Definable triangulations with regularity conditions

Czapla, Małgorzata

doi:10.2140/gt.2012.16.2067

Cited by 7 publications

(1 citation statement)

References 12 publications

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“…In [4], the author made use of this theorem to show existence of triangulations inducing Whitney stratifications. In [18,19] the author of the present paper used the latter theorem, together with some improvements that are presented in section 4.5 below, so as to compute the L p cohomology of differential forms of bounded subanalytic manifolds, not necessarily complete.…”

Section: Introductionmentioning

confidence: 99%

On regular vectors and bi-Lipschitz triviality in o-minimal structures

Valette¹

2021

Preprint

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This article deals with o-minimal structures expanding the real field. We prove that, given a family definable in such a structure, there is a regular projection, up to a definable family of homeomorphisms that is uniformly bi-Lipschitz, in the sense that the Lipschitz constants can be bounded independently of the parameters. This is a parameterized version of a result of the author. In the polynomially bounded case, we give an elementary proof of the bi-Lipschitz version of Hardt's theorem of the author, which yields existence of definably bi-Lipschitz trivial stratifications.

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Section: Introductionmentioning

confidence: 99%

On regular vectors and bi-Lipschitz triviality in o-minimal structures

Valette¹

2021

Preprint

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Stratification Theory

Trotman¹

2020

Handbook of Geometry and Topology of Singularities I

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Stratifications, Equisingularity and Triangulation

Trotman

2020

Introduction to Lipschitz Geometry of Singularities

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This text is based on 3 lectures given in Cuernavaca in June 2018 about stratifications of real and complex analytic varieties and subanalytic and definable sets. The first lecture contained an introduction to Whitney stratifications, Kuo-Verdier stratifications and Mostowski's Lipschitz stratifications. The second lecture concerned equisingularity along strata of a regular stratification for the different regularity conditions: Whitney, Kuo-Verdier, and Lipschitz, including thus the Thom-Mather first isotopy theorem and its variants. (Equisingularity means continuity along each stratum of the local geometry at the points of the closures of the adjacent strata.) A short discussion follows of equisingularity for complex analytic sets including Zariski's problem about topological invariance of the multiplicity of complex hypersurfaces and its bilipschitz counterparts. In the real subanalytic (or definable) case we mention that equimultiplicity along a stratum translates as continuity of the density at points on the stratum, and quote the relevant results of Comte and Valette generalising Hironaka's 1969 theorem that complex analytic Whitney stratifications are equimultiple along strata. The third lecture provided further evidence of the tameness of Whitney stratified sets and of Thom maps, by describing triangulation theorems in the different categories, and including definable and Lipschitz versions. While on the subject of Thom maps we indicate examples of their use in complex equisingularity theory and in the definition of Bekka's (c)-regularity. Some very new results are described as well as old ones.

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Definable triangulations with regularity conditions

Cited by 7 publications

References 12 publications

On regular vectors and bi-Lipschitz triviality in o-minimal structures

On regular vectors and bi-Lipschitz triviality in o-minimal structures

Stratification Theory

Stratifications, Equisingularity and Triangulation

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