On \(Wh_ 2\) of a Bieberbach group
type: Item
instance of: scholarly article
title: On \(Wh_ 2\) of a Bieberbach group
zbMATH Open document ID: 0589.57022
DOI: 10.1016/0040-9383(84)90014-4
author: Andrew Nicas
published in: Topology
publication date: 1984-01-01T00:00:00Z
review text: It is shown that for a Bieberbach group \(\Gamma\), the second Whitehead group \(Wh_ 2(\Gamma)\) is trivial, at least if 2-torsion is ignored. The special role of 2-torsion is explained by the fact that the author needs to represent elements of \(Wh_ 2(\Gamma)\) geometrically in terms of block homeomorphisms (not just pseudo-isotopies) - this works only away from 2; using such geometric representation the author shows: Let \(b\in Wh_ 2(\Gamma)_{odd}\). Then there is an integer \(N_ b\) such that if \(f: \Gamma\) \(\to \Gamma\) is ''s-expansive'' and \(s>N_ b\) then \(f^*(b)=0\) in \(Wh_ 2(\Gamma)_{odd}\) (where \(f^*\) denotes the transfer induced from the monomorphism f). This result and others are then used to apply Frobenius induction in order to prove the vanishing theorem.
Mathematics Subject Classification ID: 57Q60
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 57Q10
Mathematics Subject Classification ID: 57R52
Mathematics Subject Classification ID: 55Q35
Mathematics Subject Classification ID: 57R80
zbMATH DE Number: 3947234
zbMATH Keywords: Bieberbach group
zbMATH Keywords: second Whitehead group
zbMATH Keywords: block homeomorphisms
reviewed by: Friedhelm Waldhausen
MaRDI profile type: Publication
full work available at URL: 0040-9383(84)90014-4
OpenAlex ID: W2050735559
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description: scientific article; zbMATH DE number 3947234