Link concordances as surfaces in 4-space

18 pagesFixing two concordant links in 3-space, we study the set of all concordances between them, as knotted annuli in 4-space. We show that, when regarded up to surface-link concordance or link-homotopy, the set $\mathcal{C}(L)$ of concordances from a link $L$ to itself forms a group, and we analyse the interplay between these groups when we let the link $L$ vary. We define and study Milnor-type invariants of $\mathcal{C}(L)$, which are integers defined modulo some indeterminacy given by Milnor invariants of $L$. For a slice link $L$, we show that these invariants classify $\mathcal{C}(L)$ up to link-homotopy, and yield a surjection from the concordance classes of $\mathcal{C}(L)$ to the group of conjugating-automorphisms of the $k$th nilpotent quotient of the free group, for all $k\ge 1$