Rational homotopy of the little cubes operads and graph complexes

I will report on a joint work with Victor Turchin and Thomas Willwacher about the rational homotopy of the little cubes operads (equivalently, the little discs operads).

The little cubes operads (and the equivalent little discs operads) were introduced by Boardman-Vogt and May for the study of iterated loop spaces. The study of the little cubes operads has been completely renewed during the last decade and new applications of these objects have been discovered in various fields of algebra and topology. To cite one application, one can prove that the spaces of compactly supported embeddings of Euclidean spaces modulo immersions have a description in terms of mapping spaces associated to the little discs operads. This result represents the outcome of a series of works by Sinha, Arone-Turchin, Dwyer-Hess and Boavida-Weiss on the Goodwillie-Weiss calculus of functors.

The goal of my talk is to explain that the rational homotopy of mapping spaces associated to the little discs operads can be determined by graph complexes. This computation can also be performed for the spaces of homotopy automorphisms of the little discs operads. The homology of the graph complex associated to these homotopy automorphism spaces just reduces to the Grothendieck-Teichmüller group in the case of the 2-dimensional little discs.
The proof of these results relies on a study of the rational homotopy of the little discs operads which I will also explain in my talk.