Topological stability through extremely tame retractions

Suppose that F : (R × R, 0) → (R × R, 0) is a smoothly stable, R-level preserving germ which unfolds f : (R, 0) → (R, 0); then f is smoothly stable if and only if we can find a pair of smooth retractions r : (R, 0)→ (R, 0) and s : (R, 0)→ (R, 0) such that f ◦ r = s ◦F . Unfortunately, we do not know whether f will be topologically stable if we can find a pair of continuous retractions r and s. The class of extremely tame (E-tame) retractions, introduced by du Plessis and Wall, are defined by their nice geometric properties, which are sufficient to ensure that f is topologically stable. In this article, we present the E-tame retractions and their relation with topological stability, survey recent results by the author concerning their construction, and illustrate the use of our techniques by constructing E-tame retractions for certain germs belonging to the Eand Z-series of singularities.