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

TL;DR

This paper establishes a new injectivity criterion for module homomorphisms over group rings, advancing the understanding of homology cobordisms and knot/link concordance in topology.

Contribution

It introduces a generalized criterion for module injectivity that unifies and extends previous results in the context of homology cobordisms and knot/link concordance.

Findings

Provides a new criterion for module injectivity over group rings.

Unifies multiple previous results into a single generalized theorem.

Enhances tools for studying knot and link concordance via algebraic methods.

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.