The versality discriminant and local topological equivalence of mappings
Annales de l'Institut Fourier, Volume 40 (1990) no. 4, pp. 965-1004.

We will extend the infinitesimal criteria for the equisingularity (i.e. topological triviality) of deformations f of germs of mappings f 0 :k s , 0k t , 0 to non-finitely determined germs (these occur generically outside the “nice dimensions” for Mather, even among topologically stable mappings). The failure of finite determinacy is described geometrically by the “versality discriminant”, which is the set of points where f 0 is not stable (i.e. viewed as an unfolding it is not versal). The criterion asserts that algebraic filtration conditions on the infinitesimal deformations together with topological triviality of f in a “conical neighborhood” of the versality discriminant imply topological triviality of f itself.

Nous étendons le critère infinitésimal par l’équisingularité (i.e. trivialité topologique) des déformations f des germes d’applications f 0 :k s , 0k t , 0 à des germes qui ne sont pas de détermination finie (ils apparaissent génériquement en dehors des “bonnes dimensions” de Mather, même parmi les applications topologiquement stables). On décrit géométriquement le caractère non fini par le “discriminant versel”, qui représente l’ensemble des points où f 0 n’est pas stable (i.e. non versel lorsqu’on le regarde comme un déploiement). Ce critère affirme que des conditions de filtration algébrique sur les déformations infinitésimales associées à la trivialité topologique de f dans un “voisinage conique” du discriminant versel entraîne la trivialité topologique de f elle-même.

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     title = {The versality discriminant and local topological equivalence of mappings},
     journal = {Annales de l'Institut Fourier},
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Damon, James. The versality discriminant and local topological equivalence of mappings. Annales de l'Institut Fourier, Volume 40 (1990) no. 4, pp. 965-1004. doi : 10.5802/aif.1244. http://archive.numdam.org/articles/10.5802/aif.1244/

[D1] J. Damon, Finite Determinacy and Topological Triviality I. Invent. Math., 62 (1980), 299-324. | MR | Zbl

II. Sufficient Conditions and Topological Stability, Compositio Math., 47 (1982), 101-132. | Numdam | Zbl

[D2] J. Damon, Topological Triviality and Versality for Subgroups of A and K, Memoirs of A.M.S., 389 (1988). | MR | Zbl

[D3] J. Damon, Topological invariants of µ-constant deformations of complete intersection singularities, Quart. J. Math., 40 (1989), 139-160. | MR | Zbl

[DGaf] J. Damon and T. Gaffney, Topological Triviality of Deformation of Functions and Newton Filtrations, Invent. Math., 72 (1983), 335-358. | MR | Zbl

[DGal] J. Damon and A. Galligo, Universal Topological Stratification for the Pham Example, preprint.

[Gel] T. Gaffney, Properties of Finitely Determined Germs, Thesis, Brandeis Univ., 1975.

[K] A. G. Kouchnirenko, Polyèdres de Newton et Nombres de Milnor, Invent. Math., 32 (1976), 1-31. | MR | Zbl

[LeR] D. T. Le and C. P. Ramanujam, Invariance of Minor's number implies the invariance of topological type, Amer. J. Math., 98 (1976), 67-78. | Zbl

[Lo] E. Looijenga, Semi-universal Deformation of a Simple Elliptic Hypersurface Singularity: I. Unimodularity, Topology, 16 (1977), 257-262. | MR | Zbl

[M1] J. Mather, Stability of C∞ Mappings V: Transversality, Advances in Math., 4 (1970), 301-336. | MR | Zbl

[M2] J. Mather, Generic projections, Ann. of Math., (2) 98 (1973), 226-245. | MR | Zbl

[T] B. Teissier, Cycles Évanescents, Sections Planes, et Conditions de Whitney, Singularités à Cargèse, Asterisque 7, 8 (1973), 285-362. | Zbl

[V] A. N. Varchenko, A lower bound for the codimension of the stratum µ-constant in terms of the mixed Hodge structure, Vest. Mosk. Univ. Mat., 37 (1982), 29-31. | MR | Zbl

[Wi] K. Wirthmüller, Universell Topologisch Triviale Deformationen, thesis, Univ. of Regensburg.

[Wa] C. T. C. Wall, private communication.

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