On the cohomology of spaces of links and braids via configuration space integrals

TL;DR

This paper investigates the cohomology of spaces of string links and braids in higher dimensions using configuration space integrals, revealing their role in generating finite type invariants in three dimensions.

Contribution

It establishes a chain map from diagram complexes to differential forms for n>3 and shows these integrals produce all finite type invariants when n=3.

Findings

Configuration space integrals define a chain map for n>3.

All finite type invariants of string links and braids are obtained via these integrals in n=3.

The approach links diagram complexes with the cohomology of link and braid spaces.

Abstract

We study the cohomology of spaces of string links and braids in $\mathbb{R}^n$ for $n\geq 3$ using configuration space integrals. For $n>3$, these integrals give a chain map from certain diagram complexes to the deRham algebra of differential forms on these spaces. For $n=3$, they produce all finite type invariants of string links and braids.