Configuration space integrals and the cohomology of the space of homotopy string links

TL;DR

This paper extends configuration space integrals to the space of homotopy string links, including the classical case in three dimensions, and shows they encode universal finite type invariants and Milnor invariants.

Contribution

It refines the construction of configuration space integrals to apply to homotopy string links, including in three dimensions, and relates these integrals to finite type and Milnor invariants.

Findings

Integrals represent a universal finite type invariant of homotopy string links.

Configuration space integrals provide explicit formulas for Milnor invariants.

The construction applies to the classical case in three dimensions.

Abstract

Configuration space integrals have in recent years been used for studying the cohomology of spaces of (string) knots and links in $\mathbb{R}^n$ for $n>3$ since they provide a map from a certain differential algebra of diagrams to the deRham complex of differential forms on the spaces of knots and links. We refine this construction so that it now applies to the space of homotopy string links -- the space of smooth maps of some number of copies of $\mathbb{R}$ in $\mathbb{R}^n$ with fixed behavior outside a compact set and such that the images of the copies of $\R$ are disjoint -- even for $n=3$. We further study the case $n=3$ in degree zero and show that our integrals represent a universal finite type invariant of the space of classical homotopy string links. As a consequence, we obtain configuration space integral expressions for Milnor invariants of string links.