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Stratification Theory from the Weighted Point of View

Published online by Cambridge University Press:  20 November 2018

Toshizumi Fukui  and
Laurentiu Paunescu
Toshizumi Fukui
Affiliation:
Department of Mathematics, Faculty of Science, Saitama University, 255 Shimo-Okubo, Urawa 338, Japan
Laurentiu Paunescu
Affiliation:
School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia
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Abstract

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In this paper, we investigate stratification theory in terms of the defining equations of strata and maps (without tube systems), offering a concrete approach to show that some given family is topologically trivial. In this approach, we consider a weighted version of $\left( w \right)$-regularity condition and Kuo’s ratio test condition.

Keywords

32B9914P2532Cxx58A35
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 53 , Issue 1 , 01 February 2001 , pp. 73 - 97
Copyright
Copyright © Canadian Mathematical Society 2001

References

[1] Bekka, K., Regular quasi-homogeneous stratification. In: Stratification, singularities and differential equations II Stratifications and topology of singular space, Travaux en cours 55, Hermann, 1997, 1–14.Google Scholar
[2] Bekka, K. and Koike, S., The Kuo condition, an inequality of Thom's type and(c)-regularity. Topology 37(1998), 45–62.Google Scholar
[3] Brodersen, H. and Trotman, D., Whitney (b)-regularity is strictlyweaker than Kuo's ratio test for real algebraic stratifications. Math. Scand. 45(1979), 27–34.Google Scholar
[4] Bruce, J. W. and Roberts, R. M., Critical points of functions on analytic varieties. Topology 27(1988), 57–90.Google Scholar
[5] Damon, J., Deformation of sections of singularities and Gorenstein singularites. Amer. J. Math. 109(1987), 695–722.Google Scholar
[6] A. duPlessis and Wall, T., The geometry of topological stability. LondonMath. Soc. Monographs, New Series 9, Oxford Science Publication, 1995.Google Scholar
[7] Gibson, C., Wirthmüller, K., duPlessis, A. and Looijenga, E., Topological stability of smooth mappings. Lecture Notes in Math. 552, 1976.Google Scholar
[8] Gaffney, T., Integral closure of modules and Whitney equisingularities. Invent.Math. 107(1992), 301–322.Google Scholar
[9] Hartman, P., Ordinary differential equations, Second edition. Birkhäuser, 1982.Google Scholar
[10] Koike, S., On condition (af) of a stratified mapping. Ann. Inst. Fourier (Grenoble) 33(1983), 177–184.Google Scholar
[11] Kuo, T.-C., On C0-sufficiency of jets of potential functions. Topology 8(1969), 167–171.Google Scholar
[12] Kuo, T.-C., The ratio test for analytic Whitney stratification. In: Proceedings of Liverpool Singularities Symposium, I, Lecture Notes in Math. 192, Springer-Verlag, 1971, 141–149.Google Scholar
[13] Kuo, T.-C., Characterization of v-sufficiency of jets. Topology 11(1972), 115–131.Google Scholar
[14] Kuo, T.-C., On Thom-Whitney Stratification theory. Math. Ann. 234(1978), 97–107.Google Scholar
[15] Kuo, T.-C., The modified analytic trivialization of singularities. J. Math. Soc. Japan 32(1980), 605–614.Google Scholar
[16] Mather, J., Stratifications and mappings. In: Dynamical systems, (ed. Peixoto, M.), Academic Press, 1971, 195–223.Google Scholar
[17] Bierstone, E. and Milman, P., Composite differentiable functions. Ann. of Math. 116(1982), 541–558.Google Scholar
[18] Paunescu, L., A weighted version of the Kuiper-Kuo-Bochnak-Łojasiewicz theorem. J. AlgebraicGeom. 2(1993), 69–79.Google Scholar
[19] Satake, I., Linear Algebra. Marcel Dekker, Inc. 1975.Google Scholar
[20] Shiota, M., Geometry of subanalytic and semialgebraic sets. Progress in Math. 150, Birkhäuser, 1997.Google Scholar
[21] Sun, B., Stratifications and sufficiency of weighted jet. Thesis, Hawaii University, 1996.Google Scholar
[22] Teissier, B., Varietes polaires I, Invariants polaires des singularités d’hypersurfaces. Invent.Math. 40(1977), 267–292.Google Scholar
[23] Teissier, B., Varietes polaires II, Multiplicites polaires, sections planes, et conditions deWhitney. Algebraic geometry, Proceedings, La Rábida, 1981, Lecture Notes in Math. 961, Springer-Verlag, 1982, 314–500.Google Scholar
[24] Thom, R., Ensembles et morphismes stratifié. Bull. Amer.Math. Soc. 75(1969), 240–284.Google Scholar
[25] Trotman, D., Comparing regularity conditions on stratifications. In: Proceedings of Symposia in Pure Mathematics 40–2, (1983), 575–586.Google Scholar
[26] Verdier, J.-L., Stratifications de Whitney et théor`eme de Bertini-Sard. Invent.Math. 36(1976), 295–312.Google Scholar
[27] Wall, C. T. C., Finite determinacy of smooth map-germs. Bull. LondonMath. Soc. 13(1981), 481–539.Google Scholar