# On the topological triviality along moduli of deformations of ${J}_{k,0}$ singularities

Annales Polonici Mathematici (2000)

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

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topJaworski, 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

top- [1] V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps, Vol. 1, Birkhäuser, 1985. Zbl1297.32001
- [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] 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] J. Damon and A. Galligo, Universal topological stratification for the Pham example, Bull. Soc. Math. France 121 (1993), 153-181. Zbl0784.32029
- [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] 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] P. Hartman, Ordinary Differential Equations, Birkhäuser, Boston, 1982. Zbl0476.34002
- [8] P. Jaworski, Decompositions of hypersurface singularities of type ${J}_{k,0}$, Ann. Polon. Math. 59 (1994), 117-131. Zbl0819.32013
- [9] P. Jaworski, On the versal discriminant of the ${J}_{k,0}$ singularities, ibid. 63 (1996), 89-99. Zbl0848.32028
- [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] E. Looijenga, Semi-universal deformation of a simple elliptic hypersurface singularity, I: Unimodularity, Topology 16 (1977), 257-262. Zbl0373.32004
- [12] K. Wirthmüller, Universell topologische triviale Deformationen, Ph.D. thesis, Univ. of Regensburg, 1979.

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