2, 299 – 324. · Zbl 0489.58003 · doi:10.1007/BF01389162 · doi.org
[4] James Damon, Newton filtrations, monomial algebras and nonisolated and equivariant singularities, Singularities, Part 1 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 267 – 276.
[5] Henry C. King, Topological type in families of germs, Invent. Math. 62 (1980/81), no. 1, 1 – 13. · Zbl 0477.58010 · doi:10.1007/BF01391660 · doi.org
[6] Henry C. King, Topological type of isolated critical points, Ann. Math. (2) 107 (1978), no. 2, 385 – 397. · Zbl 0354.57002
[7] Satoshi Koike and Wojciech Kucharz, Sur les réalisations de jets non suffisants, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 8, A457 – A459 (French, with English summary). · Zbl 0404.58002
[8] Lê Dũng Tráng and C. P. Ramanujam, The invariance of Milnor’s number implies the invariance of the topological type, Amer. J. Math. 98 (1976), no. 1, 67 – 78. · Zbl 0351.32009 · doi:10.2307/2373614 · doi.org
[9] Mutsuo Oka, On the topology of the Newton boundary. II. Generic weighted homogeneous singularity, J. Math. Soc. Japan 32 (1980), no. 1, 65 – 92. · Zbl 0417.14004 · doi:10.2969/jmsj/03210065 · doi.org
[10] Peter B. Percell and Peter N. Brown, Finite determination of bifurcation problems, SIAM J. Math. Anal. 16 (1985), no. 1, 28 – 46. · Zbl 0565.58026 · doi:10.1137/0516003 · doi.org
[11] Manifolds — Amsterdam 1970, Proceedings of the Nuffic Summer School on Manifolds, Amsterdam, August 17-29, vol. 1970, Springer-Verlag, Berlin-New York, 1971.
[12] J. G. Timourian, The invariance of Milnor’s number implies topological triviality, Amer. J. Math. 99 (1977), no. 2, 437 – 446. · Zbl 0373.32003 · doi:10.2307/2373829 · doi.org
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