Formality of the Little N-disks Operad

The little N-disks operad, B, along with its variants, is an important tool in homotopy theory. It is defined in terms of configurations of disjoint N-dimensional disks inside the standard unit disk in Rn and it was initially conceived for detecting and understanding N-fold loop spaces. Its many uses now stretch across a variety of disciplines including topology, algebra, and mathematical physics. In this paper, the authors develop the details of Kontsevich's proof of the formality of little N-disks operad over the field of real numbers. More precisely, one can consider the singular chains C* (BR) on B as well as the singular homology H*((BR) on B. These two objects are operads in the category of chain complexes. The formality then states that there is a zig-zag of quasi-isomorphisms connecting these two operads. The formality also in some sense holds in the category of commutative differential graded algebras. The authors additionally prove a relative version of the formality for the inclusion of the little m-disks operad in the little N-disks operad when N 2m 1.