Geometry & Topology

Definable triangulations with regularity conditions

Małgorzata Czapla

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We prove that every definable in an o-minimal structure set has a definable triangulation which is locally Lipschitz and weakly bi-Lipschitz on the natural simplicial stratification of a simplicial complex. We also distinguish a class T of regularity conditions and give a universal construction of a definable triangulation with a T condition of a definable set. This class includes the Whitney (B) and the Verdier conditions.

Article information

Geom. Topol., Volume 16, Number 4 (2012), 2067-2095.

Received: 26 February 2011
Revised: 18 March 2012
Accepted: 23 April 2012
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 14P05: Real algebraic sets [See also 12D15, 13J30] 14P10: Semialgebraic sets and related spaces 32B20: Semi-analytic sets and subanalytic sets [See also 14P15] 32B25: Triangulation and related questions

Weakly Lipschitz mapping definable triangulation Whitney (B) condition Verdier condition


Czapla, Małgorzata. Definable triangulations with regularity conditions. Geom. Topol. 16 (2012), no. 4, 2067--2095. doi:10.2140/gt.2012.16.2067.

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  • H Brodersen, D Trotman, Whitney $\mathrm{(b)}$–regularity is weaker than Kuo's ratio test for real algebraic stratifications, Math. Scand. 45 (1979) 27–34
  • M Coste, An introduction to o-minimal geometry, doctoral thesis, Pisa (2000)
  • M Coste, M Reguiat, Trivialités en famille, from: “Real algebraic geometry (Rennes, 1991)”, Lecture Notes in Math. 1524, Springer, Berlin (1992) 193–204
  • M Czapla, Invariance of regularity conditions under definable, locally Lipschitz, weakly bi–Lipschitz mappings, Ann. Polon. Math. 97 (2010) 1–21
  • J Dieudonné, Foundations of modern analysis, Pure and Applied Mathematics, Vol. X, Academic Press, New York (1960)
  • J Dugundji, A Granas, Fixed point theory, vol. I, Polish Scientific Publishers, Warsaw (1982)
  • R Engelking, K Sieklucki, Geometria i topologia. Część II: Topologia, Biblioteka Matematyczna 54, Państwowe Wydawnictwo Naukowe (PWN), Warsaw (1980)
  • R M Hardt, Triangulation of subanalytic sets and proper light subanalytic maps, Invent. Math. 38 (1976/77) 207–217
  • H Hironaka, Subanalytic sets, from: “Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki”, Kinokuniya, Tokyo (1973) 453–493
  • H Hironaka, Triangulations of algebraic sets, from: “Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974)”, Amer. Math. Soc., Providence, R.I. (1975) 165–185
  • T L Loi, Whitney stratification of sets definable in the structure $\mathbb{R}_{\exp}$, from: “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, Triangulation of semi-analytic sets, Ann. Scuola Norm. Sup. Pisa 18 (1964) 449–474
  • S Łojasiewicz, Ensembles semi-analytiques, preprint, IHES, Bur-sur-Yvette (1965)
  • S Łojasiewicz, Stratifications et triangulations sous-analytiques, from: “Geometry Seminars, 1986 (Italian) (Bologna, 1986)”, Univ. Stud. Bologna (1988) 83–97
  • S Łojasiewicz, J Stasica, K Wachta, Stratifications sous-analytiques. Condition de Verdier, Bull. Polish Acad. Sci. Math. 34 (1986) 531–539
  • M Shiota, Geometry of subanalytic and semialgebraic sets, Progress in Mathematics 150, Birkhäuser, Boston, MA (1997)
  • M Shiota, Whitney triangulations of semialgebraic sets, Ann. Polon. Math. 87 (2005) 237–246
  • G Valette, Lipschitz triangulations, Illinois J. Math. 49 (2005) 953–979
  • L van den Dries, Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series 248, Cambridge Univ. Press (1998)
  • L van den Dries, C Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996) 497–540
  • H Whitney, Tangents to an analytic variety, Ann. of Math. 81 (1965) 496–549