[Paper Review] Vassiliev invariants I: Braid groups and rational homotopy theory

Louis Funar|ArXiv.org|Oct 6, 1995

Homotopy and Cohomology in Algebraic Topology3 citations

TL;DR

This paper establishes a connection between Vassiliev invariants, braid groups, and rational homotopy theory using Chen's iterated integrals and Malcev completion. It identifies the inclusion of pure braid groups into their Malcev completion as the universal Kontsevich-Vassiliev invariant, extending this morphism to the full braid group and describing its multiplication law, thereby providing a homotopical framework for Vassiliev invariants via algebraic topology tools.

ABSTRACT

We get a detailed account of Vassiliev type invariants starting with Chen's theory of iterated integrals and Malcev's completion of discrete groups. The canonical injection of the group of pure braids into its completion is identified with the universal Kontsevich-Vassiliev invariant.Further we discuss the extension of this morphism to the whole braid group and the multiplication law for the last one.

Motivation & Objective

• To establish a homotopical interpretation of Vassiliev invariants using rational homotopy theory.
• To connect the theory of iterated integrals with the structure of braid groups and their completions.
• To extend the universal Kontsevich-Vassiliev invariant from pure braid groups to the full braid group.
• To describe the algebraic structure of the multiplication law in the completed braid group.

Proposed method

• Utilizes Chen's theory of iterated integrals to construct invariants from differential forms on configuration spaces.
• Applies Malcev completion to discrete groups, particularly the braid group, to obtain a pro-unipotent group structure.
• Identifies the canonical injection of the pure braid group into its Malcev completion as the universal Kontsevich-Vassiliev invariant.
• Extends this morphism to the full braid group using algebraic and topological techniques.
• Analyzes the multiplication law in the completed braid group via the structure of the completed group algebra.
• Employs rational homotopy theory to relate cohomological invariants to Vassiliev-type finite-type invariants.

Experimental results

Research questions

• RQ1 How can Vassiliev invariants be systematically constructed using iterated integrals and rational homotopy theory?
• RQ2 What is the role of the Malcev completion of the braid group in realizing universal Vassiliev invariants?
• RQ3 How does the universal Kontsevich-Vassiliev invariant extend from pure braid groups to the full braid group?
• RQ4 What algebraic structure governs the multiplication in the completed braid group?
• RQ5 How do iterated integrals on configuration spaces give rise to finite-type invariants of knots and links?

Key findings

• The inclusion of the pure braid group into its Malcev completion realizes the universal Kontsevich-Vassiliev invariant.
• The morphism from the pure braid group to its completion is identified as the universal source of Vassiliev invariants.
• The extension of this morphism to the full braid group is constructed algebraically, preserving the universal property.
• The multiplication law in the completed braid group is described through the completed group algebra structure.
• The framework provides a systematic link between finite-type invariants and rational homotopy theory via iterated integrals.
• The approach offers a cohomological interpretation of Vassiliev invariants through the lens of Malcev completion and Chen's iterated integrals.