The manifold of isospectral symmetric tridiagonal matrices and realization of cycles by aspherical manifolds

  • A. A. Gaifullin
Article

Abstract

We consider the classical N. Steenrod’s problem of realization of cycles by continuous images of manifolds. Our goal is to find a class \( \mathcal{M}_n \) of oriented n-dimensional closed smooth manifolds such that each integral homology class can be realized with some multiplicity by an image of a manifold from the class \( \mathcal{M}_n \). We prove that as the class \( \mathcal{M}_n \) one can take a set of finite-fold coverings of the manifold M n of isospectral symmetric tridiagonal real (n + 1) × (n + 1) matrices. It is well known that the manifold M n is aspherical, its fundamental group is torsion-free, and its universal covering is diffeomorphic to ℝ n . Thus, every integral homology class of an arcwise connected space can be realized with some multiplicity by an image of an aspherical manifold with a torsion-free fundamental group. In particular, for any closed oriented manifold Q n , there exists an aspherical manifold that has torsion-free fundamental group and can be mapped onto Q n with nonzero degree.

References

  1. 1.
    N. Bourbaki, Éléments de mathématique, Fasc. 34: Groupes et algèbres de Lie, Chapitres IV, V et VI (Hermann, Paris, 1968; Mir, Moscow, 1972).Google Scholar
  2. 2.
    V. M. Buchstaber, “Modules of Differentials of the Atiyah-Hirzebruch Spectral Sequence. I, II,” Mat. Sb. 78(2), 307–320 (1969) [Math. USSR, Sb. 7, 299–313 (1969)]; Mat. Sb. 83 (1), 61–76 (1970) [Math. USSR, Sb. 12, 59–75 (1970)].MathSciNetGoogle Scholar
  3. 3.
    V. M. Buchstaber and T. E. Panov, Torus Actions in Topology and Combinatorics (MTsNMO, Moscow, 2004) [in Russian].Google Scholar
  4. 4.
    E. B. Vinberg, “Discrete Linear Groups Generated by Reflections,” Izv. Akad. Nauk SSSR, Ser. Mat. 35(5), 1072–1112 (1971) [Math. USSR, Izv. 5, 1083–1119 (1971)].MATHMathSciNetGoogle Scholar
  5. 5.
    A. A. Gaifullin, “Local Formulae for Combinatorial Pontryagin Classes,” Izv. Ross. Akad. Nauk, Ser. Mat. 68(5), 13–66 (2004) [Izv. Math. 68, 861–910 (2004)].MathSciNetGoogle Scholar
  6. 6.
    A. A. Gaifullin, “Explicit Construction of Manifolds Realising Prescribed Homology Classes,” Usp. Mat. Nauk 62(6), 167–168 (2007) [Russ. Math. Surv. 62, 1199–1201 (2007)].MathSciNetGoogle Scholar
  7. 7.
    A. A. Gaifullin, “Realisation of Cycles by Aspherical Manifolds,” Usp. Mat. Nauk 63(3), 157–158 (2008) [Russ. Math. Surv. 63, 562–564 (2008)].MathSciNetGoogle Scholar
  8. 8.
    S. P. Novikov, “Homotopic Properties of Thom Complexes,” Mat. Sb. 57(4), 407–442 (1962).MathSciNetGoogle Scholar
  9. 9.
    R. Thom, “Quelques propriétés globales des variétés différentiables,” Comment. Math. Helv. 28, 17–86 (1954).MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    M. W. Davis, “Groups Generated by Reflections and Aspherical Manifolds not Covered by Euclidean Space,” Ann. Math., Ser. 2, 117(2), 293–324 (1983).Google Scholar
  11. 11.
    M. W. Davis, “Some Aspherical Manifolds,” Duke Math. J. 55(1), 105–139 (1987).MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    M. W. Davis and T. Januszkiewicz, “Convex Polytopes, Coxeter Orbifolds and Torus Actions,” Duke Math. J. 62(2), 417–451 (1991).MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    S. Eilenberg, “On the Problems of Topology,” Ann. Math., Ser. 2, 50, 247–260 (1949).MathSciNetCrossRefGoogle Scholar
  14. 14.
    M. Ferri, “Una rappresentazione delle n-varietà topologiche triangolabili mediante grafi (n + 1)-colorati,” Boll. Unione Mat. Ital. B, Ser. 5, 13(1), 250–260 (1976).MATHMathSciNetGoogle Scholar
  15. 15.
    M. Ferri, C. Gagliardi, and L. Grasselli, “A Graph-Theoretical Representation of PL-Manifolds—a Survey on Crystallizations,” Aequationes Math. 31(2–3), 121–141 (1986).MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    M. Pezzana, “Diagrammi di Heegaard e triangolazione contratta,” Boll. Unione Mat. Ital., Ser. 4, 12,Suppl. (3), 98–105 (1975).MathSciNetGoogle Scholar
  17. 17.
    D. Sullivan, “Singularities in Spaces,” in Proc. Liverpool Singularities Symposium II (Springer, Berlin, 1971), Lect. Notes Math. 209, pp. 196–206.CrossRefGoogle Scholar
  18. 18.
    C. Tomei, “The Topology of Isospectral Manifolds of Tridiagonal Matrices,” Duke Math. J. 51(4), 981–996 (1984).MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    G. M. Ziegler, Lectures on Polytopes (Springer, Berlin, 1995), Grad. Texts Math. 152.MATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2008

Authors and Affiliations

  • A. A. Gaifullin
    • 1
  1. 1.Moscow State UniversityMoscowRussia

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