Stratified transversality by isotopy

Murolo, C.; Trotman, D.; Du Plessis, A.

doi:10.1090/S0002-9947-03-03236-7

Stratified transversality by isotopy
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by C. Murolo, D. J. A. Trotman and A. A. Du Plessis

Trans. Amer. Math. Soc. 355 (2003), 4881-4900

DOI: https://doi.org/10.1090/S0002-9947-03-03236-7

Published electronically: July 28, 2003

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Abstract:

For $\mathcal {X}$ an abstract stratified set or a $(w)$-regular stratification, hence for any $(b)$-, $(c)$- or $(L)$-regular stratification, we prove that after stratified isotopy of $\mathcal {X}$, a stratified subspace $\mathcal {W}$ of $\mathcal {X}$, or a stratified map $h : \mathcal {Z} \to \mathcal {X}$, can be made transverse to a fixed stratified map $g: \mathcal {Y} \to \mathcal {X}$.

References

Ralph Abraham and Joel Robbin, Transversal mappings and flows, W. A. Benjamin, Inc., New York-Amsterdam, 1967. An appendix by Al Kelley. MR 0240836
K. Bekka, Sur les propriétés topologiques et métriques des espaces stratifiés, thesis, University of Paris-Sud, Orsay, 1988.
K. Bekka, C-régularité et trivialité topologique, Singularity theory and its applications, Part I (Coventry, 1988/1989) Lecture Notes in Math., vol. 1462, Springer, Berlin, 1991, pp. 42–62 (French). MR 1129023, DOI 10.1007/BFb0086373
K. Bekka, Isotopy theorem, preprint, University of Liverpool, 1991.
Karim Bekka and Claudio Murolo, Homologie d’espaces stratifiés, C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), no. 9, 703–708 (French, with English and French summaries). MR 1797755, DOI 10.1016/S0764-4442(00)01655-4
Hans Brodersen and David Trotman, Whitney ${\rm (}b)$-regularity is weaker than Kuo’s ratio test for real algebraic stratifications, Math. Scand. 45 (1979), no. 1, 27–34. MR 567430, DOI 10.7146/math.scand.a-11822
William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620, DOI 10.1007/978-3-662-02421-8
Christopher G. Gibson, Klaus Wirthmüller, Andrew A. du Plessis, and Eduard J. N. Looijenga, Topological stability of smooth mappings, Lecture Notes in Mathematics, Vol. 552, Springer-Verlag, Berlin-New York, 1976. MR 0436203
Claude Godbillon, Géométrie différentielle et mécanique analytique, Hermann, Paris, 1969 (French). MR 0242081
M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics, Vol. 14, Springer-Verlag, New York-Heidelberg, 1973. MR 0341518
M. Goresky, Cohomology and homology of stratified objects, thesis, Brown University, 1976.
R. Mark Goresky, Whitney stratified chains and cochains, Trans. Amer. Math. Soc. 267 (1981), no. 1, 175–196. MR 621981, DOI 10.1090/S0002-9947-1981-0621981-X
Mark Goresky and Robert MacPherson, Intersection homology theory, Topology 19 (1980), no. 2, 135–162. MR 572580, DOI 10.1016/0040-9383(80)90003-8
Mark Goresky and Robert MacPherson, Stratified Morse theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 14, Springer-Verlag, Berlin, 1988. MR 932724, DOI 10.1007/978-3-642-71714-7
T.-C. Kuo, The ratio test for analytic Whitney stratifications, Proceedings of Liverpool Singularities—Symposium, I (1969/70), Lecture Notes in Mathematics, Vol. 192, Springer, Berlin, 1971, pp. 141–149. MR 0279333
Ta Lê Loi, Verdier and strict Thom stratifications in o-minimal structures, Illinois J. Math. 42 (1998), no. 2, 347–356. MR 1612771
John N. Mather, Stability of $C^{\infty }$ mappings. II. Infinitesimal stability implies stability, Ann. of Math. (2) 89 (1969), 254–291. MR 259953, DOI 10.2307/1970668
John N. Mather, Stratifications and mappings, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971) Academic Press, New York, 1973, pp. 195–232. MR 0368064
J. Mather, Notes on topological stability, Mimeographed notes, Harvard University, 1970.
C. McCrory, Poincaré duality in spaces with singularities, Thesis, Brandeis University, 1972.
Clint McCrory, Stratified general position, Algebraic and geometric topology (Proc. Sympos., Univ. California, Santa Barbara, Calif., 1977) Lecture Notes in Math., vol. 664, Springer, Berlin, 1978, pp. 142–146. MR 518413
Claudio Murolo, Whitney homology, cohomology and Steenrod squares, Ricerche Mat. 43 (1994), no. 2, 175–204. MR 1324747
Claudio Murolo, The Steenrod $p$-powers in Whitney cohomology, Topology Appl. 68 (1996), no. 2, 133–151. MR 1374078, DOI 10.1016/0166-8641(95)00043-7
C. Murolo, Semidifférentiabilité, transversalité et homologie de stratifications régulières, thesis, University of Provence, 1997.
Tomasz Brzeziński and Shahn Majid, Quantum geometry of algebra factorisations and coalgebra bundles, Comm. Math. Phys. 213 (2000), no. 3, 491–521. MR 1785427, DOI 10.1007/PL00005530
Claudio Murolo and David Trotman, Relèvements continus de champs de vecteurs, Bull. Sci. Math. 125 (2001), no. 4, 253–278 (French, with English and Chinese summaries). MR 1831981, DOI 10.1016/S0007-4497(00)01072-1
Hiroko Natsume, The realization of abstract stratified sets, Kodai Math. J. 3 (1980), no. 1, 1–7. MR 569526
L. Noirel, Plongements sous-analytiques d’espaces stratifiés de Thom-Mather, thesis, University of Provence, 1996.
Adam Parusiński, Lipschitz stratification, Global analysis in modern mathematics (Orono, ME, 1991; Waltham, MA, 1992) Publish or Perish, Houston, TX, 1993, pp. 73–89. MR 1278751
A. A. du Plessis, Continuous controlled vector fields, Singularity theory (Liverpool, 1996) London Math. Soc. Lecture Note Ser., vol. 263, Cambridge Univ. Press, Cambridge, 1999, pp. xviii–xix, 189–197. MR 1709353
Michael Teufel, Abstract prestratified sets are $(b)$-regular, J. Differential Geometry 16 (1981), no. 3, 529–536 (1982). MR 654642
R. Thom, Ensembles et morphismes stratifiés, Bull. Amer. Math. Soc. 75 (1969), 240–284 (French). MR 239613, DOI 10.1090/S0002-9904-1969-12138-5
David J. A. Trotman, Geometric versions of Whitney regularity for smooth stratifications, Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 4, 453–463. MR 565466
Jean-Louis Verdier, Stratifications de Whitney et théorème de Bertini-Sard, Invent. Math. 36 (1976), 295–312 (French). MR 481096, DOI 10.1007/BF01390015
Andrei Verona, Stratified mappings—structure and triangulability, Lecture Notes in Mathematics, vol. 1102, Springer-Verlag, Berlin, 1984. MR 771120, DOI 10.1007/BFb0101672
Hassler Whitney, Local properties of analytic varieties, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 205–244. MR 0188486