The \(D(2)\)-problem for dihedral groups of order \(4n\)
type: Item
instance of: scholarly article
title: The \(D(2)\)-problem for dihedral groups of order \(4n\)
zbMATH Open document ID: 1262.57002
author: Seamus O'Shea
published in: Algebraic \& Geometric Topology
publication date: 2013-01-07T00:00:00Z
review text: The following question was first posed by Wall and it is known as the \(D(2)\)-problem: Let \(X\) be a finite connected \(3\)-dimensional \(CW\)-complex, with universal cover \(\widetilde{X},\) such that \(H_{3}(\widetilde {X};\mathbb{Z})=H^{3}(X;\mathcal{B})=0\) for all coefficient systems \(\mathcal{B}\) on \(X.\) Is it true that \(X\) is homotopy equivalent to a finite \(2\)-dimensional \(CW\)-complex? It is said that the \(D(2)\) property holds for a finitely presented group \(G\) if the above question is answered in the affirmative for every \(X\) with \(\pi_{1}(X)\simeq G.\) It is also said that torsion-free cancellation holds for a group ring \(\mathbb{Z}(G)\) if \(X\oplus M\simeq X\oplus N\Longrightarrow M\simeq N\) for any \(\mathbb{Z}(G)\)-lattices \(X,\) \(M\) and \(N.\) In the present work it is proven that the \(D(2)\)-propery holds for \(D_{4n},\) the dihedral group of order \(4n,\) provided that \(\mathbb{Z}(D_{4n})\) satisfies torsion-free cancellation.
reviewed by: Charalampos Charitos
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 55P15
zbMATH DE Number: 6122388
zbMATH Keywords: \(D(2)\)-problem
zbMATH Keywords: algebraic 2-complex
zbMATH Keywords: \(k\)-invariant
MaRDI profile type: Publication
full work available at URL: agt.2012.12.2287
OpenAlex ID: W2043300088
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DOI: 10.2140/AGT.2012.12.2287
description: scientific article; zbMATH DE number 6122388