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Infinitesimal calculations in fundamental groups

Nir Gadish, Aydin Ozbek, Dev Sinha, Ben Walter

TL;DR

The paper develops a unified, computable framework linking Harrison cohomology to the Malcev Lie algebra of fundamental groups, via Hopf invariants that pair universally with π1. It recasts Chen-type invariants into a combinatorial, algebraic language of letter braiding/linking, yielding concrete algorithms to test when words sit in k-fold commutator subgroups and enabling calculation in rational completions. Central results include a universal Lie pairing, a lifting criterion for descent to quotients, and a suite of graphical/cochain models that realize invariants on surfaces, braids, and two-complexes. Beyond residually nilpotent settings, the work contemplates completed Harrison theories and outlines extensive future directions, including Milnor invariants, Johnson filtrations, and extensions to integer/finite-field regimes.

Abstract

We show that Hopf invariants, defined by evaluation in Harrison cohomology of the commutative cochains of a space, calculate the logarithm map from a fundamental group to its Malcev Lie algebra. They thus present the zeroth Harrison cohomology as a universal dual object to the Malcev Lie algebra. This structural theorem supports explicit calculations in algebraic topology, geometric topology, and combinatorial group theory. In particular, we give the first algorithm to determine whether a power of a word is a k-fold nested commutator while encoding commutator structure in any group presented by generators and relations.

Infinitesimal calculations in fundamental groups

TL;DR

The paper develops a unified, computable framework linking Harrison cohomology to the Malcev Lie algebra of fundamental groups, via Hopf invariants that pair universally with π1. It recasts Chen-type invariants into a combinatorial, algebraic language of letter braiding/linking, yielding concrete algorithms to test when words sit in k-fold commutator subgroups and enabling calculation in rational completions. Central results include a universal Lie pairing, a lifting criterion for descent to quotients, and a suite of graphical/cochain models that realize invariants on surfaces, braids, and two-complexes. Beyond residually nilpotent settings, the work contemplates completed Harrison theories and outlines extensive future directions, including Milnor invariants, Johnson filtrations, and extensions to integer/finite-field regimes.

Abstract

We show that Hopf invariants, defined by evaluation in Harrison cohomology of the commutative cochains of a space, calculate the logarithm map from a fundamental group to its Malcev Lie algebra. They thus present the zeroth Harrison cohomology as a universal dual object to the Malcev Lie algebra. This structural theorem supports explicit calculations in algebraic topology, geometric topology, and combinatorial group theory. In particular, we give the first algorithm to determine whether a power of a word is a k-fold nested commutator while encoding commutator structure in any group presented by generators and relations.
Paper Structure (33 sections, 24 theorems, 85 equations, 13 figures)

This paper contains 33 sections, 24 theorems, 85 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Infinitesimal group theory and rational homotopy
  3. Combinatorial group theory
  4. Geometric topology
  5. Relation to other work and future directions
  6. Definitions
  7. Symbols, Eil trees, and cofree Lie coalgebras
  8. Harrison complex and Hopf invariants
  9. Central Example: Handlebodies and linking of letters for free groups
  10. First handlebody example
  11. Equating topology and combinatorics of letter linking
  12. Configuration braiding of trees and words
  13. Structural theorems
  14. Lie pairing
  15. The Lifting Criterion
  16. ...and 18 more sections

Key Result

Theorem 1.2

The letter braiding functions that are well-defined on a presented group $\Gamma = \bigl\langle S\,|\,R\bigr\rangle$ form the maximal Lie coalgebra of functions on words in $S$ that vanish on $R$, and they pair perfectly with the Malcev Lie algebra of $\Gamma$ when finitely generated.

Figures (13)

  • Figure 1:
  • Figure 2:
  • Figure 3: Topological measures of the fundamental group. All produce letter linking invariants in the eigen-setting
  • Figure 4: Basic Thom forms on the circle.
  • Figure 5: A commutator in a handlebody
  • ...and 8 more figures

Theorems & Definitions (87)

  • Theorem 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Definition 1.7
  • Definition 1.8
  • Definition 1.9
  • Example 1.10
  • Definition 1.11
  • ...and 77 more