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Mat. Sb., 2016, Volume 207, Number 11, Pages 53–81 (Mi msb8714)  

This article is cited in 2 scientific papers (total in 2 papers)

Small covers of graph-associahedra and realization of cycles

A. A. Gaifullin

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: An oriented connected closed manifold $M^n$ is called a $\mathrm{URC}$-manifold if for any oriented connected closed manifold $N^n$ of the same dimension there exists a nonzero-degree mapping of a finite-fold covering $\widehat{M}^n$ of $M^n$ onto $N^n$. This condition is equivalent to the following: for any $n$-dimensional integral homology class of any topological space $X$, a multiple of it can be realized as the image of the fundamental class of a finite-fold covering $\widehat{M}^n$ of $M^n$ under a continuous mapping $f\colon \widehat{M}^n\to X$. In 2007 the author gave a constructive proof of Thom's classical result that a multiple of any integral homology class can be realized as an image of the fundamental class of an oriented smooth manifold. This construction yields the existence of $\mathrm{URC}$-manifolds of all dimensions. For an important class of manifolds, the so-called small covers of graph-associahedra corresponding to connected graphs, we prove that either they or their two-fold orientation coverings are $\mathrm{URC}$-manifolds. In particular, we obtain that the two-fold covering of the small cover of the usual Stasheff associahedron is a $\mathrm{URC}$-manifold. In dimensions 4 and higher, this manifold is simpler than all the previously known $\mathrm{URC}$-manifolds.
Bibliography: 39 titles.

Keywords: realization of cycles, domination relation, $\mathrm{URC}$-manifold, small cover, graph-associahedron.

Funding Agency Grant Number
Russian Science Foundation 14-11-00414
The work was supported by the Russian Science Foundation (project no. 14-11-00414).


DOI: https://doi.org/10.4213/sm8714

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English version:
Sbornik: Mathematics, 2016, 207:11, 1537–1561

Bibliographic databases:

UDC: 517.98
MSC: 57N65, 52B20, 52B70, 05E45, 20F55
Received: 11.04.2016 and 24.08.2016

Citation: A. A. Gaifullin, “Small covers of graph-associahedra and realization of cycles”, Mat. Sb., 207:11 (2016), 53–81; Sb. Math., 207:11 (2016), 1537–1561

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. T. E. Panov, Ya. A. Veryovkin, “Polyhedral products and commutator subgroups of right-angled Artin and Coxeter groups”, Sb. Math., 207:11 (2016), 1582–1600  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    2. D. S. Ulyumdzhiev, “Betti numbers of small covers and their two-fold coverings”, Siberian Math. J., 59:3 (2018), 551–555  mathnet  crossref  crossref  mathscinet  isi  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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