Geodesic
The Manifold of Isospectral Symmetric Tridiagonal Matrices and Realization of Cycles by Aspherical Manifolds
Informatics and Automation, Geometry, topology, and mathematical physics. I, Tome 263 (2008), pp. 44-63 
A. A. Gaifullin. The Manifold of Isospectral Symmetric Tridiagonal Matrices and Realization of Cycles by Aspherical Manifolds. Informatics and Automation, Geometry, topology, and mathematical physics. I, Tome 263 (2008), pp. 44-63. http://geodesic.mathdoc.fr/item/TRSPY_2008_263_a3/
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     author = {A. A. Gaifullin},
     title = {The {Manifold} of {Isospectral} {Symmetric} {Tridiagonal} {Matrices} and {Realization} of {Cycles} by {Aspherical} {Manifolds}},
     journal = {Informatics and Automation},
     pages = {44--63},
     year = {2008},
     volume = {263},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2008_263_a3/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 $(n+1)\times(n+1)$ 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 $\mathbb R^n$. 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.

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