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On the Pham example and the universal topological stratification of singularities. (English) Zbl 0675.58008
Singularities, Banach Cent. Publ. 20, 161-168 (1988).
[For the entire collection see Zbl 0653.00009.]
In the study of topological stable mappings, the Thom-Mather stratification is very important. The existence of such a stratification has been proven by Mather. But it is difficult to determine the Thom- Mather stratification even in the case of unimodal hypersurface singularities. Part of this difficulty was foreseen by Pham around 1970. He found an example of a complex curve singularity $$f_ 0(x,y)=y^ 3+x^ 9$$ which has a two parameter family of deformations $$f_ 1(x,y,s,t)=y^ 3+tyx^ 6+syx^ 7+x^ 9$$ such that the family is topologically trivial. However, the versal deformation of $$f_ 0$$ is topologically not a product along the t-axis.
In this paper the authors determine how the (s,t) parameter space is stratified by the topological type of the versal deformation. They believe this provides a framework for understanding the other bimodal singularities and an approach to investigating the higher-modality singularities.
Reviewer: S.Izumiya

##### MSC:
 58C25 Differentiable maps on manifolds 58K99 Theory of singularities and catastrophe theory 57N80 Stratifications in topological manifolds
##### Keywords:
Thom-Mather stratification; versal deformation