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 $\ddot{I}€$ be a group and let $M\to N$ be a homomorphism between projective $\mathbb{Z}[\ddot{I}€]$-modules such that ${\mathbb{Z}}_p{\otimes }_{\mathbb{Z}[\ddot{I}€]}M\to {\mathbb{Z}}_p{\otimes }_{\mathbb{Z}[\ddot{I}€]}N$ is injective; for which other right $\mathbb{Z}[\ddot{I}€]$-modules $V$ is the induced map $V{\otimes }_{\mathbb{Z}[\ddot{I}€]}M\to V{\otimes }_{\mathbb{Z}[\ddot{I}€]}N$ also injective? Our main theorem gives a new criterion which combines and generalizes many previous results.

15 pages, added reference to the work of Letsche