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Formality of the little $N$-disks operad

About this Title

Pascal Lambrechts, Université catholique de Louvain, IRMP 2 Chemin du Cyclotron B-1348 Louvain-la-Neuve, Belgium and Ismar Volić, Department of Mathematics, Wellesley College, Wellesley, Massachusetts 02482

Publication: Memoirs of the American Mathematical Society
Publication Year: 2014; Volume 230, Number 1079
ISBNs: 978-0-8218-9212-1 (print); 978-1-4704-1669-0 (online)
DOI: https://doi.org/10.1090/memo/1079
Published electronically: November 14, 2013
Keywords: Operad formality, little cubes operad, Fulton-MacPherson operad, trees, configuration space integrals
MSC: Primary 55P62; Secondary 18D50

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Table of Contents

Chapters

• Acknowledgments
• 1. Introduction
• 2. Notation, linear orders, weak partitions, and operads
• 3. CDGA models for operads
• 4. Real homotopy theory of semi-algebraic sets
• 5. The Fulton-MacPherson operad
• 6. The CDGAs of admissible diagrams
• 7. Cooperad structure on the spaces of (admissible) diagrams
• 8. Equivalence of the cooperads $\mathcal {D}$ and $\mathrm {H}^*(\mathrm {C}[\bullet ])$
• 9. The Kontsevich configuration space integrals
• 10. Proofs of the formality theorems
• Index of notation

Abstract

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, we 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. We additionally prove a relative version of the formality for the inclusion of the little $m$-disks operad in the little $N$-disks operad when $N\geq 2m+1$.

The formality of the little $N$-disks operad has already had many important applications. For example, it was used in a solution of the Deligne Conjecture, in Tamarkin’s proof of Kontsevich’s deformation quantization conjecture, and in the work of Arone, Lambrechts, Turchin, and Volić on determining the rational homotopy type of spaces of smooth embeddings of a manifold in a large euclidean space, such as the space of knots in $\mathbb {R}^N$, $N\geq 4$.