Topological stability through extremely tame retractions

Aasa Feragen

doi:10.1016/j.topol.2011.09.020

Topological stability through extremely tame retractions

Aasa Feragen

Datalogisk Institut

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.

Originalsprog	Engelsk
Tidsskrift	Topology and Its Applications
Vol/bind	159
Udgave nummer	2
Sider (fra-til)	457-465
Antal sider	9
ISSN	0166-8641
DOI	https://doi.org/10.1016/j.topol.2011.09.020
Status	Udgivet - 1 feb. 2012
Begivenhed	1st International Workshop on Singularities in Generic Geometry and Applications - Valencia, Spanien Varighed: 23 mar. 2009 → 27 mar. 2009 Konferencens nummer: 1

Konference

Konference	1st International Workshop on Singularities in Generic Geometry and Applications
Nummer	1
Land/Område	Spanien
By	Valencia
Periode	23/03/2009 → 27/03/2009

Adgang til dokumentet

10.1016/j.topol.2011.09.020

Topological stability through extremely tame retractionsForlagets udgivne version, 327 KB

Citationsformater

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title = "Topological stability through extremely tame retractions",

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 thatf ¿ 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. ",

author = "Aasa Feragen",

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doi = "10.1016/j.topol.2011.09.020",

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T1 - Topological stability through extremely tame retractions

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N2 - 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 thatf ¿ 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.

AB - 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 thatf ¿ 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.

U2 - 10.1016/j.topol.2011.09.020

DO - 10.1016/j.topol.2011.09.020

M3 - Conference article

SN - 0166-8641

VL - 159

SP - 457

EP - 465

JO - Topology and its Applications

JF - Topology and its Applications

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T2 - 1st International Workshop on Singularities in Generic Geometry and Applications

Y2 - 23 March 2009 through 27 March 2009

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