Formality of the little \(N\)-disks operad (Q2925671)

The little \(N\)-disks operad originated in topology as a means of encoding loop space structures, but its applications have been extensive. In 1999, Kontsevich gave a sketch of the proof that this operad is formal over the real numbers, or that its operad of singular chains with coefficients in \(\mathbb R\) is equivalent to its homology. This result has had a number of applications, ranging from the Deligne Conjecture to work of the authors and others on the homotopy types of spaces of knots. In this manuscript, the authors give a detailed proof of the formality of the little \(N\)-disks operad. Furthermore, their result is stronger than that of Kontsevich in multiple ways. First, their version of the little \(N\)-disks operad includes an operation of arity 0. Second, their formality result is given in the context of commutative differential graded algebras (CDGAs), rather than chain complexes. Third, they also prove formality relatively, by considering the inclusion of the little \(m\)-disks operad into the little \(N\)-disks operad for sufficiently large \(N\).NEWLINENEWLINEBecause the authors wish to establish formality within CDGAs, a clarification is needed. Sullivan's functor \(A_{PL}\) from the category of topological spaces to the category CDGAs gives a functor from the category of topological operads to the category of CDGA cooperads; the authors here use the semi-algebraic version \(\Omega_{PA}\). However, applying this functor to each space in a topological operad does not produce a cooperad of CDGAs, since it is not comonoidal. A modification of the definition of a CDGA model for a topological operad is necessary to have the correct framework for the desired result.NEWLINENEWLINEThe sketch of the proof is as follows. The authors use the Fulton-MacPherson operad \(C[\bullet]\), which is known to be weakly equivalent, as a topological operad, to the little \(N\)-disks operad. Each component space \(C[n]\) is a compactification of the space of normalized ordered configurations of \(n\) points in \(\mathbb R^N\). An important feature of these spaces is that they are compact semi-algebraic manifolds with boundary, and the canonical projection maps \(C[n+k] \rightarrow C[n]\) are semi-algebraic bundles whose fibers are compact manifolds. The functor \(\Omega_{PA}\) is applied to each space \(C[n]\) to obtain a CDGA.NEWLINENEWLINEThe heart of the proof, then, is the Kontsevich configuration space integral map \(I : \mathcal D(n) \rightarrow \Omega_{PA}(C[n])\) for each \(n \geq 0\), where each \(\mathcal D(n)\) is a CDGA of admissible diagrams. The construction of such diagrams is inspired by relations on differential forms. As \(n\) varies, these CDGAs assemble to form a cooperad \(\mathcal D\) which is shown to be quasi-isomorphic to the cohomology of the little \(N\)-disks operad. Then the map \(I\) is shown to be a quasi-isomorphism, establishing the formality result.NEWLINENEWLINEMuch attention is given to the precise definitions of all the operads and cooperads involved, and the properties that are used in the proof. The arguments are well-written, and this paper will be useful for anyone wanting to understand this formality result.