Finiteness conditions and \(\text{PD}_{r}\)-group covers of \(\text{PD}_{n}\)-complexes
type: Item
instance of: scholarly article
title: Finiteness conditions and \(\text{PD}_{r}\)-group covers of \(\text{PD}_{n}\)-complexes
zbMATH Open document ID: 1118.57020
author: Dessislava H. Kochloukova
author: Jonathan A. Hillman
published in: Mathematische Zeitschrift
publication date: 2007-05-31T00:00:00Z
review text: This paper shows that an infinite cyclic covering \(M'\) of an \(n\)-dimensional Poincaré duality space is an \((n-1)\)-dimensional Poincaré duality space, if and only if \(M'\) is homotopy equivalent to a complex with finite \([(n-1)/2]\)-skeleton and \(\pi_{1}(M')\) is finitely presentable. It is further shown that this is best possible in that it makes the minimal finiteness assumptions on the covering space. A corresponding result is shown for covering spaces whose covering group \(G\) is a PD\(_{r}\)-group (that is \(H^{q}(G;Z[G]) \cong Z\) if \(q=r\) and is 0 otherwise) under slightly stronger finiteness assumptions.
Mathematics Subject Classification ID: 57P10
Mathematics Subject Classification ID: 55R99
zbMATH DE Number: 5159756
zbMATH Keywords: PD-group
zbMATH Keywords: covering space
zbMATH Keywords: finitely dominated complex
zbMATH Keywords: PD-space
reviewed by: Jonathan Hodgson
MaRDI profile type: Publication
full work available at URL: s00209-006-0058-3
OpenAlex ID: W2024019564
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DOI: 10.1007/S00209-006-0058-3
description: scientific article; zbMATH DE number 5159756