Relative knot concordance

Abstract

Motivated by the relative-slice problem in $4$-manifold topology, we define for a knot $R$ the notion of a relative knot concordance group $\mathcal{C}_R$ and analyze the structure of its algebraic analogue $\mathcal{AC}_R$ modulo torsion. Using Milnor signatures, we show that $\mathcal{AC}_R$ has infinite rank for every $R$. We also show that any prescribed finite set of natural numbers can be realized as orders of elements in a single relative concordance group.