Diagram complexes, formality, and configuration space integrals for braids

TL;DR

This paper develops a diagram complex and uses configuration space integrals to study the cohomology of braid spaces in higher dimensions, extending previous results and establishing new algebraic tools.

Contribution

It introduces a new diagram complex for braids, constructs a quasi-isomorphism to de Rham cochains, and connects these with configuration space integrals and existing cohomology results.

Findings

Established a quasi-isomorphism between diagram complexes and de Rham cochains.

Extended cohomology results of braid spaces to a CDGA suitable for integration.

Provided a surjection from long links to braid space cohomology.

Abstract

We use rational formality of configuration spaces and the bar construction to study the cohomology of the space of braids in dimension four or greater. We provide a diagram complex for braids and a quasi-isomorphism to the de Rham cochains on the space of braids. The quasi-isomorphism is given by a configuration space integral followed by Chen's iterated integrals. This extends results of Kohno and of Cohen and Gitler on the cohomology of the space of braids to a CDGA suitable for integration. We show that this integration is compatible with Bott-Taubes configuration space integrals for long links via a map between two diagram complexes. As a corollary, we get a surjection in cohomology from the space of long links to the space of braids. We also discuss to what extent our results apply to the case of classical braids.