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

Abstract

Abstract. In the study of homology cobordisms, knot concordance and link concordance, the following technical problem arises frequently: let π be a group and let M → N be a homomorphism between projectiveN also injective? Our main theorem gives a new criterion which combines and generalizes many previous results.

Links not concordant to the Hopf link

“…This theorem is reminiscent of 'Casson-Gordon type' obstruction theorems in the context of knot concordance, especially of the main theorem of [CG86]. The proof of Theorem 1•2 is indeed guided by the knot theoretic case coupled with the main result of [FP12]. In particular we will show that if a link L is concordant to H and if χ satisfies the conditions of the theorem, then there exists a 4-manifold…”

“…Generalising their work to the setting of link concordance requires overcoming several delicate technical issues. One particularly difficult step (Theorem 2•1 in this paper) appeared in an earlier paper by the authors [FP12].…”

Two-solvable and two-bipolar knots with large four-genera

“…Note that in the above circumstance H φ 1 pM K q is a torsion Qpξ m qrt˘1s-module, by the corollary to [CG86, Lemma 4]; see also [FP12]. In Section 6.1, we will need to know that this form is sesquilinear [Pow16].…”

Embedding spheres in knot traces

“…From the work of Friedl and Powell [FP12, Proposition 4.1] (applied with, in their notation, ) we have that is an isomorphism. Then for , we have natural identifications and the desired result follows.…”