Selecta Mathematica

, Volume 1, Issue 3, pp 597–621 | Cite as

Stratified Picard-Lefschetz theory

  • Victor A. Vassiliev


The monodromy action in the homology of level sets of Morse functions on stratified singular analytic varieties is studied. The local variation operators in both the standard and the intersection homology groups defined by the loops around the critical values of such functions are reduced to similar operators in the homology groups of the transversal slices of the corresponding strata.


Analytic Variety Local Variation Variation Operator Similar Operator Homology Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Arnold]
    V.I. Arnold.A topological proof of transcendence of Abelian integrals in Newton's Principia. Quant,12 (1987), 1–15. See also V. I. Arnold.Huygens and Barrow, and Newton and Hooke — the first steps of the mathematical analysis and catastrophe theory, from the evolvents to quasicrystals, Nauka, Moscow, 1989; English translation: Birkhäuser.Google Scholar
  2. [ABG]
    M. F. Atiyah, R. Bott and L. Gårding.Lacunas for hyperbolic differential operators with constant coefficients. I, II. Acta Math.124 (1970), 109–189 and131 (1973), 145–206.Google Scholar
  3. [AV]
    V. I. Arnold and V. A. Vassiliev.Newton's Principia read 300 years later. Notices Amer. Math. Soc.,36:9 (1989), 1148–1154.Google Scholar
  4. [AVG 1,2]
    V. I. Arnold, A. N. Varchenko and S. M. Gusein-Zade.Singularities of differentiable maps. Nauka, Moscow, vol. 1, 2, 1982 and 1984; Engl. transl.: Birkhäuser, Basel, 1985 and 1988.Google Scholar
  5. [AVGL 1,2]
    V. I. Arnold, V. A. Vassiliev, V. V. Goryunov and O. V. Lyashko.Singularities I, II. Dynamical systems, VINITI, Moscow, vol. 6 and 39, 1988 and 1989; English transl.: Encycl. Math. Sci., vol. 6 and 39, Springer-Verlag, Berlin and New York, 1993.Google Scholar
  6. [GM 80]
    M. Goresky and R. MacPherson.Intersection homology theory. Topology,19 (1980), 135–162.Google Scholar
  7. [GM 86]
    M. Goresky and R. MacPherson.Stratified Morse theory. Springer-Verlag, Berlin and New York, 1986.Google Scholar
  8. [Milnor]
    J. Milnor.Singular points of complex hypersurfaces. Princeton Univ. Press, Princeton, NJ, and Univ. of Tokyo Press, Tokyo, 1968.Google Scholar
  9. [Newton]
    I. Newton.Philosophiae Naturalis Principia Mathematica. London, 1687.Google Scholar
  10. [Pham]
    F. Pham.Introduction à l'étude topologique des singularités de Landau. Gauthier-Villars, Paris, 1967.Google Scholar
  11. [V]
    V. A. Vassiliev.Ramified integrals, singularities and lacunas. Kluwer Academic Publishers, Dorderecht, Boston, London; 1994.Google Scholar

Copyright information

© Birkhäuser Verlag 1995

Authors and Affiliations

  • Victor A. Vassiliev
    • 1
    • 2
  1. 1.Steklov Mathematical InstituteMoscowGSP-1/Russia
  2. 2.Independent Moscow UniversityUSSR

Personalised recommendations