Algebraic concordance of links

Abstract

Algebraic concordance of knots can be understood from the perspective of Seifert matrices, Blanchfield forms, and homology surgery. We initiate a systematic study of algebraic concordance for links from each of these viewpoints. The present article is concerned with algebraic concordance from the perspective of homology surgery and Blanchfield forms, whereas a companion article by the third named author focuses on C-complexes and generalised Seifert matrices. The outcome of the present work consists of two obstructions to $μ$-component links being concordant. The first obstruction, called the homology surgery invariant, takes values in the Witt group of hermitian forms over the field of fractions $Q$ of $\mathbb{Z}[\mathbb{Z}^μ]$. The second obtruction, called the Blanchfield invariant, takes values in a Witt group of $Q/\mathbb{Z}[\mathbb{Z}^μ]$-valued hermitian linking forms. For $μ\le 2$, we describe these invariants in terms of generalised Seifert matrices.