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Formality of the Little $$N$$-disks Operad
Pascal Lambrechts, Université Catholique de Louvain, Louvain-la-Neuve, Belgium, and Ismar Volić, Wellesley College, Massachusetts
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Memoirs of the American Mathematical Society
2013; 116 pp; softcover
Volume: 230
ISBN-10: 0-8218-9212-6
ISBN-13: 978-0-8218-9212-1
List Price: US$75 Individual Members: US$45
Institutional Members: US\$60
Order Code: MEMO/230/1079

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 $$N\geq2m+1$$.

• Introduction
• Notation, linear orders, weak partitions, and operads
• Equivalence of the cooperads $$\mathcal{D}$$ and $$\mathrm {H}^*(\mathrm{C}[\bullet])$$