On the constructions of free and locally standard \(\mathbb{Z}_{2}\)-torus actions on manifolds
type: Item
instance of: scholarly article
title: On the constructions of free and locally standard \(\mathbb{Z}_{2}\)-torus actions on manifolds
zbMATH Open document ID: 1245.57031
author: Li Yu
published in: Osaka Journal of Mathematics
publication date: 2012-04-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1001.0289
full work available at URL: https://projecteuclid.org/euclid.ojm/1332337243
reviewed by: Robert D. Little
Mathematics Subject Classification ID: 57R22
Mathematics Subject Classification ID: 57R91
Mathematics Subject Classification ID: 57S17
zbMATH DE Number: 6029032
zbMATH Keywords: free \(\mathbb{Z}_2\)-torus actions
zbMATH Keywords: \(\mathbb{Z}_2\)-core
zbMATH Keywords: glue-back construction
MaRDI profile type: Publication
arXiv ID: 1001.0289
cites work: Q4819371
cites work: Convex polytopes, Coxeter orbifolds and torus actions
cites work: Equivariant classification of 2-torus manifolds
cites work: On the classification of \(O(n)\)-manifolds
cites work: Groups generated by reflections and aspherical manifolds not covered by Euclidean space
cites work: Q4537714
cites work: Q4107635
cites work: Topological types of 3-dimensional small covers
cites work: The orientability of small covers and coloring simple polytopes
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review text: If \((\mathbb{Z}_2)^m\) acts freely and smoothly on a closed manifold \(M^n\), then the orbit space \(Q^n\) is also a closed \(n\)-manifold with \(M^n\) a regular covering of \(Q^n\) with deck transformation group \((\mathbb{Z}_2)^m\). This paper offers a construction which begins with a monochromy homomorphism defined on \(H_{n-1}(Q^n,\mathbb{Z}_2)\) as generated by codimension one submanifolds in general position. The \(\mathbb{Z}_2\)-core of \(Q^n\), \(V^n\), is the complement of tubular neighborhoods of the generators with the natural involutions on the boundaries deformed to commute at intersections. The boundary of \(V^n\) is tessellated by \((n-1)\)-dimensional facets. The union of the facets corresponding to a particular generator is called a panel. Each panel is invariant under the induced involution.NEWLINENEWLINE The first major result of this paper asserts that any \(Q^n\) contains a \(\mathbb{Z}_2\)-core and the second describes a glue-back construction in which \(V^n\) and a mapping from the set of panels into \((\mathbb{Z}_2)^m\), called a \((\mathbb{Z}_2)^m\)-coloring, produce an associated \(M^n\).
description: scientific article; zbMATH DE number 6029032