Combinatorial constructions of three-dimensional small covers
type: Item
instance of: scholarly article
title: Combinatorial constructions of three-dimensional small covers
zbMATH Open document ID: 1253.57010
author: Yasuzo Nishimura
published in: Pacific Journal of Mathematics
publication date: 2012-07-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1104.1744
full work available at URL: p10.xhtml
reviewed by: Zhi Lü
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57M60
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 52B10
zbMATH DE Number: 6053324
zbMATH Keywords: small cover
zbMATH Keywords: equivariant surgery
zbMATH Keywords: connected sum
zbMATH Keywords: 3-polytope
MaRDI profile type: Publication
OpenAlex ID: W2962924211
arXiv ID: 1104.1744
DOI: 10.2140/PJM.2012.256.177
review text: The notion of small covers was introduced by \textit{M. W. Davis} and \textit{T. Januszkiewicz} [Duke Math. J. 62, No. 2, 417--451 (1991; Zbl 0733.52006)]. An \(n\)-dimensional small cover is an \(n\)-dimensional closed smooth manifold with a locally standard \(({\mathbb Z}_2)^n\)-action whose orbit space is a simple convex polytope, and it is a topological real version of a nonsingular compact toric variety. The paper under review studies the topological type of 3-dimensional small covers.NEWLINENEWLINEThe main result of the paper under review is that each 3-dimensional small cover can be constructed from \(T^3\), \({\mathbb R}P^3\) and \(S^1\times {\mathbb R}P^2\) with two different \(({\mathbb Z}_2)^3\)-actions by using only two operations on small covers. This result is a generalization of the results in [\textit{I. V. Izmest'ev}, Math. Notes 69, No. 3, 340--346 (2001); translation from Mat. Zametki 69, No. 3, 375--382 (2001; Zbl 0991.57016)] and [\textit{Y. Nishimura}, ``Equivariant surgeries of small covers'', RIMS Kokyuroku 1393, 44--47 (2004)], and in addition, it is also an improvement of the results in [\textit{S. Kuroki}, Chin. Ann. Math., Ser. B 31, No. 3, 393--410 (2010; Zbl 1217.57010)] and [\textit{Z. Lü} and \textit{L. Yu}, Forum Math. 23, No. 2, 245--284 (2011; Zbl 1222.52015)].
description: scientific article; zbMATH DE number 6053324