Stable modules and Wall's D(2)-problem
type: Item
instance of: scholarly article
title: Stable modules and Wall's D(2)-problem
zbMATH Open document ID: 1031.57007
published in: Commentarii Mathematici Helvetici
publication date: 2003-10-12T00:00:00Z
review text: Wall's D(2) problem asks whether a finite three dimensional cell complex such that \(H^{3}(X;{\mathcal B}) = 0\) for all local coefficient systems \(\mathcal B\) on \(X\) is homotopy equivalent to a finite \(2\)-dimensional complex. For finitely presented groups \(G\) one defines an algebraic \(2\)-complex over \(G\) to be an exact sequence \({\mathbb E}\) of modules and homomorphisms over \({\mathbb Z}[G]\) of the form: \[ {\mathbb E} = ( 0 \rightarrow J \rightarrow E_{2} \rightarrow E_{1} \rightarrow E_{0} \rightarrow {\mathbb Z} \rightarrow 0) \] where each \(E_{r}\) is finitely generated over \({\mathbb Z}[G]\). \textbf{Alg}\(_{G}\) denotes the category of such algebraic 2-complexes. The paper shows that for finite groups \(G\), the \(D(2)\) property holds if and only if each algebraic \(2\)-complex \(E \in \)\textbf{Alg}\(_{G}\) admits a geometric realization by a finite based \(2\)-complex. The proof relies on techniques that apply to fundamental groups, the Yoneda lemma, and the Swan-Jacobinski cancellation theorem.
Mathematics Subject Classification ID: 57M20
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 18G10
Mathematics Subject Classification ID: 18E25
Mathematics Subject Classification ID: 16E05
zbMATH DE Number: 1990084
zbMATH Keywords: algebraic 2-complex
zbMATH Keywords: stable module
zbMATH Keywords: \(k\)-invariant
zbMATH Keywords: geometrical realization
zbMATH Keywords: cohomologically 2-dimensional 3-complex
author: F. E. A. Johnson
reviewed by: Jonathan Hodgson
MaRDI profile type: Publication
full work available at URL: s000140300001
OpenAlex ID: W1964960066
DOI: 10.1007/S000140300001
description: scientific article; zbMATH DE number 1990084