We prove the validity over $R$ of a CDGA model of configuration spaces for
simply connected manifolds of dimension at least $4$, answering a conjecture of
Lambrechts--Stanley.
We get as a result that the real homotopy type of such configuration spaces only depends
on a Poincaré duality model of the manifold.
We moreover prove that our model is compatible with the action of the Fulton--MacPherson operad
when the manifold is framed, by relying on Kontsevich's proof of the formality of the little disks
operads.
We use this more precise result to get a complex computing factorization homology of
framed manifolds.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/6791