On codimension two embeddings up to link‐homotopy

We consider knotted annuli in 4‐space, called 2‐string links, which are knotted surfaces in codimension two that are naturally related, via closure operations, to both 2‐links and 2‐torus links. We classify 2‐string links up to link‐homotopy by means of a 4‐dimensional version of Milnor invariants. The key to our proof is that any 2‐string link is link‐homotopic to a ribbon one; this allows to use the homotopy classification obtained in the ribbon case by P. Bellingeri and the authors. Along the way, we give a Roseman‐type result for immersed surfaces in 4‐space. We also discuss the case of ribbon k‐string links, for k⩾3.