Goussarov–Habiro theory for string links and the Milnor–Johnson correspondence

We study the Goussarov–Habiro finite type invariants theory for framed string links in homology balls. Their degree 1 invariants are computed: they are given by Milnorʹs triple linking numbers, the mod 2 reduction of the Sato–Levine invariant, Arf and Rochlinʹs μ invariant. These invariants are seen to be naturally related to invariants of homology cylinders through the Milnor–Johnson correspondence: in particular, an analogue of the Birman–Craggs homomorphism for string links is computed. The relation with Vassiliev theory is studied.