Betti numbers of locally standard 2-torus manifolds
type: Item
instance of: scholarly article
title: Betti numbers of locally standard 2-torus manifolds
zbMATH Open document ID: 1301.57026
author: Junda Chen
author: Zhi Lü
published in: Chinese Annals of Mathematics. Series B
publication date: 2014-09-12T00:00:00Z
review text: A locally standard \(2\)-torus manifold \(M\) is a smooth connected closed \(n\)-manifold with a locally standard \((\mathbb{Z}_{2})^{n}\)-action, i.e., locally looks like the standard \((\mathbb{Z}_{2})^{n}\)-action on \(\mathbb{R}^{n}\). It is well known that the orbit space \(Q\) of a locally standard \(2\)-torus manifold admits the structure of an \(n\)-dimensional manifold with corners. Due to the result by \textit{Z. Lü} and \textit{M. Masuda} [Colloq. Math. 115, No. 2, 171--188 (2009; Zbl 1165.57023)], all locally standard torus manifolds can be recovered by the following three data up to equivariant homeomorphism: (1) \(n\)-dimensional manifold with corner \(Q\); (2) a characteristic function \(\lambda:\mathcal{F}(Q)\to (\mathbb{Z}_{2})^{n}\), where \(\mathcal{F}(Q)\) is the set of facets of \(Q\): (3) a principal \((\mathbb{Z}_{2})^{n}\)-bundle \(\xi\) over \(Q\) (this may be regarded as an element in \(H^{1}(Q;(\mathbb{Z}_{2})^{n})\)). Denote the locally standard \(2\)-torus manifold which recovers from these data by the symbol \(M(Q;\lambda,\xi)\). For example, the small cover (defined in [\textit{M. W. Davis} and \textit{T. Januszkiewicz}, Duke Math. J. 62, No. 2, 417--451 (1991; Zbl 0733.52006)]) is a locally standard \(2\)-torus manifold and can be denoted by \(M(P;\lambda)\) for a simple convex polytope \(P\) and a characteristic function \(\lambda\). Note that if \(P\) is a simple convex polytope then \(H^{1}(P;(\mathbb{Z}_{2})^{n})=\{0\}\); hence, the principal \((\mathbb{Z}_{2})^{n}\)-bundle over \(P\) is always trivial. In the paper under review, the authors compute the Betti numbers of a locally standard \(2\)-torus manifold \(M(Q;\lambda,\xi)\) such that the boundary of \(Q\) is isomorphic to the boundary of a simple convex polytope \(P\). As a consequence, they show that the Betti numbers of \(M(Q;\lambda,\xi)\) can be computed by the \(h\)-vectors of \(Q\) and the Betti numbers of \(E\), where \(E\) is the total space of the principal \((\math…
reviewed by: Shintaro Kuroki
Mathematics Subject Classification ID: 57S17
zbMATH DE Number: 6343671
zbMATH Keywords: locally standard 2-torus action
zbMATH Keywords: manifold with corners
zbMATH Keywords: Betti number
MaRDI profile type: Publication
full work available at URL: s11401-014-0845-9
OpenAlex ID: W2026126792
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DOI: 10.1007/S11401-014-0845-9
description: scientific article; zbMATH DE number 6343671