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Homotopical variations and high-dimensional Zariski-van Kampen theorems

Authors: D. Chéniot and C. Eyral
Journal: Trans. Amer. Math. Soc. 358 (2006), 1-10
MSC (2000): Primary 14F35; Secondary 14D05, 32S50, 55Q99
Published electronically: August 25, 2005
MathSciNet review: 2171220
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Abstract: We give a new definition of the homotopical variation operators occurring in a recent high-dimensional Zariski-van Kampen theorem, a definition which opens the way to further generalizations of theorems of this kind.

References [Enhancements On Off] (What's this?)

  • [AGV] V.I. Arnol'd, S.M. Gusein-Zade, A.N. Varchenko, Singularities of differentiable maps, Vol. II, Monodromy and asymptotics of integrals (Birkhäuser Boston, Inc., Boston, MA, 1988). MR 0966191 (89g:58024)
  • [C1] D. Cheniot, Une démonstration du théorème de Zariski sur les sections hyperplanes d’une hypersurface projective et du théorème de Van Kampen sur le groupe fondamental du complémentaire d’une courbe projective plane, Compositio Math. 27 (1973), 141–158 (French). MR 0366922
  • [C2] Denis Chéniot, Topologie du complémentaire d’un ensemble algébrique projectif, Enseign. Math. (2) 37 (1991), no. 3-4, 293–402 (French). MR 1151752
  • [C3] D. Chéniot, Vanishing cycles in a pencil of hyperplane sections of a non-singular quasi-projective variety, Proc. London Math. Soc. (3) 72 (1996), no. 3, 515–544. MR 1376767,
  • [CL] D. Chéniot and A. Libgober, Zariski-van Kampen theorem for higher-homotopy groups, J. Inst. Math. Jussieu 2 (2003), no. 4, 495–527. MR 2006797,
  • [GM1] M. Goresky and R. MacPherson, ``Stratified Morse theory'', Singularities, Part 1 (Arcata, CA, 1981), Proceedings of Symposia in Pure Mathematics 40 (American Mathematical Society, Providence, RI, 1983) 517-533. MR 0713089 (84k:58017)
  • [GM2] M. Goresky and R. MacPherson, Stratified Morse theory (Springer-Verlag, New-York, 1988). MR 0932724 (90d:57039)
  • [HL] H.A. Hamm and D.T. Lê, ``Lefschetz theorems on quasi-projective varieties'', Bull. Soc. Math. France 113 (1985) 123-142. MR 0820315 (87i:32017)
  • [HW] P. J. Hilton and S. Wylie, Homology theory: An introduction to algebraic topology, Cambridge University Press, New York, 1960. MR 0115161
  • [LT] D.T. Lê and B. Teissier, ``Cycles évanescents, sections planes et conditions de Whitney II'', Singularities, Part 2 (Arcata, CA, 1981), Proceedings of Symposia in Pure Mathematics 40 (American Mathematical Society, Providence, RI, 1983) 65-103. MR 0713238 (86c:32005)
  • [Li] A. Libgober, Homotopy groups of the complements to singular hypersurfaces. II, Ann. of Math. (2) 139 (1994), no. 1, 117–144. MR 1259366,
  • [M] William S. Massey, A basic course in algebraic topology, Graduate Texts in Mathematics, vol. 127, Springer-Verlag, New York, 1991. MR 1095046
  • [Sp] Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
  • [St] Norman Steenrod, The Topology of Fibre Bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N. J., 1951. MR 0039258
  • [vK] E.R. van Kampen, ``On the fundamental group of an algebraic curve'', Amer. J. Math. 55 (1933) 255-260.
  • [Wh] Hassler Whitney, Tangents to an analytic variety, Ann. of Math. (2) 81 (1965), 496–549. MR 0192520,
  • [Za] O. Zariski, ``On the problem of existence of algebraic functions of two variables possessing a given branch curve'', Amer. J. Math. 51 (1929) 305-328.

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Additional Information

D. Chéniot
Affiliation: LATP, URA CNRS 225, Centre de Mathématiques et Informatique, Université de Provence, 39 rue F. Joliot-Curie, 13453 Marseille cédex 13, France

C. Eyral
Affiliation: Department of Mathematics, The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34014 Trieste, Italy

Keywords: Homotopy groups of algebraic varieties, pencils of hyperplanes, monodromies
Received by editor(s): December 9, 2002
Published electronically: August 25, 2005
Article copyright: © Copyright 2005 American Mathematical Society

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