HL-homotopy of handlebody-links and Milnor's invariants

A handlebody-link is a disjoint union of embeddings of handlebodies in S3 and an HL-homotopy is an equivalence relation on handlebody-links generated by self-crossing changes. The second author and Ryo Nikkuni classified the set of HL-homotopy classes of 2-component handlebody-links completely using the linking numbers for handlebody-links. In this paper, we construct a family of invariants for HL-homotopy classes of general handlebody-links, by using Milnor's μ‾-invariants. Moreover, we give a bijection between the set of HL-homotopy classes of almost trivial handlebody-links and tensor product space modulo some general linear actions, especially for 3- or more component handlebody-links. Through this bijection we construct invariants of HL-homotopy classes which can be used to distinguish the classes.