An injectivity theorem for Casson-Gordon type representations relating to the concordance of knots and links
type: Item
instance of: scholarly article
zbMATH Open document ID: 1251.57008
DOI: 10.4134/BKMS.2012.49.2.395
author: Stefan Friedl
author: Mark Powell
published in: Bulletin of the Korean Mathematical Society
publication date: 2012-04-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1011.4678
full work available at URL: http://www.mathnet.or.kr/mathnet/kms_content.php
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 57Q60
Mathematics Subject Classification ID: 57M10
Mathematics Subject Classification ID: 55S35
zbMATH DE Number: 6023969
zbMATH Keywords: injectivity
zbMATH Keywords: link concordance
zbMATH Keywords: obstruction theory
zbMATH Keywords: linear representation
zbMATH Keywords: \(p\)-group
zbMATH Keywords: factorization
zbMATH Keywords: Strebel's class
zbMATH Keywords: Maschke's theorem
Wikidata QID: Q122256375
MaRDI profile type: Publication
arXiv ID: 1011.4678
title: An injectivity theorem for Casson-Gordon type representations relating to the concordance of knots and links
review text: The injectivity theorem with important topological applications mentioned in the title hinges on homological methods in group theory and looks rather technical. Roughly speaking, it allows the following type of conclusion: Given a map \( i: S \rightarrow Y\) inducing isomorphisms for \({\mathbb Z}_p\)-homology and \(i_\varphi: S_\varphi \rightarrow Y_\varphi\) an induced map of \(k\)-fold coverings, then conditions are named under which \(i_\varphi\) induces isomorphisms for (say) rational coefficients though not necessarily for \({\mathbb Z}_p\)-coefficients. In the simplest cases, \(Y\) is the complement of a knot or a slice disk and \(S\) a \(1\)-linked \(1\)-sphere. The general result reads as follows: For \(\pi\) a finitely generated group, \(p\) a prime number and \(f: M \rightarrow N\) a morphism of projective left \({\mathbb Z}[\pi]\)-modules such that NEWLINE\[NEWLINE Id \otimes f: {\mathbb Z}_p \otimes_{{\mathbb Z}[\pi]} M \longrightarrow {\mathbb Z}_p \otimes_{{\mathbb Z}[\pi]} N NEWLINE\]NEWLINE is injective, let \(\phi: \pi \to H\) be an epimorphism onto a torsion-free abelian group and \(\alpha: \pi \to GL(k,Q)\) a representation (\(Q\) a field of characteristic zero). If \(\alpha|ker(\phi)\) factors through a \(p\)-group, then NEWLINE\[NEWLINE Id \otimes f: Q[H]^k \otimes_{{\mathbb Z}[\pi]} M \longrightarrow Q[H]^k \otimes_{{\mathbb Z}[\pi]} N NEWLINE\]NEWLINE is also injective. This improves on earlier studies requiring \(\alpha\) itself to factor through a \(p\)-group. The result is an essential ingredient in obstruction theory for homology cobordisms.
reviewed by: Dieter Erle
description: scientific article; zbMATH DE number 6023969