Strict \(C^1\)-triangulations in o-minimal structures
type: Item
instance of: scholarly article
title: Strict \(C^1\)-triangulations in o-minimal structures
zbMATH Open document ID: 1423.32011
DOI: 10.12775/TMNA.2018.033
author: Małgorzata Czapla
author: Wiesław Pawłucki
published in: Topological Methods in Nonlinear Analysis
publication date: 2019-05-07T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.tmna/1532484293
review text: The authors prove a \(C^1\)-triangulation theorem in o-minimal structures. Let \(R\) be a real closed field and consider any o-minimal expansion of \(R\). It is shown that for any closed bounded definable subset \(A\) of \(R^n\) and any finite collection of definable subsets \(B_1,\ldots,B_r\) of \(A\) there exists a definable triangulation \(h : | \mathcal K | \to A\) of \(A\) compatible with \(B_1,\ldots,B_r\), where \(\mathcal K\) is a simplicial complex in \(R^n\), \(h\) is a \(C^1\)-embedding of each simplex \(\Delta \in \mathcal K\), and \(h\) extends to a definable \(C^1\)-mapping defined on a definable open neighborhood of the polyhedron \(|\mathcal K|\) in \(R^n\). This theorem improves a recent result of \textit{T. Ohmoto} and \textit{M. Shiota} [J. Topol. 10, No. 3, 765--775 (2017; Zbl 1376.14060)] in several different aspects; also the method of proof is different. The proof proceeds in two steps. First the existence of a definable \(C^1\)-triangulation of \(A\) compatible with \(B_1,\ldots,B_r\), where \(h : | \mathcal K | \to R^n\) is Lipschitz and \(\{h|_\Delta : \Delta \in \mathcal K\}\) is a \(C^1\)-stratification of \(h\) satisfying Whitney's condition (A), is shown. Then \(h\) is modified in some tubular neighborhoods of simplices in order to obtain a \(C^1\)-extension to a definable open neighborhood of \(|\mathcal K|\).
reviewed by: Armin Rainer
Mathematics Subject Classification ID: 32B25
Mathematics Subject Classification ID: 14P10
Mathematics Subject Classification ID: 03C64
Mathematics Subject Classification ID: 32B20
zbMATH DE Number: 7051690
zbMATH Keywords: triangulation
zbMATH Keywords: \(C^1\)-mapping
zbMATH Keywords: o-minimal structure
MaRDI profile type: Publication
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OpenAlex ID: W2883222042
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description: scientific article; zbMATH DE number 7051690