An Injectivity Theorem for Casson-Gordon Type Representations relating to the Concordance of Knots and Links

Abstract

In the study of homology cobordisms, knot concordance and link concordance, the following technical problem arises frequently: let $\pi$ be a group and let $M \to N$ be a homomorphism between projective $\Z[\pi]$-modules such that $\Z_p \otimes_{\Z[\pi]} M\to \Z_p \otimes_{\Z[\pi]} N$ is injective; for which other right $\Z[\pi]$-modules $V$ is the induced map $V \otimes_{\Z[\pi]} M\to V\otimes_{\Z[\pi]}N$ also injective? Our main theorem gives a new criterion which combines and generalizes many previous results.