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

Abstract

We generalize Milnor link invariants to surface-links in 4-space, possibly with boundary. To this end, we introduce the notion of cut-diagram, which is a 2-dimensional analogue of Gauss diagrams. To each cut-diagram, we associate a group extending the fundamental group of the exterior of a surface-link, and we extract Milnor-type invariants from its successive nilpotent quotients. We show that this yields concordance invariants for surface-links, and that some even are link-homotopy invariants. We give several concrete applications, including realization and classification results. The theory of cut-diagrams is further investigated, heading towards a combinatorial approach to surfaces in 4-space.

40 pages. v.5: entirely revised version, with new organization; some applications and examples added

References in corpus (6)

• Whitehead double and Milnor invariants
• Homotopy classification of ribbon tubes and welded string links
• On codimension two embeddings up to link-homotopy
• Characterization of the reduced peripheral system of links
• Milnor's concordance invariants for knots on surfaces
• Combinatorial approach to Milnor invariants of welded links