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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Volume 263, Pages 44–63 (Mi tm782)  

This article is cited in 6 scientific papers (total in 6 papers)

The Manifold of Isospectral Symmetric Tridiagonal Matrices and Realization of Cycles by Aspherical Manifolds

A. A. Gaifullin

M. V. Lomonosov Moscow State University
Full-text PDF (302 kB) Citations (6)
References:
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 $(n+1)\times(n+1)$ 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.
Received in April 2008
English version:
Proceedings of the Steklov Institute of Mathematics, 2008, Volume 263, Pages 38–56
DOI: https://doi.org/10.1134/S0081543808040044
Bibliographic databases:
UDC: 515.164
Language: Russian
Citation: A. A. Gaifullin, “The Manifold of Isospectral Symmetric Tridiagonal Matrices and Realization of Cycles by Aspherical Manifolds”, Geometry, topology, and mathematical physics. I, Collected papers. Dedicated to Academician Sergei Petrovich Novikov on the occasion of his 70th birthday, Trudy Mat. Inst. Steklova, 263, MAIK Nauka/Interperiodica, Moscow, 2008, 44–63; Proc. Steklov Inst. Math., 263 (2008), 38–56
Citation in format AMSBIB
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\paper The Manifold of Isospectral Symmetric Tridiagonal Matrices and Realization of Cycles by Aspherical Manifolds
\inbook Geometry, topology, and mathematical physics.~I
\bookinfo Collected papers. Dedicated to Academician Sergei Petrovich Novikov on the occasion of his 70th birthday
\serial Trudy Mat. Inst. Steklova
\yr 2008
\vol 263
\pages 44--63
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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  • This publication is cited in the following 6 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Труды Математического института имени В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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