Topological stability through extremely tame retractions (Q664654)

Smooth stability of a smooth map germ \(f:(\mathbb R^n,0) \to (\mathbb R^p,0)\) is equivalent to the following property. For every unfolding \(F:(\mathbb R^{n+d},0) \to (\mathbb R^{p+d},0)\) there is a pair of smooth retraction germs \(r:(\mathbb R^{n+d},0)\to (\mathbb R^n,0)\) and \(s:(\mathbb R^{p+d},0)\to (\mathbb R^p,0)\) such that \(f\circ r=s \circ F\). The blind translation of the above characterization does not hold for topological stability, only if one considers so-called tame retractions. The paper under review studies a special class of tame retractions, the extremely tame retractions. By proving properties (e.g. functoriality) of extremely tame retractions the author builds up a nice toolbox to study them, and shows several non-trivial examples, involving the E and Z series of singularities.