Configuration spaces consist in ordered collections of points in a given ambient manifold. Kontsevich and Tamarkin proved that the configuration spaces of Euclidean n-spaces are rationally formal, i.e., that their rational homotopy type is completely encoded by their cohomology. Their proofs use ideas from the theory of operads, and they prove the stronger result that the operads associated to configuration spaces of Euclidean -spaces, called the little -cubes operads and denoted , are formal. Based on these two proofs, we have computed (partially in joint work with Campos, Ducoulombier, Lambrechts, and Willwacher) the rational homotopy types of configuration spaces and framed configuration spaces of compact manifolds.
Voronov’s Swiss-Cheese operads encode the action of an -algebra on an -algebra. Livernet and Willwacher proved that an enlarged version of this operad which encodes morphisms (rather than actions) is not formal. In this talk, I will present two results: 1. the higher codimensional version of the Swiss-Cheese operad which encodes a central derivation from an -algebra to an -algebra, is formal for ; 2. Voronov’s original version of the codimension 1 Swiss-Cheese operad is not formal (in joint work with Vieira).