On the topological triviality along moduli of deformations of J k , 0 singularities

Piotr Jaworski

Annales Polonici Mathematici (2000)

  • Volume: 75, Issue: 3, page 193-212
  • ISSN: 0066-2216

Abstract

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It is well known that versal deformations of nonsimple singularities depend on moduli. However they can be topologically trivial along some or all of them. The first step in the investigation of this phenomenon is to determine the versal discriminant (unstable locus), which roughly speaking is the obstacle to analytic triviality. The next one is to construct continuous liftable vector fields smooth far from the versal discriminant and to integrate them. In this paper we extend the results of J. Damon and A. Galligo, concerning the case of the Pham singularity ( J 3 , 0 in Arnold’s classification) (see [2, 3, 4]), and deal with deformations of general J k , 0 singularities.

How to cite

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Jaworski, Piotr. "On the topological triviality along moduli of deformations of $J_{k,0}$ singularities." Annales Polonici Mathematici 75.3 (2000): 193-212. <http://eudml.org/doc/208395>.

@article{Jaworski2000,
abstract = {It is well known that versal deformations of nonsimple singularities depend on moduli. However they can be topologically trivial along some or all of them. The first step in the investigation of this phenomenon is to determine the versal discriminant (unstable locus), which roughly speaking is the obstacle to analytic triviality. The next one is to construct continuous liftable vector fields smooth far from the versal discriminant and to integrate them. In this paper we extend the results of J. Damon and A. Galligo, concerning the case of the Pham singularity ($J_\{3,0\}$ in Arnold’s classification) (see [2, 3, 4]), and deal with deformations of general $J_\{k,0\}$ singularities.},
author = {Jaworski, Piotr},
journal = {Annales Polonici Mathematici},
keywords = {$J_\{k,0\}$ singularities; topological trivialization; moduli of singularities; singularities},
language = {eng},
number = {3},
pages = {193-212},
title = {On the topological triviality along moduli of deformations of $J_\{k,0\}$ singularities},
url = {http://eudml.org/doc/208395},
volume = {75},
year = {2000},
}

TY - JOUR
AU - Jaworski, Piotr
TI - On the topological triviality along moduli of deformations of $J_{k,0}$ singularities
JO - Annales Polonici Mathematici
PY - 2000
VL - 75
IS - 3
SP - 193
EP - 212
AB - It is well known that versal deformations of nonsimple singularities depend on moduli. However they can be topologically trivial along some or all of them. The first step in the investigation of this phenomenon is to determine the versal discriminant (unstable locus), which roughly speaking is the obstacle to analytic triviality. The next one is to construct continuous liftable vector fields smooth far from the versal discriminant and to integrate them. In this paper we extend the results of J. Damon and A. Galligo, concerning the case of the Pham singularity ($J_{3,0}$ in Arnold’s classification) (see [2, 3, 4]), and deal with deformations of general $J_{k,0}$ singularities.
LA - eng
KW - $J_{k,0}$ singularities; topological trivialization; moduli of singularities; singularities
UR - http://eudml.org/doc/208395
ER -

References

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  1. [1] V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps, Vol. 1, Birkhäuser, 1985. Zbl1297.32001
  2. [2] J. Damon, On the Pham example and the universal topological stratification of singularities, in: Singularities, Banach Center Publ. 20, PWN-Polish Scientific Publ., Warszawa, 1988, 161-167. Zbl0675.58008
  3. [3] J. Damon, A-equivalence and the equivalence of sections of images and discriminants, in: Singularity Theory and its Applications, Part 1 (Coventry 1988/1989), Lecture Notes in Math. 1492, Springer, Berlin, 1991, 93-121. Zbl0822.32005
  4. [4] J. Damon and A. Galligo, Universal topological stratification for the Pham example, Bull. Soc. Math. France 121 (1993), 153-181. Zbl0784.32029
  5. [5] A. du Plessis and C. T. C. Wall, Topological stability, in: Singularities (Lille, 1991), London Math. Soc. Lecture Note Ser. 201, Cambridge Univ. Press, Cambridge, 1994, 351-362. 
  6. [6] A. du Plessis and C. T. C. Wall, The Geometry of Topological Stability, London Math. Soc. Monogr. (N.S.) 9, Oxford University Press, New York, 1995. Zbl0870.57001
  7. [7] P. Hartman, Ordinary Differential Equations, Birkhäuser, Boston, 1982. Zbl0476.34002
  8. [8] P. Jaworski, Decompositions of hypersurface singularities of type J k , 0 , Ann. Polon. Math. 59 (1994), 117-131. Zbl0819.32013
  9. [9] P. Jaworski, On the versal discriminant of the J k , 0 singularities, ibid. 63 (1996), 89-99. Zbl0848.32028
  10. [10] P. Jaworski, On the uniqueness of the quasihomogeneity, in: Geometry and Topology of Caustics - Caustics '98, Banach Center Publ. 50, Inst. Math., Polish Acad. Sci., Warszawa, 1999, 163-167. Zbl0967.32026
  11. [11] E. Looijenga, Semi-universal deformation of a simple elliptic hypersurface singularity, I: Unimodularity, Topology 16 (1977), 257-262. Zbl0373.32004
  12. [12] K. Wirthmüller, Universell topologische triviale Deformationen, Ph.D. thesis, Univ. of Regensburg, 1979. 

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