Milnor Invariants

Study group - Low Dimensional Topology - MPIM Bonn - 2019


  • The second block of talks took place from 2019-11-04 to 2019-11-07.
  • Tere are some handwritten notes, but they might contain errors so please use them carefully and at your own risk!

List of Talks

Talk 0: Organizational meeting

Friday, 2019-06-07, 11:00-12:00, MPIM Seminar room

  • First examples
  • Historical overview
  • Distribution of talks
Milnor Invariants - Organizational meeting.pdf

Part 1: General Introduction & Massey Products

2019-07-03 and 2019-07-04

Talk 1.1: Basic Introduction - Part I

Wednesday, 03-07-2019, 14:00 -15:00, Danica

  • Motivation: Link homotopy
  • Examples
  • Questions asked by Milnor

of the study group

Talk 1.2: Basic Introduction - Part II

Wednesday, 03-07-2019, 15:15 -16:15, Ben

  • Chen-Milnor theorem on the nilpotent quotients of a link exterior fundamental group
  • Magnus expansion of a longitude
  • Indeterminacy and relations
  • Examples
2019-07-03 Milnor Invariants - Introduction Part II.pdf

Talk 1.3: Massey products

Thursday, 04-07-2019, 14:00 -15:00, José

2019-07-04 Massey Products [Jose].pdf

Talk 1.4: Connection between Massey products of the link exterior to Milnor invariants

Thursday, 04-07-2019, 15:15 -16:15, Danica


Talk 1.5: Central series of groups and concordance invariance

Thursday, 04-07-2019, 16:30 -17:30, Ben

Based on:[Stallings] Homology and central series of groups[Casson] Link Cobordism and Milnor's Invariant
  • Introduction to concordance
  • Stallings' theorem on lower central series
  • I-equivalence
2019-07-04 Milnor Invariants - Stallings' Theorem on Central Series and Concordance Invariance - DRAFT.pdf

Part 2: Whitney Towers, Cochran's Derivatives and Conant-Schneiderman-Teichner

November 2019

Talk 2.1: Richard Porter in the MPIM Topology Seminar: "Holonomy Lie algebras, lifting theorems, Massey products, and hyperplane arrangements"

Monday, 04-11-2019, 15:00 - 16:00, Richard Porter

Abstract: In joint work with Alex Suciu, we explore finitely generated groups by studying the nilpotent towers and the various Lie algebras attached to such groups. The main goal is to relate an isomorphism extension problem in the Postnikov tower to the existence of certain commuting diagrams. This recasts a result of G. Rybnikov in a more general framework and leads to a generalization of Massey triple products and an application to hyperplane arrangements, whereby we show that all the nilpotent quotients of a decomposable hyperplane arrangement are combinatorially determined.

Event link:

Talk 2.2: Milnor invariants in the language of trees

Tuesday, 05-11-2019, 14:00 - 15:00, Benjamin Ruppik

We'll start the second part of the study group by recalling the perspective on the Milnor invariants presented in July: They try to decide (inductively) how deep longitudes of link components lie in the lower central series of the link group.

Milnor's algebraic algorithm for calculating presentations of quotients G/G_{n} of the link groups is computationally very expensive, and because of this prior to Tim Cochran's work not many explicit examples were available. Cochran's perspective via iterated intersection of Seifert surfaces yields a machine that given any desired Milnor invariant can cook up a link which realizes this.

In this talk I'll explain how trees can be used to package all Milnor invariants of a given length in one piece, and how you can realize any given first non-vanishing Milnor invariant by iterated Bing-doubling.

Based on:[Cochran] Derivatives of links: Massey products and Milnor’s concordance invariants (see Sources)[Conant, Schneiderman, Teichner] Higher-order intersections in low-dimensional topology
2019-11-05 Recap & Milnor Invariants in the Language of Trees [Ben].pdf

Talk 2.3: Triple linking numbers and surface systems

Tuesday, 05-11-2019, 15:10 - 16:10, Mihail Arabadji

Talk 2.4: Introduction to Whitney Towers

Tuesday, 05-11-2019, 16:30 - 17:30, Danica Kosanović

Talk 2.5: Peter Teichner in the Seminar on configuration spaces and diffeomorphisms: "Milnor invariants, Whitney towers and the Kontsevich integral"

Thursday, 07-11-2019, 16:30 - 18:00, Peter Teichner

Dates and Times

The study group is split into two parts:

  1. The first bock on the connection between Milnor invariants and Massey products took place Wednesday and Thursday in the first week of July (2019-07-03 and 2019-07-04).
  2. The second block on the work of Cochran and Conant-Schneiderman-Teichner will be in the first week of November (2019-11-04 to 2019-11-06).

Up-to-date information is in the 'Bonn low-dimensional topology' Google calendar; you can also check the MPIM calendar!


Danica: danica (at) mpim-bonn (dot) mpg (dot) de, Website

Ben: bruppik (at) mpim-bonn (dot) mpg (dot) de, Website