The versality discriminant and local topological equivalence of mappings
Using infinitesimal methods, criteria for the equisingularity (i.e. topological triviality) of deformations f of germs of mappings \(f_ 0: k^ 2,0\to k^ t,0\) have been established by the author for weighted homogeneous germs [Invent. Math. 62, 299-324 (1980; Zbl 0489.58003)] and more generally deformations of germs which have finite codimension in a graded sense relative to an algebraic filtration [Mem. Am. Math. Soc. 389, 1-106 (1988; Zbl 0665.58005)]. The criteria are stated in terms of filtration conditions on the infinitesimal deformations. Such results depend on the finite determinacy of \(f_ 0\). 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). In this paper the infinitesimal criteria is extended to non-finitely determined germs (these occur generically outside the ``nice dimensions'' of Mather, even among topologically stable mappings). The criteria asserts that algebraic filtration conditions on the infinitesimal deformations together with topological triviality of f in a ``conical neighborhood'' of the versality discriminant implies topological triviality of f itself. Several forms of the result are given for both the weighted homogeneous and general cases; and applications are indicated to the determination of topologically stable germs, including the joint work with A. Galligo on the Pham example.
scientific article; zbMATH DE number 4153504
scientific article