Skip to main content
SpringerLink
Log in
Book cover

Proceedings of the International Congress of Mathematicians pp 209–226Cite as

Birkhäuser

Topology of Discriminants and Their Complements

  • Conference paper

Abstract

The general notion of a discriminant is as follows. Consider any function space \(\mathcal{F}\), finite dimensional or not, and some class of singularities S that the functions from \(\mathcal{F}\) can take at the points of the issue manifold. The corresponding discriminant variety ∑(S) ⊂ \(\mathcal{F}\) is the space of all functions that have such singular points. For example, let \(\mathcal{F}\) be the space of (real or complex) polynomials of the form

$${x^d} + {a_1}{x^{d - 1}} + \cdots + {a^{d,}}$$
(1)

and S = {a multiple root}.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J.F. Adams, On the cobar construction, Colloque de topologie algebrique (Lou-vain, 1956). — Paris a.o., George Thone, 1957, 81–87.

    Google Scholar 

  2. F. Aikardi, Tree-like curves, in: Singularities and Bifurcations (V. I. Arnold, ed.), Adv. in Soy. Math., 21, Amer. Math. Soc., Providence, RI, 1994, 1–31.

    Google Scholar 

  3. V.I. Arnold, The cohomology ring of the group of colored braids, Mat. Zametki (Math. Notes) 1969, 5:2, 227–231.

    Google Scholar 

  4. V.I. Arnold, On some topological invariants of algebraic functions, Trudy Moskov. Mat. Obshch. 1970, 21, 27–46, Engl. transl.: Trans. Moscow Math. Soc. 1970, 21, 30–52.

    Google Scholar 

  5. V.I. Arnold, Some unsolved problems in the singularity theory, Trudy Sem. S.L. Soboleva, 1, Novosibirsk, 1976, 5–15, Engl. transl. in: Singularities, Part I, Proc. Sympos. Pure Math. 40 (Arcata, California), AMS, Providence, RI, 1983.

    Google Scholar 

  6. V.I. Arnold, On some problems in singularity theory, in: Geometry and Analysis, V.K. Patodi Memorial Volume, Bombay, 1979, 1–10.

    Article  MathSciNet  Google Scholar 

  7. V.I. Arnold, The spaces of functions with mild singularities, Functional Anal. Appl. 1989, 23:3.

    Google Scholar 

  8. V.I. Arnold, Vassiliev’s theory of knots and discriminants, Proceedings of the First European Congress of Mathematicians, Paris, 1992, Birkhäuser, Basel and Boston, 1994.

    Google Scholar 

  9. V.I. Arnold, Plane curves, their invariants, perestroikas and classifications, in: Singularities and Bifurcations (V. I. Arnold, ed.), Adv. in Sov. Math. 21, Amer. Math. Soc., Providence, RI, 1994, 33–91.

    Chapter  Google Scholar 

  10. V.I. Arnold, V.A. Vassiliev, V.V. Goryunov, and O.V. Lyashko, Singularities I, Moscow, VINITI, 1–256. Engl. transl.: Encycl. Math. Sci., 6, Springer, Berlin and New York, 1993.

    Google Scholar 

  11. D. Bar-Natan, On the Vassiliev knot invariants, Topology, to appear.

    Google Scholar 

  12. D. Bar-Natan, Vassiliev homotopy strin.q link invariants, preprint, 1993.

    Google Scholar 

  13. Joan S. Birman, New points of view in knot and link theory, BAMS, 1993, 253–287.

    Google Scholar 

  14. Joan S. Birman and X.-S. Lin, Knot polynomials and Vassiliev’s invariants, Invent. Math. 1993, 111, 225–270.

    Article  MathSciNet  Google Scholar 

  15. A. Björner and V. Welker, The homology of “k-equal” manifolds and related partition lattices, Report No. 39, 1991/1992, Institut Mittag-Leffler, 1992.

    Google Scholar 

  16. E. Brieskorn, Sur les groupes de tresses (d’apres V.I. Arnold), Sém. Bourbaki, 1971/72, No. 401, Lecture Notes in Math. 317, Springer, Berlin and New York, 1973, 21–44.

    Google Scholar 

  17. P. Cartier, Construction combinatoire des invariants de Vassiliev-Kontsevich des noeuds, C.R. Acad. Sci. Paris 316 Série I 1993, 1205–1210.

    MATH  Google Scholar 

  18. K.T. Chen, Iterated path integrals, Bull. of Amer. Math. Soc., 1977, 83:5, 831–879.

    Article  MathSciNet  Google Scholar 

  19. F.R. Cohen, R.L. Cohen, B.M. Mann, and R.J. Milgram, The topology of rational functions and divisors of surfaces, Acta Math. 1991, 166, 163–221.

    Article  MathSciNet  Google Scholar 

  20. F.R. Cohen, T.J. Lada, and J.P. May, The homology of iterated loop spaces, Lecture Notes in Math. 533, Springer, Berlin and New York, 1976.

    Google Scholar 

  21. R. Fenn and P. Taylor, Introducing doodles, in: Topology of lowdimensional manifolds (R. Fenn, ed.), Lecture Notes in Math., 722, Springer, Berlin and New York, 1977, 37–43.

    Google Scholar 

  22. P. Freyd, D. Yetter, J. Hoste, W. Lickorish, K. Millet, and A. Ocneanu, A new polynomial invariant for knots and links, Bull. Amer. Math. Soc. 1985, 12, 183–193.

    Article  MathSciNet  Google Scholar 

  23. M. Goresky and R. MacPherson, Stratified Morse Theory, Springer, Berlin a.o., 1986.

    Google Scholar 

  24. M. Gromov, Partial Differential Relations, Springer, Berlin a.o., 1986.

    Book  Google Scholar 

  25. K. Igusa, Higher singularities of smooth functions are unnecessary, Ann. of Math. (2) 1984, 119, 1–58.

    Article  MathSciNet  Google Scholar 

  26. V.F.R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 1987, 126, 335–388.

    Article  MathSciNet  Google Scholar 

  27. M. Kontsevich, Vassiliev’s knot invariants, Adv. in Soy. Math., 16, part 2, Amer. Math. Soc., Providence, RI, 1993, 137–150.

    Google Scholar 

  28. X.-S. Lin, Milnor link invariants are all of finite type, Columbia Univ., preprint, 1992.

    Google Scholar 

  29. A.B. Merkov, in: Singularities and Bifurcations (V. I. Arnold, ed.), Adv. in Sov. Math., 21, Amer. Math. Soc., Providence, RI, 1994, 199–211.

    Google Scholar 

  30. R.J. Milgram, Iterated loop spaces, Ann. of Math. (2) 1966, 84, 386–403.

    Article  MathSciNet  Google Scholar 

  31. J. Milnor, Isotopy of links, in: Algebraic geometry and topology, Princeton Univ. Press, Princeton, NJ, 1957, 280–306.

    Google Scholar 

  32. J. Milnor, Morse theory, Ann. of Math. Stud., Princeton Univ. Press, Princeton NJ, 1963.

    Google Scholar 

  33. D. Mumford, Algebraic Geometry I. Complex Projective Varieties, Springer-Verlag, Berlin and New York, 1976.

    MATH  Google Scholar 

  34. S. Piunikhin, Combinatorial expression for universal Vassiliev link invariant, Harvard Univ., preprint, March 1993.

    Google Scholar 

  35. G.B. Segal, Configuration spaces and iterated loop spaces, Invent. Math. 1973, 21:3, 213–221.

    Article  MathSciNet  Google Scholar 

  36. G.B. Segal, Topology of spaces of rational functions, Acta Math. 1979, 143, 39–72.

    Article  MathSciNet  Google Scholar 

  37. S. Smale, The classification of immersions of spheres in Euclidean space, Ann. of Math. (2), 1959, 69:2, 327–344.

    Article  MathSciNet  Google Scholar 

  38. V. Snaith, A stable decomposition of S2 n S n X, J. London Math. Soc. (2) 1974, 577–583.

    Article  Google Scholar 

  39. V.A. Vassiliev, Stable cohomology of complements of the discriminants of the singularities of smooth functions, Russian Math. Surveys (Uspekhi) 1987, 42:2, 219–220 and Current Problems of Math., 33, Moscow, VINITI, 1988, 3–29; Engl. transl.: J. Soviet Math. 1990, 52:4, 3217–3230.

    Google Scholar 

  40. V.A. Vassiliev, Topology of spaces of functions without complicated singularities, Functional Anal. Appl., 1989, 23:4, 24–36.

    MathSciNet  Google Scholar 

  41. V.A. Vassiliev, Topology of complements to discriminants and loop spaces, in: Theory of Singularities and its Applications (V.I. Arnold ed.), Adv. in Sov. Math., 1, Amer. Math. Soc., Providence, RI, 1990, 9–21.

    Chapter  Google Scholar 

  42. V.A. Vassiliev, Cohomology of knot spaces, ibid, 23–69.

    Google Scholar 

  43. V.A. Vassiliev, A geometric realization of the homologies of classical Lie groups, and complexes, S-dual to flag manifolds, Algebra i Analiz 3, 1991, no. 4, 108–115; Engl. transl. in: St. Petersburg Math. J., 1992, 3, no. 4, 809–815.

    Google Scholar 

  44. V.A. Vassiliev, Stable homotopy type of the complement to affine plane arrangement, preprint, 1991.

    Google Scholar 

  45. V.A. Vassiliev, Complements of discriminants of smooth maps: Topology and applications. Revised ed., Transl. of Math. Monographs, 98, Amer. Math. Soc., Providence RI, 1994.

    Google Scholar 

  46. V.A. Vassiliev, Complexes of connected graphs, in: Proc. of I.M.Gelfand’s seminar, Birkhäuser, Basel, 1993, 223–235.

    Google Scholar 

  47. V.A. Vassiliev, Invariants of ornaments, in: Singularities and Bifurcations (V. I. Arnold, ed.), Adv. in Soy. Math., 21, AMS, Providence, RI, 1994, 225–262.

    Chapter  Google Scholar 

  48. V.A. Vassiliev, Ramified integrals, singularities and lacunas, Kluwer Academic Publishers, Dordrecht a.o., 1994, 304 pages.

    MATH  Google Scholar 

  49. P. Vogel, Invariants de Vassiliev des noeuds, Sem. Bourbaki, 45-eme annee, 1992–1993, 769, March 1993.

    Google Scholar 

  50. G.M. Ziegler and R.T. Zivaljevic, Homotopy type of arrangements via diagrams of spaces, Report No. 10 (1991/1992), Inst. Mittag-Leffler, 1991.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Independent University of Moscow, P.O. Box 230, Moscow, 117463, Russia

    Victor A. Vassiliev

  2. Steklov Mathematical Institute, Vavilova st. 42, Moscow, 117966, GSP-1, Russia

    Victor A. Vassiliev

Authors
  1. Victor A. Vassiliev
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Département de Mathématiques, EPFL, 1015, Laussane, Switzerland

    S. D. Chatterji

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Birkhäser Verlag, Basel, Switzerland

About this paper

Cite this paper

Vassiliev, V.A. (1995). Topology of Discriminants and Their Complements. In: Chatterji, S.D. (eds) Proceedings of the International Congress of Mathematicians. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-9078-6_16

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-9078-6_16

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9897-3

  • Online ISBN: 978-3-0348-9078-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation