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

Mathematics > Geometric Topology [Submitted on 29 Sep 2021 (v1), last revised 18 Jun 2026 (this version, v5)] Title:Milnor-type invariants for surface-links and cut-diagrams View PDF HTML (experimental)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. Submission history From: Jean-Baptiste Meilhan [view email][v1] Wed, 29 Sep 2021 17:15:37 UTC (276 KB) [v2] Tue, 22 Nov 2022 07:33:53 UTC (285 KB) [v3] Tue, 7 Nov 2023 13:22:28 UTC (283 KB) [v4] Sat, 29 Nov 2025 10:20:14 UTC (298 KB) [v5] Thu, 18 Jun 2026 12:16:40 UTC (929 KB) References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.