Link concordances as surfaces in 4-space and the 4-dimensional Milnor invariants

Fixing two concordant links in $3$--space, we study the set of all embedded concordances between them, as knotted annuli $4$--space. When regarded up to surface-concordance or link-homotopy, $\mathcal{C}(L)$ from a link $L$ itself forms group. In order investigate these groups, define Milnor-type invariants $\mathcal{C}(L)$, which are integers defined modulo certain indeterminacy given by Milnor $L$. We show particular that, for slice $L$, classify link-homotopy.