Infinitely many pairwise homotopy inequivalent 2-complexes \(K_ i\) with fixed \(\pi_ 1 K_ i\) and \(\chi (K_ i)\)
type: Item
instance of: scholarly article
title: Infinitely many pairwise homotopy inequivalent 2-complexes \(K_ i\) with fixed \(\pi_ 1 K_ i\) and \(\chi (K_ i)\)
zbMATH Open document ID: 0805.57002
DOI: 10.1016/0022-4049(93)90021-K
author: Martin Lustig
published in: Journal of Pure and Applied Algebra
publication date: 1995-01-31T00:00:00Z
review text: The examples of Dunwoody and the reviewer, that the homotopy type of a finite 2-complex is not determined by the fundamental group and the Euler characteristic, resulted in finitely many distinct classes for a given \(\pi_ 1\) and \(\chi\). In the present paper, an infinite such family is constructed: The idea takes up Dunwoody's result that for any \(i\geq 0\), \(s^{2i+1}\) and \(t^{3i+1}\) form a generating set for the trefoil group \(G= \langle s,t\mid s^ 2= t^ 3\rangle\), and that the corresponding relation modules are pairwise non-isomorphic for distinct primes \(i\). The essential step then is a nice commutator trick that realizes these modules as \(H_ 2\)-modules of covering spaces of complexes which are extensions of the standard complex of \(\langle s,t\mid s^ 2= t^ 3\rangle\).
Mathematics Subject Classification ID: 57M20
zbMATH DE Number: 427666
zbMATH Keywords: homotopy type of a finite 2-complex
zbMATH Keywords: fundamental group
zbMATH Keywords: Euler characteristic
zbMATH Keywords: generating set for the trefoil group
zbMATH Keywords: relation modules
MaRDI profile type: Publication
cites work: Non-Free Projective Modules for Torsion-Free Groups
cites work: The Homotopy Type of a Two-Dimensional Complex
cites work: Q3855825
cites work: Q4145882
cites work: Über den Homotopietyp zweidimensionaler CW-Komplexe und Elementartransformationen bei Darstellungen von Gruppen durch Erzeugende und definierende Relationen.
cites work: Q5513066
description: scientific article; zbMATH DE number 427666