Topological stability through extremely tame retractions

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Suppose that F : (Rn ×Rd, 0)¿(Rp ×Rd, 0) is a smoothly stable, Rd-level preserving germ which unfolds f : (Rn, 0)¿(Rp, 0); then f is smoothly stable if and only if we can find a pair of smooth retractions r : (Rn+d, 0)¿(Rn, 0) and s : (Rp+d, 0)¿(Rp, 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 E- and Z-series of singularities.
Original languageEnglish
JournalTopology and Its Applications
Volume159
Issue number2
Pages (from-to)457-465
Number of pages9
ISSN0166-8641
DOIs
Publication statusPublished - 2012
Event1st International Workshop on Singularities in Generic Geometry and Applications - Valencia, Spain
Duration: 23 Mar 200927 Mar 2009
Conference number: 1

Conference

Conference1st International Workshop on Singularities in Generic Geometry and Applications
Number1
CountrySpain
CityValencia
Period23/03/200927/03/2009

ID: 33348249