We study the real homotopy type of configuration spaces on smooth closed manifolds. In this talk, I will give an introduction to the problem of understanding configuration spaces and present an algebraic model of these spaces using graph complexes similar to Kontsevich's.

We will study the natural action from the (framed) little discs operad on the space of (framed) configuration of points, with applications to Manifold Calculus in mind and we will see that under some conditions the same type of graphs are able to capture the real homotopy type of this operadic module.

(Joint work with Julien Ducoulombier, Najib Idrissi and Thomas Willwacher)