A bilipschitz version of Hardt's theorem

Guillaume Valette

doi:10.1016/j.crma.2005.05.004

Topology

A bilipschitz version of Hardt's theorem

Presented by: Bernard Malgrange

Guillaume Valette ¹

¹ Instytut Matematyki, Uniwersytet Jagielloński, ul. Reymonta 4, 30 059 Kraków, Poland

Comptes Rendus. Mathématique, Volume 340 (2005) no. 12, pp. 895-900.

Abstract
Résumé

In this Note we give a sketch of the proof of a theorem which is a bilipschitz version of Hardt's theorem. Given a family definable in an o-minimal structure Hardt's theorem states the existence (for generic parameters) of a trivialization which is definable in the o-minimal structure. We show that, for a polynomially bounded o-minimal structure, there exists such an isotopy which is bilipschitz. The proof is inspired by Bochnak et al. [Géométrie Algébrique Réelle, Springer-Verlag, 1987]. and involves the construction of ‘Lipschitz triangulations’ which are defined in this Note. The complete proof of existence will appear later.

Dans cette note on donne les grandes lignes de la preuve d'un théorème qui constitue une version bilipschitzienne du théorème de Hardt. Étant donnée famille d'ensembles définissables dans une structure o-minimale le théorème de Hardt établit l'existence d'une trivialisation topologique (pour des paramètres génériques) définissable dans la structure. On démontre que l'isotopie peut être choisie bilipschitzienne pour les structures o-minimales polynomialement bornées. La preuve consiste à démontrer l'existence de « triangulations lipschitz » simultanées (cf. Bochnak et al. [Géométrie Algébrique Réelle, Springer-Verlag, 1987]). On en donne ici l'idée et la définition ; la preuve détaillée de l'existence sera publié plus tard.

Received: 2004-07-06
Accepted: 2005-04-27
Published online: 2005-06-15

DOI: 10.1016/j.crma.2005.05.004

Author's affiliations:

Guillaume Valette ¹

¹ Instytut Matematyki, Uniwersytet Jagielloński, ul. Reymonta 4, 30 059 Kraków, Poland

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     title = {A bilipschitz version of {Hardt's} theorem},
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Guillaume Valette. A bilipschitz version of Hardt's theorem. Comptes Rendus. Mathématique, Volume 340 (2005) no. 12, pp. 895-900. doi : 10.1016/j.crma.2005.05.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.05.004/

References
Cited by

[1] L. Birbrair Local bi-Lipschitz classification of two-dimensional semialgebraic sets, Rev. Semin. Iberoam. Mat. Singul. Tordesillas, Volume 2 (1998) no. 1, pp. 29-34 (in Portuguese)

[2] J. Bochnak; M. Coste; M.-F. Roy Géométrie Algébrique Réelle, Ergeb. Math., vol. 12, Springer-Verlag, 1987

[3] M. Coste, An introduction to O-minimal geometry. Dip. Mat. Univ. Pisa, Dottorato di Ricerca in Matematica, Istituti Editoriali e Poligrafici Internazionali, Pisa, 2000

[4] M. Coste; M. Reguiat Trivialités en famille, Real Algebraic Geometry, Lecture Notes in Math., vol. 1524, Springer, 1992, pp. 193-204

[5] R.M. Hardt Semi-algebraic local-triviality in semi-algebraic mappings, Amer. J. Math., Volume 102 (1980) no. 2, pp. 291-302

[6] K. Kurdyka On a subanalytic stratification satisfying a Whitney property with exponent 1, Real Algebraic Geometry (Rennes, 1991), Lecture Notes in Math., vol. 1524, Springer, Berlin, 1992, pp. 316-322

[7] J.-M. Lions; J.-P. Rolin Théorème de préparation pour les fonctions logarithmico-exponentielles, Ann. Inst. Fourier (Grenoble), Volume 47 (1997) no. 3, pp. 859-884

[8] T. Mostowski Lipschitz equisingularity, Dissertationes Math. (Rozprawy Mat.), Volume 243 (1985), p. 46

[9] A. Parusiński Lipschitz stratification of subanalytic sets, Ann. Sci. École Norm. Sup. (4), Volume 27 (1994) no. 6, pp. 661-696

[10] A. Parusiński On the preparation theorem for subanalytic functions, New Developments in Singularity Theory (Cambridge, 2000), NATO Sci. Ser. II Math. Phys. Chem., vol. 21, Kluwer Academic, Dordrecht, 2001, pp. 193-215

[11] A. Parusiński Regular projections for sub-analytic sets, C. R. Acad. Sci. Paris Sér. I Math., Volume 307 (1988) no. 7, pp. 343-347

[12] G. Valette. Lipschitz triangulations, preprint

[13] L. van den Dries, P. Speissegger, O-minimal preparation theorems, in press

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