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Transversality theorems for the weak topology

Author: Saurabh Trivedi
Journal: Proc. Amer. Math. Soc. 141 (2013), 2181-2192
MSC (2010): Primary 58A35, 57R35; Secondary 32H02, 32Q28, 32S60
Published electronically: January 11, 2013
MathSciNet review: 3034444
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Abstract: In 1979, Trotman proved, using the techniques of the Thom transversality theorem, that under some conditions on the dimensions of the manifolds under consideration, openness of the set of maps transverse to a stratification in the strong (Whitney) topology implies that the stratification is $ (a)$-regular. Here we first discuss the Thom transversality theorem for the weak topology and then give a similiar kind of result for the weak topology under very weak hypotheses. Recently, several transversality theorems have been proved for complex manifolds and holomorphic maps. In view of these transversality theorems we also prove a result analogous to Trotman's result in the complex case.

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  • 1. A. du Plessis and H. Vosegaard, Characterisation of strong smooth stability, Math. Scand. 88 (2001), no. 2, 193-228. MR 1839573 (2002e:58074)
  • 2. A. du Plessis and C. T. C. Wall, The geometry of topological stability, London Mathematical Society Monographs, New Series, vol. 9, The Clarendon Press, Oxford University Press, New York, 1995. MR 1408432 (97k:58024)
  • 3. F. Forstnerič, Holomorphic flexibility properties of complex manifolds, Amer. J. Math. 128 (2006), no. 1, 239-270. MR 2197073 (2006k:32024)
  • 4. M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics, Vol. 14, Springer-Verlag, New York, 1973. MR 0341518 (49:6269)
  • 5. M. Goresky and R. D. 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 0932724 (90d:57039)
  • 6. M. W. Hirsch, Differential topology, Graduate Texts in Mathematics, Springer-Verlag, corrected 6th printing, 1997.
  • 7. S. Kaliman and M. Zaĭdenberg, A transversality theorem for holomorphic mappings and stability of Eisenman-Kobayashi measures, Trans. Amer. Math. Soc. 348 (1996), no. 2, 661-672. MR 1321580 (96g:32043)
  • 8. T. L. Loi, Transversality theorem in o-minimal structures, Compos. Math. 144 (2008), no. 5, 1227-1234. MR 2457526 (2010h:03052)
  • 9. G. Ricketts, Transversality and stratifications, M.Sc. Thesis, La Trobe University (1979).
  • 10. D. J. A. Trotman, Stability of transversality to a stratification implies Whitney $ (a)$-regularity, Invent. Math. 50 (1979), no. 3, 273-277. MR 520929 (80b:58015)
  • 11. H. Whitney, Tangents to an analytic variety, Ann. of Math. (2) 81 (1965), 496-549. MR 0192520 (33:745)

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Additional Information

Saurabh Trivedi
Affiliation: LATP (UMR 6632), Centre de Mathématiques et Informatique, Université de Provence, 39 rue Joliot-Curie, 13013 Marseille, France
Address at time of publication: LATP (UMR 7353), Centre de Mathématiques et Informatique, Aix-Marseille Université, 39 rue Joliot-Curie, 13453 Marseille Cedex 13, France

Keywords: Transversality, stratifications, weak topology
Received by editor(s): July 13, 2011
Received by editor(s) in revised form: September 5, 2011, and September 28, 2011
Published electronically: January 11, 2013
Communicated by: Franc Forstneric
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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