Skip to main content
SpringerLink
Log in

Topology of spaces of hyperbolic polynomials and combinatorics of resonances

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper we study the topology of the strata, indexed by number partitions λ, in the natural stratification of the space of monic hyperbolic polynomials of degreen. We prove stabilization theorems for removing an independent block or an independent relation in λ. We also prove contractibility of the ‘one-point compactifications of the strata indexed by a large class of number partitions, including λ=(k m, 1r), form≥2. Furthermore, we study the maps between the homology groups of the strata, induced by imposing additional relations (resonances) on the number partition λ, or by merging some of the blocks of λ.

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

Access this article

Log in via an institution

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. V. I. Arnol’d,Topological invariants of algebraic functions, Transactions of the Moscow Mathematical Society21 (1970), 30–52.

    Google Scholar 

  2. V. I. Arnol’d,Spaces of functions with moderate singularities, Functional Analysis and its Applications23 (1989), 169–177.

    Article  MATH  MathSciNet  Google Scholar 

  3. A. Björner and M. Wachs,Shellable non-pure complexes and posets II, Transactions of the American Mathematical Society349 (1997), 3945–3975.

    Article  MATH  MathSciNet  Google Scholar 

  4. F. R. Cohen, R. L. Cohen, B. M. Mann and R. J. Milgram,The topology of the spaces of rational functions and divisors of surfaces, Acta Mathematica166 (1991), 163–221.

    Article  MathSciNet  Google Scholar 

  5. R. Forman,Morse theory for cell complexes, Advances in Mathematics134 (1998), 90–145.

    Article  MATH  MathSciNet  Google Scholar 

  6. D. N. Kozlov,Collapsibility of △(II n )/S and some related CW complexes, Proceedings of the American Mathematical Society128 (2000), 2253–2259.

    Article  MATH  MathSciNet  Google Scholar 

  7. D. Quillen,Higher algebraic K-theory, I, Lecture Notes in Mathematics341, Springer-Verlag, Berlin, 1973, pp. 85–148.

    Google Scholar 

  8. G. Segal,Classifying spaces and spectral sequences, Publications Mathématiques de l’Institut des Hautes Études Scientifiques34 (1968), 105–112.

    Article  MATH  Google Scholar 

  9. B. Shapiro, Private communication.

  10. B. Shapiro and V. Welker,Combinatorics and topology of stratifications of the space of monic polynomials with real coefficients, Results in Mathematics33 (1998), 338–355.

    MATH  MathSciNet  Google Scholar 

  11. V. A. Vassiliev,Topology of spaces of functions without complicated singularities, Functional Analysis and its Applications23 (1989), 24–36.

    Google Scholar 

  12. V. A. Vassiliev,Cohomology of knot spaces, inAdvances in Soviet Mathematics; Theory of Singularities and its Applications (V. I. Arnol’d, ed.), American Mathematical Society, Providence, R.I., 1990, pp. 23–69.

    Google Scholar 

  13. V. A. Vassiliev,Complements of Discriminants of Smooth Maps: Topology and Applications, Translations of Mathematical Monographs, Vol. 98, American Mathematical Society, Providence, RI, 1994. Revised Edition.

    Google Scholar 

  14. V. A. Vassiliev,Homology of spaces of homogeneous polynomials in R 2 without multiple zeros, inLocal and Global Problems of Singularity Theory (Russian), Trudy Mat. Inst. Steklova221 (1998), 143–148.

  15. D. N. Kozlov,Resonance category, math.CT/0111196, preprint, 22 pages.

Download references

Author information

Author notes

  1. Dmitry N. Kozlov

    Present address: Department of Mathematics, University of Bern, CH-3012, Bern, Switzerland

Authors and Affiliations

  1. Department of Mathematics, Royal Institute of Technology, S-100 44, Stockholm, Sweden

    Dmitry N. Kozlov

Authors
  1. Dmitry N. Kozlov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dmitry N. Kozlov.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kozlov, D.N. Topology of spaces of hyperbolic polynomials and combinatorics of resonances. Isr. J. Math. 132, 189–206 (2002). https://doi.org/10.1007/BF02784511

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02784511

Keywords

Search

Navigation