Diagrams for primitive cycles in spaces of pure braids and string links

TL;DR

This paper explores the algebraic and topological structures of pure braid and string link spaces using diagrams, Chen's integrals, and graphing techniques, revealing new connections and conjectures in high-dimensional topology.

Contribution

It constructs a Hopf algebra isomorphism linking homology of pure braids to diagram cobar constructions and establishes a correspondence between Milnor invariants and Chen integrals.

Findings

Hopf algebra isomorphism between pure braid homology and diagram cobar construction

Correspondence between Milnor invariants and Chen integrals for spherical links

Graphing induces injections into homotopy groups of high-dimensional string links

Abstract

The based loop space of a configuration space of points in a Euclidean space can be viewed as a space of pure braids in a Euclidean space of one dimension higher. We continue our study of such spaces in terms of Kontsevich's CDGA of diagrams and Chen's iterated integrals. We construct a power series connection which yields a Hopf algebra isomorphism between the homology of the space of pure braids and the cobar construction on diagrams. It maps iterated Whitehead products to trivalent trees modulo the IHX relation. As an application, we establish a correspondence between Milnor invariants of Brunnian spherical links and certain Chen integrals. Finally we show that graphing induces injections of a certain submodule of the homotopy of configuration spaces into the homotopy of many spaces of high-dimensional string links. We conjecture that graphing is injective on all rational homotopy classes.