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# Georgian Mathematical Journal

Editor-in-Chief: Kiguradze, Ivan / Buchukuri, T.

Editorial Board: Kvinikadze, M. / Bantsuri, R. / Baues, Hans-Joachim / Besov, O.V. / Bojarski, B. / Duduchava, R. / Engelbert, Hans-Jürgen / Gamkrelidze, R. / Gubeladze, J. / Hirzebruch, F. / Inassaridze, Hvedri / Jibladze, M. / Kadeishvili, T. / Kegel, Otto H. / Kharazishvili, Alexander / Kharibegashvili, S. / Khmaladze, E. / Kiguradze, Tariel / Kokilashvili, V. / Krushkal, S. I. / Kurzweil, J. / Kwapien, S. / Lerche, Hans Rudolf / Mawhin, Jean / Ricci, P.E. / Tarieladze, V. / Triebel, Hans / Vakhania, N. / Zanolin, Fabio

IMPACT FACTOR 2018: 0.551

CiteScore 2018: 0.52

SCImago Journal Rank (SJR) 2018: 0.320
Source Normalized Impact per Paper (SNIP) 2018: 0.711

Mathematical Citation Quotient (MCQ) 2018: 0.27

Online
ISSN
1572-9176
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Volume 26, Issue 2

# On the construction of a covering map

Samson Saneblidze
• Corresponding author
• A. Razmadze Mathematical Institute, I. Javakhishvili Tbilisi State University, 6 Tamarashvili Str., Tbilisi 0177, Georgia
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Published Online: 2019-04-06 | DOI: https://doi.org/10.1515/gmj-2019-2016

## Abstract

Let $Y=|X|$ be the geometric realization of a path-connected simplicial set X, and let $G={\pi }_{1}\left(X\right)$ be the fundamental group. Given a subgroup $H\subset G$, let $G/H$ be the set of cosets. Using the combinatorial model $𝛀X\to 𝐏X\to X$ of the path fibration $\mathrm{\Omega }Y\to PY\to Y$ and a canonical action $\mu :𝛀X×G/H\to G/H$, we construct a covering map $G/H\to {Y}_{H}\to Y$ as the geometric realization of the associated short sequence $G/H\to 𝐏X{×}_{\mu }G/H\to X$. This construction, in particular, does not use the existence of a maximal tree in X. For a 2-dimensional X and $H=\left\{1\right\}$, it can also be viewed as a simplicial approximation of a Cayley 2-complex of G.

Keywords: Covering map; cubical set; necklical set

MSC 2010: 20F65; 57M10; 55U05; 55P35

Dedicated to Academician Nodar Berikashvili on the occasion of his 90th birthday

## References

• [1]

M. Gromov, Homotopical effects of dilatation, J. Differential Geom. 13 (1978), no. 3, 303–310.

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T. Kadeishvili and S. Saneblidze, A cubical model for a fibration, J. Pure Appl. Algebra 196 (2005), no. 2–3, 203–228.

• [3]

R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Classics Math., Springer, Berlin, 2001. Google Scholar

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J. Milnor, Construction of universal bundles. I, Ann. of Math. (2) 63 (1956), 272–284.

• [5]

M. Rivera and S. Saneblidze, A combinatorial model for the path fibration, J. Homotopy Relat. Struct. (2018), 10.1007/s40062-018-0216-4.

Accepted: 2018-12-28

Published Online: 2019-04-06

Published in Print: 2019-06-01

Funding Source: Shota Rustaveli National Science Foundation

Award identifier / Grant number: 217-614

This research was partially supported by Shota Rustaveli NSF grant 217-614.

Citation Information: Georgian Mathematical Journal, Volume 26, Issue 2, Pages 303–309, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X,

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