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

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## 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.

## Keywords

STEKLOV Institute Homology Class Coxeter Group Small Covering Fundamental Class## References

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