Semi-algebraic neighborhoods of closed semi-algebraic sets

[Voisinages semi-algébriques d’ensembles semi-algébriques fermés]

Dutertre, Nicolas

Annales de l'Institut Fourier, Tome 59 (2009), p. 429-458 / Harvested from Numdam

Résumé

Étant donné un ensemble semi-algébrique fermé (non nécessairement compact) $X$ de $\mathbb{R}^n$, nous construisons une fonction semi-algébrique $f$ positive et de classe ${\mathcal{C}}^2$ telle que $X=f^{-1}(0)$ et telle que pour $\delta >0$ suffisamment petit, l’inclusion de $X$ dans $f^{-1}([0,\delta ])$ soit une rétraction. En corollaire, nous obtenons plusieurs formules pour la caractéristique d’Euler de $X$.

Abstract

Given a closed (not necessarly compact) semi-algebraic set $X$ in $\mathbb{R}^n$, we construct a non-negative semi-algebraic ${\mathcal{C}}^2$ function $f$ such that ${X{=}f^{-1}(0)}$ and such that for $\delta >0$ sufficiently small, the inclusion of $X$ in $f^{-1}([0,\delta ])$ is a retraction. As a corollary, we obtain several formulas for the Euler characteristic of $X$.

Détail

Publié le : 2009-01-01

MR 2514870 | Zbl 1174.14051

DOI : https://doi.org/10.5802/aif.2435
Classification:  14P10,  14P25
Mots clés: voisinage tubulaire, ensembles semi-algébriques, rétraction, function semi-algébrique approchante quasirégulière, voisinage semi-algébrique approchant quasirégulier

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