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




This article is cited in 5 scientific papers (total in 5 papers)
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 finitefold 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 torsionfree, 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 torsionfree fundamental group. In particular, for any closed oriented manifold $Q^n$, there exists an aspherical manifold that has torsionfree fundamental group and can be mapped onto $Q^n$ with nonzero degree.
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Proceedings of the Steklov Institute of Mathematics, 2008, 263, 38–56
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UDC:
515.164 Received in April 2008
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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\bookinfo Collected papers. Dedicated to Academician Sergei Petrovich Novikov on the occasion of his 70th birthday
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\publ MAIK Nauka/Interperiodica
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A. A. Gaifullin, “A Minimal Triangulation of Complex Projective Plane Admitting a Chess Colouring of FourDimensional Simplices”, Proc. Steklov Inst. Math., 266 (2009), 29–48

Gaifullin A., “Universal Realisators for Homology Classes”, Geom. Topol., 17:3 (2013), 1745–1772

Baird T., Ramras D.A., “Smoothing Maps Into Algebraic Sets and Spaces of Flat Connections”, Geod. Dedic., 174:1 (2015), 359–374

A. A. Gaifullin, “Small covers of graphassociahedra and realization of cycles”, Sb. Math., 207:11 (2016), 1537–1561

Gaifullin A.A., Neretin Yu.A., “Infinite Symmetric Group, Pseudomanifolds, and Combinatorial CobordismLike Structures”, J. Topol. Anal., 10:3 (2018), 605–625

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