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.