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 $\hat{I}¼$-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 $\mathbb{Z}[{\mathbb{Z}}^{\hat{I}}¼]$. The second obtruction, called the Blanchfield invariant, takes values in a Witt group of $Q/\mathbb{Z}[{\mathbb{Z}}^{\hat{I}}¼]$-valued hermitian linking forms. For $\hat{I}¼\le 2$, we describe these invariants in terms of generalised Seifert matrices.

61 pages, 15 figures

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