Abstract
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.
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 language | English |
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| Journal | Topology and Its Applications |
| Volume | 159 |
| Issue number | 2 |
| Pages (from-to) | 457-465 |
| Number of pages | 9 |
| ISSN | 0166-8641 |
| DOIs | |
| Publication status | Published - 1 Feb 2012 |
| Event | 1st International Workshop on Singularities in Generic Geometry and Applications - Valencia, Spain Duration: 23 Mar 2009 → 27 Mar 2009 Conference number: 1 |
Conference
| Conference | 1st International Workshop on Singularities in Generic Geometry and Applications |
|---|---|
| Number | 1 |
| Country/Territory | Spain |
| City | Valencia |
| Period | 23/03/2009 → 27/03/2009 |