RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Sb.: Year: Volume: Issue: Page: Find

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

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

Full text: PDF file (927 kB)
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 517.98
MSC: 57N65, 52B20, 52B70, 05E45, 20F55

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
\Bibitem{Gai16} \by A.~A.~Gaifullin \paper Small covers of graph-associahedra and realization of cycles \jour Mat. Sb. \yr 2016 \vol 207 \issue 11 \pages 53--81 \mathnet{http://mi.mathnet.ru/msb8714} \crossref{https://doi.org/10.4213/sm8714} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3588979} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2016SbMat.207.1537G} \elib{https://elibrary.ru/item.asp?id=27350062} \transl \jour Sb. Math. \yr 2016 \vol 207 \issue 11 \pages 1537--1561 \crossref{https://doi.org/10.1070/SM8714} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000393619200003} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85011556854} 

• http://mi.mathnet.ru/eng/msb8714
• https://doi.org/10.4213/sm8714
• http://mi.mathnet.ru/eng/msb/v207/i11/p53

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. D. S. Ulyumdzhiev, “Betti numbers of small covers and their two-fold coverings”, Siberian Math. J., 59:3 (2018), 551–555
•  Number of views: This page: 427 Full text: 27 References: 41 First page: 47