The Lambrechts–Stanley Model of Configuration Spaces

Author
N. I.
Publication
In: Invent. Math. (2019) 216.1, pp. 1–68.
Online on
Online on .
Accepted on
Accepted on .
Updated on
Updated on .
Full text
 Full text.
10.1007/s00222-018-0842-9
 10.1007/s00222-018-0842-9.
arXiv:1608.08054
 arXiv:1608.08054.
hal-01438861
 hal-01438861.
MR3935037
 MR3935037.
Zbl07051043
 Zbl07051043.

Abstract

We prove the validity over 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.