Relative knot concordance

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 ${\mathcal{A}\mathcal{C}}_R$ modulo torsion. Using Milnor signatures, we show that ${\mathcal{A}\mathcal{C}}_R$ 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.