Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-25T07:41:16.960Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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