Skip to main content
Log in

Coverings over Tori and topological approach to Klein’s resolvent problem

  • Published:
Transformation Groups Aims and scope Submit manuscript

Abstract

This paper answers the question: what coverings over a topological torus can be induced from a covering over a space of dimension k? The answer to this question is then applied in algebro-geometric context to present obstructions to transforming an algebraic equation depending on several parameters to an equation depending on fewer parameters by means of a rational transformation.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. В. И. Арнольд, Тоnолгuческuе uнварuанmы алгебраuческuх функцuй. II, Функц. анализ и его прилож. 4 (1970), no. 2, 1–9. Engl. transl.: V. Arnol’d, Topological invariants of algebraic functions. II. Funct.Anal. Appl. 4 (1970), no. 2, 91–98.

  2. J.-L. Brylinski, D. A. McLaughlin, Multidimensional reciprocity laws, J. Reine Angew. Math. 481 (1996), 125–147.

    MathSciNet  MATH  Google Scholar 

  3. J. Buhler, Z. Reichstein, On the essential dimension of a finite group, Compositio Math. 106 (1997), no. 2, 159–179.

    Article  MathSciNet  MATH  Google Scholar 

  4. J. Buhler, Z. Reichstein, On Tschirnhaus transformations, in: Topics in Number theory (University Park, PA, 1997), Math. Appl., Vol. 467, Kluwer Acad. Publ., Dordrecht, 1999, pp. 127–142.

  5. Н. Чеботарёв, Проблема резольвенm u крumuческuе многообразця, Изв. АН СССР, сер. мат. 7 (1943), no. 3, 123–146. [N. Chebotarev, The problem of resolvents and critical manifolds, Izv. Akad. Nauk SSSR, Ser. Matem. 7 (1943), 123–146 (in Russian)].

  6. A. Duncan, Essential dimensions of A 7 and S 7, Math. Res. Lett. 17 (2010), no. 2, 263–266.

    MathSciNet  MATH  Google Scholar 

  7. Д. Б. Фукс, Когомологuu груnn кос mоd 2, Фунализ и его прилож. 4 (1970), no. 2, 62–73. Engl. transl.: D. B. Fuchs, Cohomology of the braid group mod 2, Funct.Anal. Appl. 4 (1970), no. 2, 143–151.

  8. V. Guillemin, A. Pollack, Differential Topology, AMS Chelsea Publishing, Providence, RI, 2010.

    MATH  Google Scholar 

  9. F. Klein, Lectures on the Icosahedron and the Solution of the Equation of the Fifth Degree, 2nd ed., Dover Publications, New York, N.Y., 1956. Russian transl.: Ф. Клейн, Лекцuu об uкосаздре u решенuu уравненuй nяmой сmеnенu, Наука, М., 1989.

  10. В. Я. Лин, Суnерnозuцuu алгебраuческuх функцuй, Функц. анализ и его прилож. 10 (1976), no. 1, 37–45. Engl. transl.: V. Lin, Superpositions of algebraic functions, Funct.Anal. Appl. 10 (1976), no. 1, 32–38.

  11. M. Mazin, Parshin residues via coboundary operators, to appear in Michigan Math. J., arXiv:0707.3748v3 (2007).

  12. M. Mazin, Geometric theory of Parshin’s residues. Toric neighborhoods of Parshin’s points, arXiv:0910.2529v1 (2009).

  13. M. Mazin, Geometric theory of Parshin’s residues, Mathematical Reports of the Canadian Academy of Science, 2010.

  14. A. Meyer, Z, Reichstein, An upper bound on the essential dimension of a central simple algebra, J. Algebra 329 (2011), 213–221.

    Article  MathSciNet  MATH  Google Scholar 

  15. J. Milnor, Morse Theory, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. Russian transl.: Дж. Милнор, Теорuя Морса, Мир, М., 1965.

  16. D. Mumford, Algebraic Geometry I: Complex Projective Varieties, Reprint of the 1976 ed., Grundlehren der mathematischen Wissenschaften, Vol. 221, Springer-Verlag, Berlin, 1995. Russian transl.: Д. Мамфорд, Алгебраuческая геомеmрuя, I. Комnлексные nроекmuвные многообразuя, Мир., М., 1979.аы

  17. Z, Reichstein, Essential dimension, in: Proceedings of the International Congress of Mathematicians, Vol. II, Hindustan Book Agency, New Delhi, 2010, pp. 162–188.

  18. Z. Reichstein, B. Youssin, Essential dimensions of algebraic groups and a resolution theorem for G-varieties, Canad. J. Math. 52 (2000), no. 5, 1018–1056. With an appendix by János Kollár and Endre Szabó.

  19. В. А. Васильев, Когомологuu груnn u сложносmь алгорumмов, Функц. анализ и его прилож. 22 (1988), no. 3, 15–24. Engl. transl.: V. Vasil’ev, Braid group cohomologies and algorithm complexity, Funct.Anal. Appl. 22 (1988), no. 3, 182–190.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yuri Burda.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Burda, Y. Coverings over Tori and topological approach to Klein’s resolvent problem. Transformation Groups 17, 921–951 (2012). https://doi.org/10.1007/s00031-012-9199-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00031-012-9199-0

Keywords

Navigation