SOME GEOMETRIC PROPERTIES OF WHITNEY STRATIFICATIONS

Authors

  • David Angelo Trotman Aix-Marseille Université, France

DOI:

https://doi.org/10.37569/DalatUniversity.12.2.921(2022)

Keywords:

Fibering, Stratifications, Triangulation, Whitney conditions.

Abstract

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

Downloads

Download data is not yet available.

References

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. https://doi.org/10.1007/BFb0086373

Czapla, M. (2012). Definable triangulations with regularity conditions. Geometry & Topology, 16(4), 2067-2095. https://doi.org/10.2140/gt.2012.16.2067

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. https://doi.org/10.1017/CBO9780511569265.013

Feldman, E. A. (1965). The geometry of immersions. I. Transactions of the American Mathematical Society, 120, 185-224. https://doi.org/10.1090/S0002-9947-1965-0185602-6

Goresky, R. M. (1981). Whitney stratified chains and cochains. Transactions of the American Mathematical Society, 267, 175-196. https://doi.org/10.1090/S0002-9947-1981-0621981-X

Halupczok, I., & Yin, Y. (2018). Lipschitz stratifications in power-bounded o-minimal fields. Journal of the European Mathematical Society, 20(11), 2717-2767. https://doi.org/10.4171/JEMS/823

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. https://doi.org/10.1007/978-1-4684-9449-5

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. https://doi.org/10.1090/S0273-0979-2012-01383-6

Murolo, C., du Plessis, A., & Trotman. D. (2017). On the smooth Whitney fibering conjecture, preprint. https://hal.archives-ouvertes.fr/hal-01571382

Nguyen, N., & Valette, G. (2016). Lipschitz stratifications in o-minimal structures. Annales Scientifique de l' École Normale Superieure, 49(4), 399-421. https://doi.org/10.24033/asens.2286

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. https://doi.org/10.5802/aif.3208

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. https://doi.org/10.1215/ijm/1436275489

Parusiński, A. (1994). Lipschitz stratification of subanalytic sets. Annales Scientifiques de l’École Normale Superieure, 27(6), 661-696. https://doi.org/10.24033/asens.1703

Parusiński, A., & Păunescu, L. (2017). Arc-wise analytic stratification, Whitney fibering conjecture and Zariski equisingularity. Advances in Mathematics, 309, 254-305. https://doi.org/10.1016/j.aim.2017.01.016

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. https://doi.org/10.1007/BFb0076157

Shiota, M. (1996). Geometry of subanalytic and semialgebraic sets. Birkhäuser. https://doi.org/10.1007/978-1-4612-2008-4

Shiota, M. (2005). Whitney triangulations of semialgebraic sets. Annales Polonici Mathematici, 87, 237-246. https://doi.org/10.4064/ap87-0-20

Ta, L. L. (1998). Verdier and strict Thom stratifications in o-minimal structures. Illinois Journal of Mathematics, 42(2), 347-356. https://doi.org/10.1215/ijm/1256045049

Thom, R. (1969). Ensembles et morphismes stratifiés [Sets and stratified morphisms]. Bulletin of the American Mathematical Society, 75, 240-284. https://doi.org/10.1090/S0002-9904-1969-12138-5

Trivedi, S., & Trotman, D. (2014). Detecting Thom faults in stratified mappings. Kodai Mathematical Journal, 37(2), 341-354. https://doi.org/10.2996/kmj/1404393891

Trotman, D. (1978/79). Stability of transversality to a stratification implies Whitney (a)-regularity. Inventiones Mathematicae, 50(3), 273-277. https://doi.org/10.1007/BF01410081

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. https://doi.org/10.1007/978-3-030-53061-7_4

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. https://doi.org/10.1090/conm/697/14061

Verdier, J.-L. (1976). Stratifications de Whitney et théorème de Bertini-Sard [Whitney stratifications and the Bertini-Sard theorem]. Inventiones Mathematicae, 36, 295-312. https://doi.org/10.1007/BF01390015

Whitney, H. (1957). Elementary structure of real algebraic varieties. Annals of Mathematics, 66(3), 545-556. https://doi.org/10.2307/1969908

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. https://doi.org/10.1515/9781400874842-014

Whitney, H. (1965b). Tangents to an analytic variety. Annals of Mathematics, 81(3), 496-549. https://doi.org/10.2307/1970400

Downloads

Published

28-01-2022

Volume and Issues

Section

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. https://doi.org/10.37569/DalatUniversity.12.2.921(2022)