• David Angelo Trotman Aix-Marseille Université, France



Fibering, Stratifications, Triangulation, Whitney conditions.


We describe some old and new results about Whitney stratifications and state some open problems.


Download data is not yet available.


Bekka, K. (1991). C-régularité et trivialité topologique [C-regularity and topological triviality]. In D. Mond & J. Montaldi (Eds.), Singularity theory and its applications, Warwick 1989, Part I: Geometric aspects of singularities (Lecture Notes in Mathematics) (pp. 42-62). Springer.

Czapla, M. (2012). Definable triangulations with regularity conditions. Geometry & Topology, 16(4), 2067-2095.

Denkowska, Z., Wachta, K., & Stasica, J. (1985). Stratification des ensembles sous-analytiques avec les propriétés (A) et (B) de Whitney [Stratification of sub-analytical sets with Whitney properties (A) and (B)]. Universitatis Iagellonicae Acta Mathematica, 25, 183-188.

du Plessis, A. A. (1999). Continuous controlled vector fields. In W. Bruce & D. Mond, Singularity theory (Liverpool, 1996), London Mathematical Society Lecture Note Series, 263, (pp. 189-197). Cambridge University Press.

Feldman, E. A. (1965). The geometry of immersions. I. Transactions of the American Mathematical Society, 120, 185-224.

Goresky, R. M. (1981). Whitney stratified chains and cochains. Transactions of the American Mathematical Society, 267, 175-196.

Halupczok, I., & Yin, Y. (2018). Lipschitz stratifications in power-bounded o-minimal fields. Journal of the European Mathematical Society, 20(11), 2717-2767.

Hironaka, H. (1973). Subanalytic sets. In Y. Kusunoki (Ed.), Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki (pp. 453-493). Kinokuniya.

Hirsch, M. W. (1976). Differential Topology. Springer.

Łojasiewicz, S. (1965). Ensembles semi-analytiques [Semi-analytical sets]. Institut des Haute Études Scientiques Notes.

Łojasiewicz, S., Stasica, J., & Wachta, K. (1986). Stratifications sous-analytiques. Condition de Verdier [Sub-analytic stratifications. Verdier condition]. Bulletin Polish Academy of Sciences Mathematics, 34(9-10), 531-539.

Mather, J. N. (2012). Notes on topological stability. Bulletin of the American Mathematical Society (New Series), 49(4), 475-506.

Murolo, C., du Plessis, A., & Trotman. D. (2017). On the smooth Whitney fibering conjecture, preprint.

Nguyen, N., & Valette, G. (2016). Lipschitz stratifications in o-minimal structures. Annales Scientifique de l' École Normale Superieure, 49(4), 399-421.

Nguyen, N., & Valette, G. (2018). Whitney stratifications and the continuity of local Lipschitz-Killing curvatures. Annales de l’Institut Fourier (Grenoble), 68(5), 2253-2276.

Nguyen, N., Trivedi, S., & Trotman, D. (2014). A geometric proof of the existence of definable Whitney stratifications. Illinois Journal of Mathematics, 58(2), 381-389.

Parusiński, A. (1994). Lipschitz stratification of subanalytic sets. Annales Scientifiques de l’École Normale Superieure, 27(6), 661-696.

Parusiński, A., & Păunescu, L. (2017). Arc-wise analytic stratification, Whitney fibering conjecture and Zariski equisingularity. Advances in Mathematics, 309, 254-305.

Pawlucki, W. (1985). Quasiregular boundary and Stokes’s formula for a subanalytic leaf. Seminar on deformations (Lódz/Warsaw, 1982/84) Lecture Notes in Mathematics, 1165, (pp. 235-252). Springer.

Shiota, M. (1996). Geometry of subanalytic and semialgebraic sets. Birkhäuser.

Shiota, M. (2005). Whitney triangulations of semialgebraic sets. Annales Polonici Mathematici, 87, 237-246.

Ta, L. L. (1998). Verdier and strict Thom stratifications in o-minimal structures. Illinois Journal of Mathematics, 42(2), 347-356.

Thom, R. (1969). Ensembles et morphismes stratifiés [Sets and stratified morphisms]. Bulletin of the American Mathematical Society, 75, 240-284.

Trivedi, S., & Trotman, D. (2014). Detecting Thom faults in stratified mappings. Kodai Mathematical Journal, 37(2), 341-354.

Trotman, D. (1978/79). Stability of transversality to a stratification implies Whitney (a)-regularity. Inventiones Mathematicae, 50(3), 273-277.

Trotman, D. (2020). Stratification theory. In J. L. Cisneros-Molina, T. D. Lê, & J. Seade (Eds.), Handbook of geometry and topology of singularities I (pp. 231-260). Springer.

Trotman, D., & Valette, G. (2017). On the local geometry of definably stratified sets. In F. Broglia, F. Delon, M. Dickman, D. Gondard-Cozette, & V. A. Powers, (Eds.), Ordered algebraic structures and related topics. Contemporary Mathematics, 697, (pp. 349-366). American Mathematical Society.

Verdier, J.-L. (1976). Stratifications de Whitney et théorème de Bertini-Sard [Whitney stratifications and the Bertini-Sard theorem]. Inventiones Mathematicae, 36, 295-312.

Whitney, H. (1957). Elementary structure of real algebraic varieties. Annals of Mathematics, 66(3), 545-556.

Whitney, H. (1965a). Local properties of analytic varieties. In S. S. Cairns (Ed.), Differential and Combinatorial Topology: A Symposium in Honor of Marston Morse (pp. 205-244). Princeton University Press.

Whitney, H. (1965b). Tangents to an analytic variety. Annals of Mathematics, 81(3), 496-549.




Volume and Issues


Natural Sciences and Technology

How to Cite

Trotman, D. A. (2022). SOME GEOMETRIC PROPERTIES OF WHITNEY STRATIFICATIONS. Dalat University Journal of Science, 12(2), 78-85.