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 Tr. Mat. Inst. Steklova, 2008, Volume 263, Pages 44–63 (Mi tm782)

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

A. A. Gaifullin

M. V. Lomonosov Moscow State University

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.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2008, 263, 38–56

Bibliographic databases:

UDC: 515.164

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, Tr. 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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This publication is cited in the following articles:
1. A. A. Gaifullin, “A Minimal Triangulation of Complex Projective Plane Admitting a Chess Colouring of Four-Dimensional Simplices”, Proc. Steklov Inst. Math., 266 (2009), 29–48
2. Gaifullin A., “Universal Realisators for Homology Classes”, Geom. Topol., 17:3 (2013), 1745–1772
3. Baird T., Ramras D.A., “Smoothing Maps Into Algebraic Sets and Spaces of Flat Connections”, Geod. Dedic., 174:1 (2015), 359–374
4. A. A. Gaifullin, “Small covers of graph-associahedra and realization of cycles”, Sb. Math., 207:11 (2016), 1537–1561
5. Gaifullin A.A., Neretin Yu.A., “Infinite Symmetric Group, Pseudomanifolds, and Combinatorial Cobordism-Like Structures”, J. Topol. Anal., 10:3 (2018), 605–625
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