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

Benjamin Audoux, Jean-Baptiste Meilhan, Akira Yasuhara

TL;DR

<3-5 sentence high-level summary> The paper develops cut-diagrams as a combinatorial framework to extend Milnor invariants from classical links to surface-links in 4-space, including those with boundary. It introduces Chen–Milnor-type presentations for nilpotent quotients of cut-diagram groups and defines Milnor loop- and arc-invariants via Magnus expansions of longitudes, proving concordance and, for non-repeated invariants, link-homotopy invariance. The authors establish isotopy and concordance invariance through explicit diagrammatic moves and provide extensive applications: realization results, Spun-link invariants, concordance classifications, and obstructions to ribbonness, connecting to Orr invariants and k-slice theory. The work offers a robust, algorithmic, and combinatorial pathway to study higher-dimensional surface-links in 4-space.

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.

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

TL;DR

<3-5 sentence high-level summary> The paper develops cut-diagrams as a combinatorial framework to extend Milnor invariants from classical links to surface-links in 4-space, including those with boundary. It introduces Chen–Milnor-type presentations for nilpotent quotients of cut-diagram groups and defines Milnor loop- and arc-invariants via Magnus expansions of longitudes, proving concordance and, for non-repeated invariants, link-homotopy invariance. The authors establish isotopy and concordance invariance through explicit diagrammatic moves and provide extensive applications: realization results, Spun-link invariants, concordance classifications, and obstructions to ribbonness, connecting to Orr invariants and k-slice theory. The work offers a robust, algorithmic, and combinatorial pathway to study higher-dimensional surface-links in 4-space.

Abstract

We generalize Milnor link invariants to all types of surface-links in --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.

Paper Structure

This paper contains 39 sections, 36 theorems, 88 equations, 14 figures.

Table of Contents

  1. Cut-diagrams
  2. An intuitive approach
  3. Abstract cut-diagrams
  4. The $1$--dimensional case
  5. Peripheral systems of cut-diagrams
  6. The group of a cut-diagram
  7. Generic path and associated words
  8. Loop-longitudes
  9. Arc-longitudes
  10. Peripheral systems
  11. Chen homomorphisms and a Chen--Milnor Theorem for cut-diagrams
  12. Road networks and the Chen homomorphisms
  13. Nilpotent quotients
  14. Chen--Milnor type presentations
  15. Milnor invariants for cut-diagrams and surface-links
  16. ...and 24 more sections

Key Result

Theorem 1

Milnor loop- and arc-invariants are well-defined concordance invariants of surface-links. Moreover, if the indexing sequence contains no repetition, the corresponding Milnor invariants are also invariant under link-homotopy.

Figures (14)

  • Figure 1: From a surface-link diagram to a $2$--dimensional cut-diagram. Here, and in forthcoming figures, regions are named with capital letters, and labels are given by circled nametags
  • Figure 2: From surface-link diagrams to cut-diagrams: triple point
  • Figure 3: From surface-link diagrams to cut-diagrams: branch point
  • Figure 4: Illustrations of the labeling conditions
  • Figure 5: Diagrams that violate the first (left) and second (right) labeling condition
  • ...and 9 more figures

Theorems & Definitions (108)

  • Theorem
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 2.1
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5
  • ...and 98 more