Milnor-type invariants for surface-links and cut-diagrams

TL;DR

This paper extends Milnor invariants to surface-links in 4-space using cut-diagrams, providing new invariants, classification results, and insights into surface-link concordance and ribbon properties.

Contribution

It introduces a novel generalization of Milnor invariants for surface-links via cut-diagrams, enabling new invariants and classification methods.

Findings

Milnor-type invariants are invariant under concordance.

New classification results up to link-homotopy.

Criteria for surface-links to be ribbon.

Abstract

We generalize Milnor link invariants to all types of surface-links in $4$--space (possibly with boundary). This is achieved by using the notion of cut-diagram, which is a 2-dimensional generalization of Gauss diagrams, associated to surface-links. We define a notion of group for cut-diagrams, which generalizes the fundamental group of the complement, and we extract Milnor-type invariants from the successive nilpotent quotients of this group. We show that these are invariant under concordance. We give several concrete applications of the resulting Milnor concordance invariants for surface-links, comparing their relative strength with previously known concordance invariants, and providing realization results. We also obtain several classification results up to link-homotopy, as well as a criterion for a surface-link to be ribbon. The theory of cut-diagrams is also further investigated, heading towards a combinatorial approach to the study of surfaces in 4-space.