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Lipschitz stratification of subanalytic sets. (English) Zbl 0819.32007

After the pioneering work of T. Mostowski [Diss. Math. 243, 46 pp. (1985; Zbl 0578.32020)], who showed the existence of Lipschitz stratifications in the complex analytic case, the author shows the existence of Lipschitz stratifications in the real, for subanalytic sets. This is by no means easy. The techniques of Mostowski do not apply in the real, although some ideas do.
The decomposition done by the author is similar to triangulation but technically more complicated. Some of the techniques existed already in [S. Łojasiewicz, “Ensembles semi-analytiques, multigraphic”, I.H.E.S., Bures-sur-Yvette (1965)], like \(L\)-regular sets and regular projections, but they are modernized and regular projections are chosen in a very subtle way (following T. Mostowski [loc. cit.]). Flattening techniques are delicate and everything is nicely put together.
The reviewer does not like the author’s treating of the bibliography. It is not recalled in any consistent way (references are often lacking), for instance [the reviewer, S. Łojasiewicz and J. Stasica, Bull. Acad. Pol. Sci., Ser. Sci. Math. 27, 529-536 (1979; Zbl 0435.32006)] should be quoted for the fiber cutting lemma, as it is done in [E. Bierstone and P. D. Milman, Publ. Math., Inst. Hautes Etud. Sci. 67, 5-42 (1988; Zbl 0674.32002)].
Except for this, the paper is really well written and very clear, despite the technicallity of the subject.

MSC:

32B20 Semi-analytic sets, subanalytic sets, and generalizations
32B25 Triangulation and topological properties of semi-analytic andsubanalytic sets, and related questions
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References:

[1] S. BANACH , Wstep do teorii funkcji rzeczywistych, Monografie matematyczne , Warszawa-Wroclaw, 1951 (in Polish). MR 13,216a
[2] E. BIERSTONE and P. D. MILMAN , Semianalytic and Subanalytic Sets (Publ. I.H.E.S., Vol. 67, 1988 , pp. 5-42). Numdam | MR 89k:32011 | Zbl 0674.32002 · Zbl 0674.32002
[3] E. BIERSTONE and P. D. MILMAN , Arc-analytic Functions (Invent. Math., Vol. 101, 1990 , pp. 411-424). MR 92a:32011 | Zbl 0723.32005 · Zbl 0723.32005
[4] H. BRODERSEN and D. TROTMAN , Whitney (b)-regularity is Weaker Than Kuo’s Ratio Test for Real Algebraic Stratifications (Math. Scand., Vol. 45, 1979 , p. 27-43). MR 81i:58008 | Zbl 0429.58001 · Zbl 0429.58001
[5] Z. DENKOWSKA , S. ŁOJASIEWICZ and J. STASICA , Certaines propriétés élémentaires des ensembles sous-analytiques (Bull. Acad. Polon. Sci. Sér. Math., Vol. 27, 1979 , pp. 529-535). MR 81i:32003 | Zbl 0435.32006 · Zbl 0435.32006
[6] C. G. GIBSON et al., Topological Stability of Smooth Mappings (Lecture Notes in Math., Vol. 552, Springer-Verlag, 1976 ). MR 55 #9151 | Zbl 0377.58006 · Zbl 0377.58006
[7] H. HIRONAKA , Introduction to Real Analytic Sets and Real Analytic maps , Instituto L. Tonelli, Pisa, 1973 . MR 57 #16665 · Zbl 0297.32008
[8] H. HIRONAKA , Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero I, II (Ann. of Math., Vol. 79, 1964 , n^\circ 2, pp. 109-326). MR 33 #7333 | Zbl 0122.38603 · Zbl 0122.38603
[9] H. HIRONAKA , M. LEJEUNE-JALABERT and B. TEISSIER , Platificateur local en géométrie analytique et aplatissement local (Astérisque, Vol. 7-8, 1973 , pp. 441-463). MR 53 #13636 | Zbl 0287.14007 · Zbl 0287.14007
[10] K. KURDYKA , Points réguliers d’un sous-analytique (Ann. Inst. Fourier, Grenoble, Vol. 38, 1988 , n^\circ 1, pp. 133-156). Numdam | MR 89g:32010 | Zbl 0619.32007 · Zbl 0619.32007
[11] K. KURDYKA , On a Subanalytic Stratification Satisfying a Whitney Property With Exponent 1, Real Algebraic Geometry, Proceedings , Rennes, 1991 , M. Coste et al., eds., (Lecture Notes in Math., Vol. 1524, 1992 , pp. 316-323). Zbl 0779.32006 · Zbl 0779.32006
[12] B. C. KOOPMAN and A. B. BROWN , On the covering of Analytic Loci by Complexes (Trans. AMS, Vol. 34, 1932 , pp. 231-251). MR 1501636 | Zbl 0004.13203 | JFM 58.1203.02 · Zbl 0004.13203
[13] T.-C. KUO , On Classification of Real Singularities (Invent. Math., Vol. 82, 1985 , pp. 257-262). MR 87d:58025 | Zbl 0587.32018 · Zbl 0587.32018
[14] S. ŁOJASIEWICZ , Ensembles semi-analytiques, multigraphic , I.H.E.S., Bures-sur-Yvette, 1965 .
[15] T. MOSTOWSKI , Lipschitz Equisingularity (Dissertationes Math., CCXLIII, 1985 , PWN, Warszawa). MR 87e:32008 | Zbl 0578.32020 · Zbl 0578.32020
[16] T. MOSTOWSKI , Tangent Cones and Lipschitz Stratifications, Singularities, Banach Center Publications (S. Łojasiewicz, ed.), Vol. XX, PWN, Warszawa, 1988 , pp. 303-322. MR 92f:32008 | Zbl 0662.32012 · Zbl 0662.32012
[17] T. MOSTOWSKI , A Criterion for Lipschitz Equisingularity (Bull. Acad. Polon. Sci. Sér. Math., Vol. 37, 1989 , pp. 109-116). MR 92b:32038 | Zbl 0761.32018 · Zbl 0761.32018
[18] T. OSHAWA , On the L2 Cohomology of Complex Spaces (Math. Z., Vol. 209, 1992 , pp. 519-530). Article | Zbl 0759.58002 · Zbl 0759.58002
[19] A. PARUSIŃSKI , Lipschitz Properties of Semi-analytic Sets (Ann. Inst. Fourier, Grenoble, Vol. 38, 1988 , n^\circ 4, pp. 1189-213). Numdam | MR 90e:32016 | Zbl 0631.32006 · Zbl 0631.32006
[20] A. PARUSIŃSKI , Lipschitz Stratification of Real Analytic Sets, Singularities, Banach Center Publications (S. Łojasiewicz, ed.), Vol. XX, PWN, Warszawa, 1988 , pp. 323-333. Zbl 0666.32011 · Zbl 0666.32011
[21] A. PARUSIŃSKI , Regular Projections for Sub-analytic Sets (C. R. Acad. Sci. Paris, Série I, t. 307, 1988 , pp. 343-347). MR 89k:32012 | Zbl 0649.32007 · Zbl 0649.32007
[22] A. PARUSIŃSKI , Subanalytic functions , preprint MPI/90-58 (Trans. AMS), to appear.
[23] A. PARUSIŃSKI , Lipschitz Stratification, Global Analysis in Modern Mathematics (K. Uhlenbeck, ed.), Proceedings of a Symposium in Honor of Richard Palais’ Sixtieth Birthday, Publish or Perish, Houston, 1993 , pp. 73-91. Zbl 0932.32010 · Zbl 0932.32010
[24] W. PAWLUCKI , Le théorème de Puiseux pour une application sous-analytique (Bull. Acad. Polon. Sci. Sér. Math., Vol. 32, 1984 , pp. 555-560). MR 86j:32015 | Zbl 0574.32010 · Zbl 0574.32010
[25] H. J. SUSSMANN , Real-analytic Desingularization and Subanalytic Sets : an Elementary Approach (Trans. A.M.S., Vol. 317, 1990 , n^\circ 2, pp. 417-461). MR 90e:32007 | Zbl 0696.32005 · Zbl 0696.32005
[26] L. SIEBENMANN and D. SULLIVAN , On Complexes that are Lipschitz Manifolds, Geometric Topology (J. Cantrell, ed.), Academic Press, New York, 1979 , pp. 503-552. MR 80h:57027 | Zbl 0478.57008 · Zbl 0478.57008
[27] J. L. VERDIER , Stratifications de Whitney et théorème de Bertini-Sard (Invent. Math., Vol. 36, 1976 , pp. 295-312). MR 58 #1242 | Zbl 0333.32010 · Zbl 0333.32010
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