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

  • A. A. Gaifullin


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.


STEKLOV Institute Homology Class Coxeter Group Small Covering Fundamental Class 
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© Pleiades Publishing, Ltd. 2008

Authors and Affiliations

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

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