The Lambrechts-Stanley Model of Configuration Spaces

TL;DR

This paper proves a model for configuration spaces of simply connected closed manifolds over the real numbers, showing the model's validity, its dependence on the manifold's homotopy type, and its compatibility with operad actions, with applications to factorization homology.

Contribution

It establishes a real CDGA model for configuration spaces that confirms a conjecture of Lambrechts--Stanley and connects the model to operad actions and factorization homology.

Findings

The model accurately represents the real homotopy type of configuration spaces.

The model is compatible with the Fulton--MacPherson operad for framed manifolds in dimension ≥ 4.

The approach uses ideas similar to Kontsevich's proof of operad formality.

Abstract

We prove the validity over $\mathbb{R}$ of a commutative differential graded algebra model of configuration spaces for simply connected closed smooth manifolds, answering a conjecture of Lambrechts--Stanley. We get as a result that the real homotopy type of such configuration spaces only depends on the real homotopy type of the manifold. We moreover prove, if the dimension of the manifold is at least $4$, that our model is compatible with the action of the Fulton--MacPherson operad (weakly equivalent to the little disks operad) when the manifold is framed. We use this more precise result to get a complex computing factorization homology of framed manifolds. Our proofs use the same ideas as Kontsevich's proof of the formality of the little disks operads.