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

TL;DR

This paper introduces new invariants for handlebody-link classes under HL-homotopy, utilizing Milnor's invariants, and establishes a bijection linking these classes to tensor spaces for multi-component links.

Contribution

It constructs a family of invariants for HL-homotopy classes of handlebody-links using Milnor's invariants and describes a bijection with tensor spaces for multi-component cases.

Findings

Constructed invariants for handlebody-link HL-homotopy classes.

Established a bijection between HL-homotopy classes and tensor spaces.

Provided a framework for classifying handlebody-links with multiple components.

Abstract

A handlebody-link is a disjoint union of embeddings of handlebodies in $S^3$ 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 link-homotopy 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 comparable invariants of HL-homotopy classes.