Formality of the Little $N$-disks Operad PDF

The little $N$-disks operad, $\mathcal B$, 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 $\mathbb{R}^N$ 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 $\operatorname{C}_*(\mathcal B; \mathbb{R})$ on $\mathcal B$ as well as the singular homology $\operatorname{H}_*(\mathcal B; \mathbb{R})$ of $\mathcal B$.

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 $N\geq2m+1$.