Topological stability through extremely tame retractions

Suppose that F : ( R n × R d , 0 ) → ( R p × R d , 0 ) is a smoothly stable, R d -level preserving germ which unfolds f : ( R n , 0 ) → ( R p , 0 ) ; then f is smoothly stable if and only if we can find a pair of smooth retractions r : ( R n + d , 0 ) → ( R n , 0 ) and s : ( R p + d , 0 ) → ( R p , 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. ass 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. s 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 E- and Z-series of singularities.