Advertisement

Journal of Soviet Mathematics

, Volume 52, Issue 4, pp 3217–3230 | Cite as

Stable cohomologies of the complements of the discriminants of deformations of singularities of smooth functions

  • V. A. Vasil'ev
Article

Abstract

Complements to discriminants of singularities of smooth functions are far generalizations of the classifying spaces of Artin and Brieskorn braid groups. A group of stable cohomologies (i.e., cohomologies preserved under adjacency of singularities) is described for these spaces. A relationship between these cohomologies and Gauss-Manin connectivity of singularities is indicated. A cellular realization of cohomologies of symmetric groups with coefficients in Z2 is described.

Keywords

Smooth Function Symmetric Group Braid Group Stable Cohomology 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literature cited

  1. 1.
    V. I. Arnol'd, “Braids of algebraic functions and cohomologies of swallowtails,” Usp. Mat. Nauk,23, No. 4, 247–248 (1968).Google Scholar
  2. 2.
    V. I. Arnol'd, “A note on branching of hyperelliptic integrals as functions of parameters,” Funkts. Anal. Prilozhen.,2, No. 3, 1–3 (1968).Google Scholar
  3. 3.
    V. I. Arnol'd, “Cohomology ring of the colored braid group,” Mat. Zametki,5, No. 2, 227–231 (1969).Google Scholar
  4. 4.
    V. I. Arnol'd, “On classes of cohomologies of an algebraic function invariant under Chirnhausen transformations,” Funkts. Anal. Prilozhen.,4, No. 1, 84–85 (1970).Google Scholar
  5. 5.
    V. I. Arnol'd, “On some topological invariants of algebraic functions,” Tr. Mosk. Mat. Obsch.,21, 27–46 (1970).Google Scholar
  6. 6.
    V. I. Arnol'd, “Topological invariants of algebraic functions, II,” Funkts. Anal. Prilozhen.,4, No. 2, 1–9 (1970).Google Scholar
  7. 7.
    V. I. Arnol'd, “Some unsolved problems of singularity theory,” Trudy Seminara S. L. Soboleva, No. 1, 5–15 (1976).Google Scholar
  8. 8.
    V. I. Arnol'd, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable Maps [in Russian], Vol. I, Nauka, Moscow (1982).Google Scholar
  9. 9.
    V. I. Arnol'd, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable Maps [in Russian], Vol. II, Nauka, Moscow (1982).Google Scholar
  10. 10.
    F. V. Vainshtein, “Cohomologies of braid groups,” Funkts. Anal. Prilozhen.,12, No. 2, 72–73 (1978).Google Scholar
  11. 11.
    A. N. Varchenko, “Asymptotic Hodge structure in vanishing cohomologies,” Izv. Akad. Nauk SSSR, Ser. Mat.,45, No. 3, 540–591 (1981).Google Scholar
  12. 12.
    A. N. Varchenko, “Asymptotics of integrals and Hodge structures,” Itogi Nauki i Tekhniki, Sovr. Probl. Mat.,22, 130–166 (1983).Google Scholar
  13. 13.
    A. M. Gabarielov, “Bifurcations, Dynkin diagrams, and modality of isolated singularities,” Funkts. Anal. Prilozhen.,8, No. 2, 7–12 (1974).Google Scholar
  14. 14.
    V. V. Goryunov, “Cohomologies of braid groups of series C and D and some stratifications,” Funkts. Anal. Prilozhen.,12, No. 2, 76–77 (1978).Google Scholar
  15. 15.
    V. V. Goryunov, “Cohomologies of braid groups of series C and D,” Tr. Mosk. Mat. Obsch.,42, 234–242 (1981).Google Scholar
  16. 16.
    V. Ya. Lin, “Artin braids and associated groups and spaces,” Itogi Nauki i Tekhniki, Algebra, Topologiya, Geometriya,17, 159–227 (1979).Google Scholar
  17. 17.
    O. V. Lyashko, “Geometry of bifurcation diagrams,” Itogi Nauki i Tekhniki, Sovr. Probl. Mat.,22, 94–129 (1983).Google Scholar
  18. 18.
    D. B. Fuks, “Mod 2 cohomologies of the braid group,” Funkts. Anal. Prilozhen.,4, No. 2, 62–73 (1970).Google Scholar
  19. 19.
    V. Arnold, “On some problems in singularity theory,” V. K. Patodi Memorial Volume, Bombay (1979), pp. 1–10.Google Scholar
  20. 20.
    E. Brieskorn, “Die Monodromie der isolierten Singularitaten von Hyperflachen,” Manuscr. Math.,2, 103–161 (1970).Google Scholar
  21. 21.
    E. Brieskorn, “Singular elements of semi-simple algebraic groups,” Actes Congres Int. Math., Nice, 1970, Vol. 2, Paris (1971), pp. 279–284.Google Scholar
  22. 22.
    E. Brieskorn, “Sur les groupes de tresses (d'apres V. I. Arnold),” Lect. Notes Math.,317, 21–44 (1973).Google Scholar
  23. 23.
    F. R. Cohen, “Braid orientations and bundles with flat connections,” Invent. Math.,46, No. 2, 99–110 (1978).Google Scholar
  24. 24.
    F. R. Cohen, T. J. Lada, and J. P. May, “The homology of iterated loop spaces,” Lect. Notes Math.,533 (1976).Google Scholar
  25. 25.
    M. Nakaoka, “Homology of the infinite symmetric group,” Ann. Math.,73, 229–257 (1961).Google Scholar
  26. 26.
    Nguyen H. V. Hung, “The mod 2 cohomology algebras of symmetric groups,” Acta Math. Vietnam.,6, 41–48 (1981).Google Scholar
  27. 27.
    Nguyen H. V. Hung, “Algebre de cohomologie du groupe symmetrique infini et classes characteristiques de Dickson,” C. R. Acad. Sci. Pari, Ser. 1, No. 297, 611–614 (1983).Google Scholar
  28. 28.
    G. B. Sega, “Configuration-spaces and iterated loop-spaces,” Invent. Math.,21, No. 3, 213–221 (1973).Google Scholar
  29. 29.
    J.-P. Serre, “Homologie singuliere des espaces fibres. Applications,” Ann. Math.,54, 425–505 (1951).Google Scholar
  30. 30.
    T. Steenbrink, “Mixed Hodge structure on the vanishing cohomology,” in: Real and Complex Singularities, Nordic Summer School, Oslo (1976), pp. 525–563.Google Scholar

Copyright information

© Plenum Publishing Corporation 1990

Authors and Affiliations

  • V. A. Vasil'ev

There are no affiliations available

Personalised recommendations