An injectivity theorem for Casson-Gordon type representations relating to the concordance of knots and links
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.
scientific article; zbMATH DE number 6023969