The versality discriminant and local topological equivalence of mappings

Damon, James

Annales de l'Institut Fourier, Tome 40 (1990) no. 4, pp. 965-1004.

Nous étendons le critère infinitésimal par l’équisingularité (i.e. trivialité topologique) des déformations $f$ des germes d’applications $f_0:k^s$, $0\to k^t$, $0$ 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.

We will extend the infinitesimal criteria for the equisingularity (i.e. topological triviality) of deformations $f$ of germs of mappings $f_0:k^s$, $0\to k^t$, $0$ 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.

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