Milnor introduced in the 50’s a family of link invariants, extracted from the peripheral system, which are invariant under “link homotopy”, i.e. under continuous deformations where distinct components remain disjoint. A full link homotopy classification of links was achieved only 40 years later by Habegger and Lin, using a refinement of Milnor invariants for “string links”. The situation is very different in higher dimension: any embedding of a disjoint union of 2–spheres in 4–space is link homotopic to the trivial one. In this talk, we consider higher order analogues of string links, and give a classification up to link homotopy using a 4–dimensional version of Milnor invariants. We then turn back to the link case, and give a 4-dimensional interpretation of Milnor’s original invariant.
Based on joint works with B. Audoux and E. Wagner.