Formality of the little N-disks operad

TL;DR

This paper details the proof of the formality of the little N-disks operad over the reals, extending to a relative version for inclusions of smaller disks, with implications for real homotopy theory.

Contribution

It provides a detailed proof of Kontsevich's formality theorem for the little N-disks operad and establishes a relative formality result for certain inclusions.

Findings

Formality of the little N-disks operad over the reals is established.

A relative formality result for the inclusion of smaller disks is proved.

The results connect operad formality with real homotopy theory.

Abstract

We develop the details of Kontsevich's proof of the formality of little N-disks operad over the field of real numbers. Formality holds in the category of operads of chain complexes and also in some sense in the category of commutative differential graded algebras, which is the category encoding "real" homotopy theory. We also prove a relative version of the formality for the inclusion of the little m-disks operad in the little N-disks operad for N>=2m+1.