Topological Methods in Nonlinear Analysis

Strict $C^1$-triangulations in o-minimal structures

Małgorzata Czapla and Wiesław Pawłucki

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Inspired by the recent articles of T. Ohmoto and M. Shiota [9], [10] on $C^1$-triangulations of semialgebraic sets, we prove here by using different methods the following theorem: Let $R$ be a real closed field and let an expansion of $R$ to an o-minimal structure be given. Then for any closed bounded definable subset $A$ of $ R^n$ and a finite family $B_1,\dots,B_r$ of definable subsets of $A$ there exists a definable triangulation $h\colon |\mathcal K|\rightarrow A$ of $A$ compatible with $B_1,\dots,B_r$ such that $\mathcal K$ is a simplicial complex in $R^n$, $h$ is a $C^1$-embedding of each (open) simplex $\Delta\in \mathcal K$ and $h$ extends to a definable $C^1$-mapping defined on a definable open neighborhood of $|\mathcal K|$ in $R^n$. This improves Ohmoto-Shiota's theorem in three ways; firstly, $h$ is a $C^1$-embedding on each simplex; secondly, the simplicial complex $\mathcal K$ is in the same space as $A$ and thirdly, our proof is performed for any o-minimal structure. The possibility to have $h$ with the first of these properties was stated by Ohmoto and Shiota as an open problem (see [9]).

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Topol. Methods Nonlinear Anal., Volume 52, Number 2 (2018), 739-747.

First available in Project Euclid: 25 July 2018

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Czapla, Małgorzata; Pawłucki, Wiesław. Strict $C^1$-triangulations in o-minimal structures. Topol. Methods Nonlinear Anal. 52 (2018), no. 2, 739--747. doi:10.12775/TMNA.2018.033.

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  • M. Coste, An Introduction to O-minimal Geometry, Dottorato di Ricerca in Matematica, Dipartimento di Matematica, Università di Pisa, Istituti Editoriali e Poligrafici Internazionali, Pisa, 2000.
  • M. Czapla, Invariance of regularity conditions under definable, locally Lipschitz, weakly bi-Lipschitz mappings, Ann. Polon. Math. 97 (2010), 1–21.
  • M. Czapla, Definable triangulations with regularity conditions, Geom.Topol. 16 (2012), no. 4, 2067–2095.
  • K. Kurdyka and W. Pawłucki, O-minimal version of Whitney's extension theorem, Studia Math. 224 (2014), no. 1, 81–96.
  • T.L. Loi, Whitney stratifications of sets definable in the structure $\mathbb R_{\rom{exp}}$, Singularities and Differential Equations (Warsaw, 1993), Banach Center Publ. 33, Polish Acad. Sci., Warsaw, 1996, 401–409.
  • T.L. Loi, Verdier and strict Thom stratifications in o-minimal structures, Illinois J. Math. 42 (1998), 347–356.
  • S. Łojasiewicz Stratifications et triangulations sous-analytiques, Geometry Seminars, 1986 (Italian) (Bologna, 1986); Univ. Stud. Bologna, 1988, 83–97.
  • S. Łojasiewicz, J. Stasica and K. Wachta, Stratifications sous-analytiques. Condition de Verdier, Bull. Polish Acad. Sci. (Math.) 34 (1986), 531–539.
  • T. Ohmoto and M. Shiota, $\Cal C^1$-triangulations of semialgebraic sets, arXiv:1505.03970v1 [math AG] 15 May 2015.
  • T. Ohmoto and M. Shiota, $\Cal C^1$-triangulations of semialgebraic sets, J. Topol. 10 (2017), 765–775.
  • A. Thamrongthanyalak, Whitney's extension theorem in o-minimal structures Ann. Polon. Math. 119 (2017), no. 1, 49–67.
  • L. van den Dries, Tame Topology and O-minimal Structures, Cambridge University Press, 1998.