Advertisement

On Equisingularity of Families of Maps (ℂn, 0) → (ℂn+1, 0)

  • Kevin Houston
Conference paper
  • 504 Downloads
Part of the Trends in Mathematics book series (TM)

Abstract

A classical theorem of Briançon, Speder and Teissier states that a family of isolated hypersurface singularities is Whitney equisingular if, and only if, the μ*-sequence for a hypersurface is constant in the family. This paper shows that the constancy of relative polar multiplicities and the Euler characteristic of the Milnor fibres of certain families of non-isolated singularities is equivalent to the Whitney equisingularity of a family of corank 1 maps from n-space to n + 1-space. The number of invariants needed is 4n − 2, which greatly improves previous general estimates.

Keywords

Whitney stratification equisingularity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    T. Gaffney, Polar multiplicities and equisingularity of map germs, Topology, 32 (1993), 185–223.CrossRefMathSciNetGoogle Scholar
  2. [2]
    T. Gaffney and D.B. Massey, Trends in Equisingularity, in Singularity Theory, eds Bill Bruce and David Mond, London Math. Soc. Lecture Notes 263, Cambridge University Press 1999, 207–248.Google Scholar
  3. [3]
    T. Gaffney and R. Gassler, Segre numbers and hypersurface singularities, J. Algebraic Geometry, 8, (1999), 695–736.MathSciNetGoogle Scholar
  4. [4]
    M. Goresky and R. Macpherson, Stratified Morse Theory, Springer Verlag, Berlin, 1988.zbMATHGoogle Scholar
  5. [5]
    V.V. Goryunov, Semi-simplicial resolutions and homology of images and discriminants of mappings, Proc. London Math. Soc. 70 (1995), 363–385.MathSciNetGoogle Scholar
  6. [6]
    K. Houston, On the topology of augmentations and concatenations of singularities, Manuscripta Mathematica 117, (2005), 383–405.CrossRefMathSciNetGoogle Scholar
  7. [7]
    V.H. Jorge Pérez, Polar multiplicities and equisingularity of map germs from ℂ3 to ℂ3, Houston Journal of Mathematics 29, (2003), 901–924.MathSciNetGoogle Scholar
  8. [8]
    V.H. Jorge Pérez, Polar multiplicities and Equisingularity of map germs from ℂ3 to ℂ4, in Real and Complex Singularities, eds. David Mond and Marcelo José Saia, Dekker Lecture Notes in Pure and Applied Mathematics Vol 232, Marcel Dekker 2003, 207–226.Google Scholar
  9. [9]
    V.H. Jorge Pérez and Marcelo Saia, Euler obstruction, Polar multiplicities and equisingularity of map germs in O(n, p), n < p, Notas do ICMC, no 157, 2002.Google Scholar
  10. [10]
    D. Massey, Numerical invariants of perverse sheaves, Duke Mathematical Journal, 73(2), (1994), 307–369.CrossRefMathSciNetGoogle Scholar
  11. [11]
    D. Massey, Lê Cycles and Hypersurface Singularities, Springer Lecture Notes in Mathematics 1615, (1995).Google Scholar
  12. [12]
    D. Mond, Vanishing cycles for analytic maps, in Singularity Theory and its Applications, SLNM 1462, D. Mond, J. Montaldi (Eds.), Springer Verlag Berlin, 1991, pp. 221–234.CrossRefGoogle Scholar
  13. [13]
    B. Teissier, Multiplicités polaires, section planes, et conditions de Whitney, in Algebraic Geometry, Proc. La Rábida, 1981, eds J.M. Aroca, R. Buchweitz, M. Giusti, and M. Merle, Springer Lecture Notes In Math 961 (1982), 314–491.Google Scholar
  14. [14]
    M. Tibăr, Bouquet decomposition of the Milnor fiber, Topology 35, (1996), 227–241.CrossRefMathSciNetGoogle Scholar
  15. [15]
    M. Tibăr, Limits of tangents and minimality of complex links, Topology 42, (2003), 629–639.CrossRefMathSciNetGoogle Scholar
  16. [16]
    C.T.C. Wall, Finite determinacy of smooth map-germs, Bull. Lond. Math. Soc., 13 (1981), 481–539.MathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • Kevin Houston
    • 1
  1. 1.School of MathematicsUniversity of LeedsLeedsUK

Personalised recommendations